529 Sentences With "polyhedral" | Random Sentence Generator

He's fashioned polyhedral lights to hang over the restaurant tables where Eliasdottir serves kimchi rice with sesame-marinated cucumbers and roasted eggplant.

A new polyhedral sample-collection mechanism acts like an "underwater Pokéball," allowing scientists to catch 'em all without destroying their soft, squishy bodies in the process.

A bold, breviloquent debut novel whose polyhedral story line plunges sans parachute into the bloody chamber of political violence unleashed during the massacre-ridden years in Peru.

Younger generations, embracing video games and smartphones as their escapism of choice, seemed indifferent or bored by D&D's make-believe world of swords and sorcery, labyrinthine rules and polyhedral dice.

Witness the words "anything" and "everything" projected onto a wall near the front, Teddy bears handcuffed to metal poles in the back, and a twisting polyhedral light fixture above the bar.

Roughly the weight of the Eiffel Tower and enclosing a volume greater than St Peter's Basilica in Rome, its polyhedral frame measures 68 metres from top to bottom and over 20503 metres in diameter.

Magna-Tiles (age 3 and up) are flat, colorful, magnetic shapes (the basic set has squares and triangles of varying sizes; expansion packs include other polygons) from which kids can build a seemingly endless array of polyhedral structures.

" Uncovering the polyhedral structure representing all possible quantum field theories would, in a sense, unify quark interactions, magnets and all observed and imagined phenomena in a single, inevitable structure—a sort of 21st-century version of Geoffrey Chew's "only possible nature consistent with itself.

In the kit you'll find a detailed rule book, campaign book, pre-made Stranger Things character sheets (including Will the Wise and Dustin the Dwarf), two spooky Demogorgon figurines (one left unpainted for those who like to customize), and a set of iconic polyhedral dice.

It is an application of the Weaire-Phelan foam, the most efficiently packed foam of equal-volume polyhedral bubbles, discovered in 1994 by Irish physicist Denis Weaire and his student Robert Phelan (first using a computer simulation, then created in a lab in 2012).

The decision to tighten border controls, made in the wake of 90,000 asylum claims in the country, last year, sparked outrage throughout the EU. In response to Austria giving migrants a red light, Danish-Icelandic artist Olafur Eliasson decided to create a "Green light," a crystalline polyhedral LED light made from recycled materials.

But for anyone who's sampled the iconic role-playing game, D&D is also an exuberant dive into imagination, world-building, teamwork and identity — especially when played in its original format on a table cluttered with rule books, maps, monster manuals, character worksheets and colorful polyhedral dice nestled among pizza boxes, Funyuns bags and half-empty Mountain Dew cans.

Hydra Allegiance Pendant — $5 See Details Guardians of the Galaxy Sterling Groot Basket Branch Ring — $53 See Details Harry Potter Luck Necklace — $32 See Details Gryffindor Charm Bracelet — $35 See Details Slytherin Hufflepuff Ravenclaw Portal 2 Wheatley Personality Core Charm Bead — $25-$70 See Details d20 Polyhedral Dice Ring — $30 See Details Sonic Stud Earrings — $18 See Details Destiny Strange Coin Medallion — $20 See Details Mario Kart LED Twinkling Bracelet — $60 See Details

Polyhedral space is a certain metric space. A (Euclidean) polyhedral space is a (usually finite) simplicial complex in which every simplex has a flat metric. (Other spaces of interest are spherical and hyperbolic polyhedral spaces, where every simplex has a metric of constant positive or negative curvature). In the sequel all polyhedral spaces are taken to be Euclidean polyhedral spaces.

The convex conjugate of a closed convex function is again a closed convex function. The convex conjugate of a polyhedral convex function (a convex function with polyhedral epigraph) is again a polyhedral convex function.

Another direction of research are developments of dynamical billiards in polyhedral spaces, e.g. of nonpositive curvature (hyperbolic billiards). Positively curved polyhedral spaces arise also as links of points (typically metric singularities) in Euclidean polyhedral spaces.

The Fan application contains functions for polyhedral complexes (which generalize simplicial complexes), planar drawings of 3-polytopes, polyhedral fans, and subdivisions of points or vectors.

Following these chapters, additional engravings depict additional polyhedral forms, including polyhedral compounds such as the stella octangula, polyhedral variations of spheres and cones, and outlined skeletons of polyhedra following those drawn by Leonardo da Vinci for Luca Pacioli's earlier book Divina proportione. In this part of the book, the shapes are arranged in a three- dimensional setting and often placed on smaller polyhedral pedestals.

In full generality, polyhedral spaces were first defined by Milka Milka, A. D. Multidimensional spaces with polyhedral metric of nonnegative curvature. I. (Russian) Ukrain. Geometr. Sb. Vyp. 5--6 1968 103–114.

The polyhedral graph formed as the Schlegel diagram of a regular dodecahedron. Schlegel diagram of truncated icosidodecahedral graph In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the 3-vertex-connected planar graphs.

All 1-dimensional polyhedral spaces are just metric graphs. A good source of 2-dimensional examples constitute triangulations of 2-dimensional surfaces. The surface of a convex polyhedron in R^3 is a 2-dimensional polyhedral space. Any PL-manifold (which is essentially the same as a simplicial manifold, just with some technical assumptions for convenience) is an example of a polyhedral space.

Tait conjectured that every cubic polyhedral graph (that is, a polyhedral graph in which each vertex is incident to exactly three edges) has a Hamiltonian cycle, but this conjecture was disproved by a counterexample of W. T. Tutte, the polyhedral but non-Hamiltonian Tutte graph. If one relaxes the requirement that the graph be cubic, there are much smaller non- Hamiltonian polyhedral graphs. The graph with the fewest vertices and edges is the 11-vertex and 18-edge Herschel graph,. and there also exists an 11-vertex non-Hamiltonian polyhedral graph in which all faces are triangles, the Goldner–Harary graph.. More strongly, there exists a constant α < 1 (the shortness exponent) and an infinite family of polyhedral graphs such that the length of the longest simple path of an n-vertex graph in the family is O(nα)...

The polyhedral framework of gcc is called Graphite.Sebastian Pop, Albert Cohen, Cedric Bastoul , Sylvain Girbal, Pierre Jouvelot, Georges-André Silber et Nicolas Vasilache. Graphite: Loop optimizations based on the polyhedral model for GCC. 4th GCC Developer's Summit.

The centered polyhedral numbers are a class of figurate numbers, each formed by a central dot, surrounded by polyhedral layers with a constant number of edges. The length of the edges increases by one in each additional layer.

In mathematics, specifically convex geometry, the normal fan of a convex polytope P is a polyhedral fan that is dual to P. Normal fans have applications to polyhedral combinatorics, linear programming, tropical geometry and other areas of mathematics.

A unique feature was a generalized polyhedral cell formulation, allowing the solver to handle any mesh type imported. The first official release included the first commercially available polyhedral mesher, offering faster model convergence compared to an equivalent tetrahedral mesh..

There are a number of problems in computational geometry which involve polyhedral terrains.

The k-SYG is constructed as follows. The space around each point p in P is partitioned into a set of polyhedral cones of opening angle \theta, meaning the angle of each pair of rays inside a polyhedral cone emanating from the apex is at most \theta, and then p connects to k points of P in each of the polyhedral cones whose projections on the cone axis is minimum.

Handbook in Computational Geometry p. 352 The polyhedral terrain is a generalization of the two-dimensional geometric object, the monotone polygonal chain. As the name may suggest, a major application area of polyhedral terrains include geographic information systems to model real-world terrains.

There are 18 convex polyhedral prisms created from 5 Platonic solids and 13 Archimedean solids as well as for the infinite families of three-dimensional prisms and antiprisms. The symmetry number of a polyhedral prism is twice that of the base polyhedron.

Balloon cell nevi are a cutaneous condition characterized histologically by large, pale, polyhedral balloon cells.

The polyhedral modelR. Allen and K. Kennedy. Optimizing Compilers for Modern Architectures. Morgan Kaufmann, 2002.

In geometry, the polyhedral group is any of the symmetry groups of the Platonic solids.

A combinatorial map is a combinatorial object modelling topological structures with subdivided objects. Historically, the concept was introduced informally by J. Edmonds for polyhedral surfaces Edmonds J., A Combinatorial Representation for Polyhedral Surfaces, Notices Amer. Math. Soc., vol. 7, 1960 which are planar graphs.

A piecewise linear function over two dimensions (top) and the polygonal areas on which it is linear (bottom) In computational geometry, a polyhedral terrain in three-dimensional Euclidean space is a polyhedral surface that intersects every line parallel to some particular line in a connected set (i.e., a point or a line segment) or the empty set.Richard Cole, Micha Sharir, "Visibility problems for polyhedral terrains" 1989, Without loss of generality, we may assume that the line in question is the z-axis of the Cartesian coordinate system. Then a polyhedral terrain is the image of a piecewise-linear function in x and y variables.

Duijvestijn provides a count of the polyhedral graphs with up to 26 edges;. The number of these graphs with 6, 7, 8, ... edges is :1, 0, 1, 2, 2, 4, 12, 22, 58, 158, 448, 1342, 4199, 13384, 43708, 144810, ... . One may also enumerate the polyhedral graphs by their numbers of vertices: for graphs with 4, 5, 6, ... vertices, the number of polyhedral graphs is :1, 2, 7, 34, 257, 2606, 32300, 440564, 6384634, 96262938, 1496225352, ... .

Use of the polyhedral model (also called the polytope model) within a compiler requires software to represent the objects of this framework (sets of integer- valued points in regions of various spaces) and perform operations upon them (e.g., testing whether the set is empty). For more detail about the objects and operations in this model, and an example relating the model to the programs being compiled, see the polyhedral model page. There are many frameworks supporting the polyhedral model.

In chemistry, the Jemmis mno rules represent a unified rule for predicting and systematizing structures of compounds, usually clusters. The rules involve electron counting. They were formulated by Eluvathingal Devassy Jemmis to explain the structures of condensed polyhedral boranes such as , which are obtained by condensing polyhedral boranes by sharing a triangular face, an edge, a single vertex, or four vertices. These rules are additions and extensions to Wade's rules and polyhedral skeletal electron pair theory.

In another approach the preserved one-dimensional trait is the orthogonal direction. This gives rise for the notion of polyhedral terrain in three dimensions: a polyhedral surface with the property that each vertical (i.e., parallel to Z axis) line intersects the surface at most by one point or segment.

This implements a suite of cache- locality optimizations as well as auto-parallelism and vectorization using a polyhedral model.

In the study of polyhedral spaces (particularly of those that are also topological manifolds) metric singularities play a central role. Let a polyhedral space be an n-dimensional manifold. If a point in a polyhedral space that is an n-dimensional topological manifold has no neighborhood isometric to a Euclidean neighborhood in R^n, this point is said to be a metric singularity. It is a singularity of codimension k, if it has a neighborhood isometric to R^{n-k} with a metric cone.

An antiprism graph is a special case of a circulant graph, Ci2n(2,1). Other infinite sequences of polyhedral graph formed in a similar way from polyhedra with regular-polygon bases include the prism graphs (graphs of prisms) and wheel graphs (graphs of pyramids). Other vertex-transitive polyhedral graphs include the Archimedean graphs.

In the former case, the point is necessarily a codimension 3 metric singularity. The general problem of topologically classifying singularities in polyhedral spaces is largely unresolved (apart from simple statements that e.g. any singularity is locally a cone over a spherical polyhedral space one dimension less and we can study singularities there).

It is interesting to study the curvature of polyhedral spaces (the curvature in the sense of Alexandrov spaces), specifically polyhedral spaces of nonnegative and nonpositive curvature. Nonnegative curvature on singularities of codimension 2 implies nonnegative curvature overall. However, this is false for nonpositive curvature. For example, consider R^3 with one octant removed.

The first "official release" of STAR-CCM+ included the world's first commercially available polyhedral meshing algorithm. The use of a polyhedral mesh has proven to be more accurate for fluid-flow problems than a hexahedral or tetrahedral mesh of a similar size (number of cells), but is considerably more difficult to create.

If a polyhedral graph does not contain a triangular face, its dual graph does contain a triangle and is also polyhedral, so one can realize the dual in this way and then realize the original graph as the polar polyhedron of the dual realization... It is also possible to realize any polyhedral graph directly by choosing the outer face to be any face with at most five vertices (something that exists in all polyhedral graphs) and choosing more carefully the fixed shape of this face in such a way that the Tutte embedding can be lifted, or by using an incremental method instead of Tutte's method to find a liftable planar drawing that does not have equal weights for all the interior edges..

In 1969, Branko Grünbaum conjectured that every 3-regular graph with a polyhedral embedding on any two-dimensional oriented manifold such as a torus must be of class one. In this context, a polyhedral embedding is a graph embedding such that every face of the embedding is topologically a disk and such that the dual graph of the embedding is simple, with no self-loops or multiple adjacencies. If true, this would be a generalization of the four color theorem, which was shown by Tait to be equivalent to the statement that 3-regular graphs with a polyhedral embedding on a sphere are of class one. However, showed the conjecture to be false by finding snarks that have polyhedral embeddings on high-genus orientable surfaces.

An animation of the Roman Surface Steiner's Roman surface is a more degenerate map of the projective plane into 3-space, containing a cross-cap. The tetrahemihexahedron is a polyhedral representation of the real projective plane. A polyhedral representation is the tetrahemihexahedron, which has the same general form as Steiner's Roman Surface, shown here.

McDonnell Douglas F-4 Phantom II showing polyhedral wing and anhedral tail Most aircraft have been designed with planar wings with simple dihedral (or anhedral). Some older aircraft such as the Vought F4U Corsair and the Beriev Be-12 were designed with gull wings bent near the root. Modern polyhedral wing designs generally bend upwards near the wingtips (also known as tip dihedral), increasing dihedral effect without increasing the angle the wings meet at the root, which may be restricted to meet other design criteria. Polyhedral is seen on gliders and some other aircraft.

The projective plane can be immersed (local neighbourhoods of the source space do not have self-intersections) in 3-space. Boy's surface is an example of an immersion. Polyhedral examples must have at least nine faces.Brehm, U.; "How to build minimal polyhedral models of the Boy surface", The mathematical intelligencer 12, No. 4 (1990), pp 51-56.

All 11 unfoldings of the cube A polyhedral net for the cube is necessarily a hexomino, with 11 hexominoes (shown at right) actually being nets. They appear on the right, again coloured according to their symmetry groups. A polyhedral net for the cube cannot contain the O-tetromino, nor the I-pentomino, the U-pentomino, or the V-pentomino.

Dragon Warriors requires the use of the whole spectrum of polyhedral dice: d4, d6, d8, d10 (d100), d12, and d20. The game's designers have since stated that utilising mechanics that involve polyhedral dice, rather than just simply common six-sided dice, was an error of judgment, and could have had an effect on the accessibility of the original series.

Margaret M. Bayer is an American mathematician working in polyhedral combinatorics. She is a professor of mathematics at the University of Kansas.

In any polyhedron that represents a given polyhedral graph G, the faces of G are exactly the cycles in G that do not separate G into two components: that is, removing a facial cycle from G leaves the rest of G as a connected subgraph. Thus, the faces are uniquely determined from the graph structure. Another strengthening of Steinitz's theorem, by Barnette and Grünbaum, states that for any polyhedral graph, any face of the graph, and any convex polygon representing that face, it is possible to find a polyhedral realization of the whole graph that has the specified shape for the designated face. This is related to a theorem of Tutte, that any polyhedral graph can be drawn in the plane with all faces convex and any specified shape for its outer face.

Another example is applying Voronoi decomposition to the atoms in the A15 phases, which forms the polyhedral approximation of the Weaire–Phelan structure.

Bacterial microcompartments are widespread, membrane- bound organelles that are made of a protein shell that surrounds and encloses various enzymes. provide a further level of organization; they are compartments within bacteria that are surrounded by polyhedral protein shells, rather than by lipid membranes. These "polyhedral organelles" localize and compartmentalize bacterial metabolism, a function performed by the membrane- bound organelles in eukaryotes.

1988 PolyLib, PPL, isl, the Cloog polyhedral code generator,Cedric Bastoul. Code Generation in the Polyhedral Model Is Easier Than You Think. PACT'13 IEEE International Conference on Parallel Architecture and Compilation Techniques (2004) and the barvinok library for counting integer solutions. Of these libraries, PolyLib and PPL focus mostly on rational values, while the other libraries focus on integer values.

Peripheral cycles appear in the theory of polyhedral graphs, that is, 3-vertex-connected planar graphs. For every planar graph G, and every planar embedding of G, the faces of the embedding that are induced cycles must be peripheral cycles. In a polyhedral graph, all faces are peripheral cycles, and every peripheral cycle is a face., Theorems 2.7 and 2.8.

In geometric topology, the side-approximation theorem was proved by . It implies that a 2-sphere in R3 can be approximated by polyhedral 2-spheres.

A polyhedral graph is the graph of a simple polyhedron if it is cubic (every vertex has three edges), and it is the graph of a simplicial polyhedron if it is a maximal planar graph. The Halin graphs, graphs formed from a planar embedded tree by adding an outer cycle connecting all of the leaves of the tree, form another important subclass of the polyhedral graphs.

For any coordination number above 2 more than one coordination geometry is possible. For example four coordinate coordination compounds can be tetrahedral, square planar, square pyramidal or see-saw shaped. The polyhedral symbol is used to describe the geometry. A configuration index is determined from the positions of the ligands and together with the polyhedral symbol is placed at the beginning of the name.

Polyhedral frameworks are designed to support compilers techniques for analysis and transformation of codes with nested loops, producing exact results for loop nests with affine loop bounds and subscripts ("Static Control Parts" of programs). They can be used to represent and reason about executions (iterations) of statements, rather than treating a statement as a single object representing properties of all executions of that statement. Polyhedral frameworks typically also allow the use of symbolic expressions. Polyhedral frameworks can be used for dependence analysis for arrays, including both traditional alias analysis and more advanced techniques such as the analysis of data flow in arrays or identification of conditional dependencies.

Applying twice aa, (rectifying a rectification) is Conway's expand operation, e, which is the same as Johnson's cantellation operation, t0,2 generated from regular polyhedral and tilings.

In this form they are depicted as spherical, polyhedral beings, with three of the faces having "eye slits", one of which can emit a destructive energy beam.

In polyhedral combinatorics (a branch of mathematics), a stacked polytope is a polytope formed from a simplex by repeatedly gluing another simplex onto one of its facets..

A related statement, due to Hans Sterk, is that Aut(X) acts on the nef cone of X with a rational polyhedral fundamental domain.Huybrechts (2016), Theorem 8.4.2.

Finally, a graph is Hamiltonian if there exists a cycle that passes through each of its vertices exactly once. Barnette's conjecture states that every cubic bipartite polyhedral graph is Hamiltonian. By Steinitz's theorem, a planar graph represents the edges and vertices of a convex polyhedron if and only if it is polyhedral. A three-dimensional polyhedron has a cubic graph if and only if it is a simple polyhedron.

In this way, Tutte embeddings can be used to find Schlegel diagrams of every convex polyhedron. For every 3-connected planar graph G, either G itself or the dual graph of G has a triangle, so either this gives a polyhedral representation of G or of its dual; in the case that the dual graph is the one with the triangle, polarization gives a polyhedral representation of G itself..

The Jemmis mno rule provides the relationship between polyhedral boranes, condensed polyhedral boranes, and β-rhombohedral boron. This is similar to the relationship between benzene, condensed benzenoid aromatics, and graphite, shown by Hückel's 4n + 2 rule, as well as the relationship between tetracoordinate tetrahedral carbon compounds and diamond. The Jemmis mno rules reduce to Hückel's rule when restricted to two dimensions and reduce to Wade's rules when restricted to one polyhedron.

In a maximal planar graph, or more generally in every polyhedral graph, the peripheral cycles are exactly the faces of a planar embedding of the graph, so a polyhedral graph is strangulated if and only if all the faces are triangles, or equivalently it is maximal planar. Every chordal graph is strangulated, because the only induced cycles in chordal graphs are triangles, so there are no longer cycles to delete.

John David Kennedy (born 1943) is a chemist and emeritus professor of inorganic chemistry at the University of Leeds.chem.leeds.ac.uk He works in the area of polyhedral borane chemistry.

Martin Beech interprets the many polyhedral compounds within Stars as corresponding to double stars and triple star systems in astronomy. Beech writes that, for Escher, the mathematical orderliness of polyhedra depicts the "stability and timeless quality" of the heavens, and similarly Marianne L. Teuber writes that Stars "celebrates Escher's identification with Johannes Kepler's neo-Platonic belief in an underlying mathematical order in the universe". Alternatively, Howard W. Jaffe interprets the polyhedral forms in Stars crystallographically, as "brilliantly faceted jewels" floating through space, with its compound polyhedra representing crystal twinning. However, R. A. Dunlap points out the contrast between the order of the polyhedral forms and the more chaotic biological nature of the chameleons inhabiting them.

Grinberg used his theorem to find non-Hamiltonian cubic polyhedral graphs with high cyclic edge connectivity. The cyclic edge connectivity of a graph is the smallest number of edges whose deletion leaves a subgraph with more than one cyclic component. The 46-vertex Tutte graph, and the smaller cubic non- Hamiltonian polyhedral graphs derived from it, have cyclic edge connectivity three. Grinberg used his theorem to find a non-Hamiltonian cubic polyhedral graph with 44 vertices, 24 faces, and cyclic edge connectivity four, and another example (shown in the figure) with 46 vertices, 25 faces, and cyclic edge connectivity five, the maximum possible cyclic edge connectivity for a cubic planar graph other than K4.

Wolsey has made seminal contributions in duality theory for integer programming, submodular optimization, the group-theoretic approach and polyhedral analysis of fixed-charge network flow and production planning models.

This paper presents an approach to measure the curve distance between two points on a polyhedral surface in the manner that simulates dragging a tapeline at the two points.

A related conjecture of Barnette states that every cubic polyhedral graph in which all faces have six or fewer edges is Hamiltonian. Computational experiments have shown that, if a counterexample exists, it would have to have more than 177 vertices.. The intersection of these two conjectures would be that every bipartite cubic polyhedral graph in which all faces have four or six edges is Hamiltonian. This was proved to be true by .

The resulting material exhibited antimicrobial efficacy for the prevention of growth of both Gram-positive and Gram-negative bacteria. Array of QAS functionalized polyhedral oligomeric silsesquioxanes (Q-POSS) have been reported. These researchers varied the alkyl chain length from –C12H25 to –C18H37 and varied the counter ion between chloride, bromide, and iodine. The first reaction was the hydrosilylation between allydimethlamine and octasilane polyhedral oligomeric silsesquioxane via Karstedt's catalyst to make a tertiaryamino-functinoalized silsesquioxane.

PLUTO is an automatic parallelization tool based on the polyhedral model. The polyhedral model for compiler optimization is a representation for programs that makes it convenient to perform high-level transformations such as loop nest optimizations and loop parallelization. Pluto transforms C programs from source to source for coarse-grained parallelism and data locality simultaneously. The core transformation framework mainly works by finding affine transformations for efficient tiling and fusion, but not limited to those.

Acta Crystallograph. B49, 28-56 The infinite sheet structures that campigliaite has are characterized by strongly bonded polyhedral sheets, which are linked in the third dimension by weaker hydrogen bonds.

A Euclidean graph in three-dimensional space is a pair (V, E), where V is a set of points (sometimes called vertices or nodes) and E is a set of edges (sometimes called bonds or spacers) where each edge joins two vertices. There is a tendency in the polyhedral and chemical literature to refer to geometric graphs as nets (contrast with polyhedral nets), and the nomenclature in the chemical literature differs from that of graph theory.

In contrast to the simplex algorithm, which finds an optimal solution by traversing the edges between vertices on a polyhedral set, interior-point methods move through the interior of the feasible region.

Because polyhedral face numberings of this type are used as "spindown life counters" in the game Magic: The Gathering, name the canonical polyhedron realization of this dual polyhedron as "the Lich's nemesis".

Proc Nat. Acad Sci USA 113: 2478-2483. 8\. Liu Y, Ishino S, Ishino Y, Pehau-Arnaudet G, Krupovic M, and Prangishvili D (2017) “A novel type of polyhedral viruses infecting hyperthermophilic archaea”.

Albert Harry Wheeler (18 January 1873, Leominster, Massachusetts – 1950) was an American mathematician, inventor, and mathematics teacher, known for physical construction (usually in paper) of polyhedral models and teaching this art to students.

The capsule, which is humifuse (Ipheion, Beauverdia) or aerocarpic, globose or prismatic, and contains many seeds (pluriseeded) which are irregular and polyhedral with a black tegmen. The embryo is linear or slightly curved.

Snyder equal-area projection is used in the ISEA (Icosahedral Snyder Equal Area) discrete global grids. The first projection studies was conducted by John P. Snyder in the 1990s. Snyder, J. P. (1992), “An Equal-Area Map Projection for Polyhedral Globes”, Cartographica, 29(1), 10-21. urn:doi:10.3138/27H7-8K88-4882-1752. It is a modified Lambert azimuthal equal-area projection, most adequate to the polyhedral globe, a truncated icosahedron with 32 same-area faces (20 hexagons and 12 pentagons).

It may be especially convenient to remove all singularities to obtain a space with a flat Riemannian metric and to study the holonomies there. One concepts thus arising are polyhedral Kähler manifolds, when the holonomies are contained in a group, conjugate to the unitary matrices. In this case, the holonomies also preserve a symplectic form, together with a complex structure on this polyhedral space (manifold) with the singularities removed. All the concepts such as differential form, L2 differential form, etc.

A burst of research on viruses of E. histolytica stems from a series of papers published by Diamond et al. from 1972 to 1979. In 1972, they hypothesized two separate polyhedral and filamentous viral strains within E. histolytica that caused cell lysis. Perhaps the most novel observation was that two kinds of viral strains existed, and that within one type of amoeba (strain HB-301) the polyhedral strain had no detrimental effect but led to cell lysis in another (strain HK-9).

The EPH approximated by a polyhedral set is described by a system of a finite number of linear inequalities, which must be constructed by the approximation technique. Mathematical theory of optimal polyhedral approximation of convex bodies was developed during recently, and its results can be applied for developing the effective methods for approximating the EPH. A large number of bi-objective slices of such approximations can be computed and displayed in the form of a decision map in several seconds.

In the mathematical field of graph theory, the Barnette–Bosák–Lederberg graph is a cubic (that is, 3-regular) polyhedral graph with no Hamiltonian cycle, the smallest such graph possible. It was discovered in the mid-1960s by Joshua Lederberg, David Barnette, and Juraj Bosák, after whom it is named. It has 38 vertices and 69 edges. Other larger non-Hamiltonian cubic polyhedral graphs include the 46-vertex Tutte graph and a 44-vertex graph found by Emanuels Grīnbergs using Grinberg's theorem.

Significant results have been obtained in understanding the reactions of transition metal organometallics, week H-bond, electronic structure of three- dimensional aromatic compounds, polyhedral boranes, carboranes, silaboranes, electron counting rules for polycondensation, and structure of boron allotropes. The latter involved an extension of the Wade's Rules for polyhedral boranes to macropolyhedral boranes and the Huckel 4n+2 Rule to three dimensions. The Jemmis mno rules for polyhedral boranes have found a place in textbooks and are being taught in Inorganic Chemistry Courses in leading educational institutions around the world. Just as the basic tenets of the structural chemistry of carbon has stood the test of time, and led to major developments in carbon, the edifice of the structural chemistry expounded by Jemmis has already begun to do so for boron.

This family of methods is introduced by [Brezzi et al]F. Brezzi, K. Lipnikov, and M. Shashkov. Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal.

It also includes in a chapter of supplementary material the translations of three related articles by Volkov and Shor, including a simplified proof of Pogorelov's theorems generalizing Alexandrov's uniqueness theorem to non-polyhedral convex surfaces.

The span wing employs polyhedral configuration with the outer wing panels exhibiting much greater dihedral. Engines used include the Jabiru 2200 four-stroke powerplant as well as Rotax, Limbach Flugmotoren and Volkswagen 1600 automotive engines.

Louis Zocchi, Technical Sergeant, USAF (retired), is a gaming hobbyist, former game distributor and publisher, and maker and seller of polyhedral game dice. In 1986, he was elected to the Charles Roberts Awards Hall of Fame.

In geometry and polyhedral combinatorics, the Kleetope of a polyhedron or higher-dimensional convex polytope is another polyhedron or polytope formed by replacing each facet of with a shallow pyramid.. Kleetopes are named after Victor Klee..

In graph theory, a branch of mathematics, the Herschel graph is a bipartite undirected graph with 11 vertices and 18 edges, the smallest non-Hamiltonian polyhedral graph. It is named after British astronomer Alexander Stewart Herschel.

The polyhedral skeletal electron pair theory or Wade's electron counting rules predict trends in the stability and structures of many metal clusters. Jemmis mno rules have provided additional insight into the relative stability of metal clusters.

The Koebe–Andreev–Thurston circle packing theorem can be interpreted as providing another strengthening of Steinitz's theorem, that every 3-connected planar graph may be represented as a convex polyhedron in such a way that all of its edges are tangent to the same unit sphere.. By performing a carefully chosen Möbius transformation of a circle packing before transforming it into a polyhedron, it is possible to find a polyhedral realization that realizes all the symmetries of the underlying graph, in the sense that every graph automorphism is a symmetry of the polyhedral realization... More generally, if G is a polyhedral graph and K is any smooth three-dimensional convex body, it is possible to find a polyhedral representation of G in which all edges are tangent to K.. Circle packing methods can also be used to characterize the graphs of polyhedra that have a circumsphere or insphere. The characterization involves edge weights, constrained by systems of linear inequalities. These weights correspond to the angles made by adjacent circles in a system of circles, made by the intersections of the faces of the polyhedron with their circumsphere or the horizons of the vertices of the polyhedron on its insphere...

Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral combinatorics falls into two distinct areas. Mathematicians in this area study the combinatorics of polytopes; for instance, they seek inequalities that describe the relations between the numbers of vertices, edges, and faces of higher dimensions in arbitrary polytopes or in certain important subclasses of polytopes, and study other combinatorial properties of polytopes such as their connectivity and diameter (number of steps needed to reach any vertex from any other vertex). Additionally, many computer scientists use the phrase “polyhedral combinatorics” to describe research into precise descriptions of the faces of certain specific polytopes (especially 0-1 polytopes, whose vertices are subsets of a hypercube) arising from integer programming problems.

The rhombic dodecahedron is a Voronoi cell of the other optimal way to stack spheres. As the Voronoi cell of a regular space pattern, it is a plesiohedron. It is the polyhedral dual of the triangular orthobicupola.

J. Comput. Phys., 257-Part B:1163–1227, 2014. It allows the approximation of elliptic problems using a large class of polyhedral meshes. The proof that it enters the GDM framework is done in [Droniou et al].

Other infinite sequences of polyhedral graph formed in a similar way from polyhedra with regular- polygon bases include the antiprism graphs (graphs of antiprisms) and wheel graphs (graphs of pyramids). Other vertex-transitive polyhedral graphs include the Archimedean graphs. If the two cycles of a prism graph are broken by the removal of a single edge in the same position in both cycles, the result is a ladder graph. If these two removed edges are replaced by two crossed edges, the result is a non-planar graph called a Möbius ladder..

The Schlegel diagram of a convex polyhedron represents its vertices and edges as points and line segments in the Euclidean plane, forming a subdivision of an outer convex polygon into smaller convex polygons (a convex drawing of the graph of the polyhedron). It has no crossings, so every polyhedral graph is also a planar graph. Additionally, by Balinski's theorem, it is a 3-vertex-connected graph. According to Steinitz's theorem, these two graph-theoretic properties are enough to completely characterize the polyhedral graphs: they are exactly the 3-vertex-connected planar graphs.

The chapters of the book organize the material into topics within graph theory, rather than being strictly chronological. The first chapter, on paths, includes maze-solving algorithms as well as Euler's work on Euler tours. Next, a chapter on circuits includes material on knight's tours in chess (a topic that long predates Euler), Hamiltonian cycles, and the work of Thomas Kirkman on polyhedral graphs. Next follow chapters on spanning trees and Cayley's formula, chemical graph theory and graph enumeration, and planar graphs, Kuratowski's theorem, and Euler's polyhedral formula.

Polyhedral bodies were discovered by transmission electron microscopy in the cyanobacterium Phormidium uncinatum in 1956. These were later observed in other cyanobacteria and in some chemotrophic bacteria that fixed carbon dioxide—many of them are sulfur reducers or nitrogen fixers (for example, Halothiobacillus, Acidithiobacillus, Nitrobacter and Nitrococcus). The polyhedral bodies were first purified from Thiobacillus neapolitanus (now Halothiobacillus neapolitanus) in 1973 and shown to contain RuBisCO, held within a rigid outer covering. The authors proposed that since these appeared to be organelles involved in carbon fixation, they should be called carboxysomes.

On the death of William Wallace in 1631, Mylne was appointed Master Mason to the Crown, and returned to Edinburgh. His first royal commission came shortly before this, in 1629, when he was tasked with the construction of a large pond at the palace of Holyroodhouse. With the assistance of his two sons, he also erected a large polyhedral sundial at Holyrood for Charles I, on the occasion of his Scottish coronation in 1633. The sundial, which bears numerous Stuart emblems, is the earliest surviving polyhedral example in the country.

The discovery of adamantane in petroleum in 1933 launched a new field of chemistry dedicated to the synthesis and properties of polyhedral organic compounds. Adamantane derivatives have found practical application as drugs, polymeric materials, and thermally stable lubricants.

Two graphs are called cospectral or isospectral if the adjacency matrices of the graphs have equal multisets of eigenvalues. enneahedra, the smallest possible cospectral polyhedral graphs Cospectral graphs need not be isomorphic, but isomorphic graphs are always cospectral.

In general, a polytope is prismatoidal if its vertices exist in two hyperplanes. For example, in four dimensions, two polyhedra can be placed in two parallel 3-spaces, and connected with polyhedral sides. 320px A tetrahedral-cuboctahedral cupola.

This structure theory has led to new advances in polyhedral combinatorics and new bounds on the chromatic number of claw-free graphs, as well as to new fixed- parameter-tractable algorithms for dominating sets in claw-free graphs.

It is part of a multidimensional family of hypercube honeycombs, with Schläfli symbols of the form {4,3,...,3,4}, starting with the square tiling, {4,4} in the plane. It is one of 28 uniform honeycombs using convex uniform polyhedral cells.

If true, this would imply a conjecture of W. T. Tutte that every bridgeless graph has a nowhere-zero 5-flow. A stronger type of embedding than a circular embedding is a polyhedral embedding, an embedding of a graph on a surface in such a way that every face is a simple cycle and every two faces that intersect do so in either a single vertex or a single edge. (In the case of a cubic graph, this can be simplified to a requirement that every two faces that intersect do so in a single edge.) Thus, in view of the reduction of the cycle double cover conjecture to snarks, it is of interest to investigate polyhedral embeddings of snarks. Unable to find such embeddings, Branko Grünbaum conjectured that they do not exist, but disproved Grünbaum's conjecture by finding a snark with a polyhedral embedding.

The Classic period Maya region featured large scale prismatic blade production, the exchange of polyhedral cores, and large scale sociopolitical and economic organization (Moholy-Nagy et al. 1984; Knight and Glascock 2009). A very common form of obsidian used to transport it and derive blades from was the polyhedral core, which was most frequently used from the Early to Late Classic (Trachman 1999). Prismatic blades made from polyhedral cores have been found at Copan and its hinterland regions; a dramatic increase in these blades during the Classic has been attributed to a royal dynasty assuming control over procurement of obsidian and production at two workshops in Copan's epicenter (Aoyama 2001). Most of this obsidian came from the Ixtepeque source to make utilitarian blades that all residents had access to, but green obsidian from central Mexico has been found in elite contexts, suggesting long distance exchange ties to Teotihuacan (Aoyama 2001).

Greetings from Info Mesa GRASP was graphics program written for Silicon Graphics computers that was used by the structural biology community to visualize macromolecules. It was most widely used software for computing and displaying polyhedral molecular surfaces during the 1990s.

Muetterties, E. L.; Balthis, J. H.; Chia, Y. T.; Knoth, W. H.; Miller, H. C. Inorg. Chem. 1964, 3, 444. Salts and Acids of B10H102− and B12H122− Muetterties was an inventor on some basic findings with the polyhedral borate anions.

In addition to the polyhedral boranes, the program explored pi-allyl, fluoroalkyl, and boron hydride complexes of the transition metals. Research also extended to stereochemically-non-rigid complexes.Muetterties, E. L.. Polytopal form and isomerism. Tetrahedron (1974), 30(12), 1595-604.

An icosahedron. Geometric combinatorics is related to convex and discrete geometry, in particular polyhedral combinatorics. It asks, for example, how many faces of each dimension a convex polytope can have. Metric properties of polytopes play an important role as well, e.g.

Thus, the Schwarz lantern demonstrates that simply connecting inscribed vertices is not enough to ensure surface area convergence. Animation of Schwarz lantern convergence (or lack thereof) for various refinement strategies. The polyhedral surface bears resemblance to a cylindrical paper lantern.

A Polyhedral's dial faces can be designed to give the time for different time-zones simultaneously. Examples include the Scottish sundial of the 17th and 18th century, which was often an extremely complex shape of polyhedral, and even convex, faces.

Cahill–Keyes map of the world. The Cahill–Keyes projection with Tissot's indicatrix of deformation. CE 2012 by Duncan Webb using Cahill–Keyes projection. The Cahill–Keyes projection is a polyhedral compromise map projection first proposed by Gene Keyes in 1975.

A polyhedral model may be represented in terms of the partition of the plane into polygonal regions, each region being associated with a plane patch which is the image of points of the region under the piecewise-linear function in question.

The structures of condensed oxyanions can be rationalized in terms of AOn polyhedral units with sharing of corners or edges between polyhedra. The phosphate and polyphosphate esters adenosine monophosphate (AMP), adenosine diphosphate (ADP) and adenosine triphosphate (ATP) are important in biology.

Some of these frameworks use one or more libraries for performing polyhedral operations. Others, notably Omega, combine everything in a single package. Some commonly used libraries are the Omega Library (and a more recent fork), piplib,Paul Feautrier. Parametric Integer Programming.

Ottawa, Canada, June 2006. Polly provides polyhedral optimizations for LLVM, and R-StreamBenoit Meister, Nicolas Vasilache, David Wohlford, Muthu Baskaran, Allen Leung and Richard Lethin. R-Stream Compiler. In Encyclopedia of Parallel Computing, David Padua Ed., pp 1756-1765, Springer, 2011.

Condensed polyhedral boranes and metallaboranes m + n + o + p − q = 2 + 20 + 0 + 0 + 0 = 22 SEPs are required; 16 BH units provide 16 pairs; four shared boron atoms provide 6 pairs, which describes why is stable as a neutral species.

There exist planar non- Hamiltonian graphs in which all faces have five or eight sides. For these graphs, Grinberg's formula taken modulo three is always satisfied by any partition of the faces into two subsets, preventing the application of his theorem to proving non-Hamiltonicity in this case . It is not possible to use Grinberg's theorem to find counterexamples to Barnette's conjecture, that every cubic bipartite polyhedral graph is Hamiltonian. Every cubic bipartite polyhedral graph has a partition of the faces into two subsets satisfying Grinberg's theorem, regardless of whether it also has a Hamiltonian cycle .

Conversely, every continuous piecewise-linear surface comes from an equilibrium stress in this way. If a finite planar graph is drawn and given an equilibrium stress in such a way that all interior edges of the drawing have positive weights, and all exterior edges have negative weights, then by translating this stress into a three-dimensional surface in this way, and then replacing the flat surface representing the exterior of the graph by its complement in the same plane, one obtains a convex polyhedron, with the additional property that its perpendicular projection onto the plane has no crossings... The Maxwell–Cremona correspondence has been used to obtain polyhedral realizations of polyhedral graphs by combining it with a planar graph drawing method of W. T. Tutte, the Tutte embedding. Tutte's method begins by fixing one face of a polyhedral graph into convex position in the plane. This face will become the outer face of a drawing of a graph.

There is a third topological polyhedral figure with 5 faces, degenerate as a polyhedron: it exists as a spherical tiling of digon faces, called a pentagonal hosohedron with Schläfli symbol {2,5}. It has 2 (antipodal point) vertices, 5 edges, and 5 digonal faces.

This is essentially the content of Hilbert's third problem – more precisely, not all polyhedral pyramids are scissors congruent (can be cut apart into finite pieces and rearranged into the other), and thus volume cannot be computed purely by using a decomposition argument.

Like most role-playing games, Supernatural requires several players; one person to be the Game Master and two to five others to play hunters such as Dean Winchester and Sam Winchester. It also requires multiple polyhedral dice ranging from four-sided to twelve-sided.

Ironclaw uses a system where attributes of characters are matched to different polyhedral dice. These attributes include a character's physical, mental, and social capabilities, in addition to the abilities associated with their species.This system was later used in Sanguine's other role-playing games, including Jadeclaw.

Amoebophilus species are ectoparasites of amoeba. The thallus is composed of an internal haustorium that can be heart-shaped, globose, or lobose. Trailing chains of four or more conidia are produced from the haustorium. Zygospores are spherical at first and become polyhedral with age.

Later, organomodified clays, TiO2 nanoparticles, silica nanoparticles, layered double hydroxides, carbon nanotubes and polyhedral silsesquioxanes were proved to work as well. Recent research has suggested that combining nanoparticles with traditional fire retardants (e.g., intumescents) or with surface treatment (e.g., plasma treatment) effectively decreases flammability.

A polyhedral compound is made of two or more polyhedra sharing a common centre. Symmetrical compounds often share the same vertices as other well-known polyhedra and may often also be formed by stellation. Some are listed in the list of Wenninger polyhedron models.

For NCPs, using a polyhedral approximation can transform the NCPs into a set of LCPs, which can then be solved by the LCP solver. Other approaches beyond these methods, such NCP- functions or cone complementarity problems (CCP) based methods are also employed to solve NCPs.

The causal agent was transmitted by Nephotettix nigropictus after an incubation of two weeks. Polyhedral particles of 65 nm diameter in the cytoplasm of phloem cells were always associated with the disease. No serologic relationship was found between this virus and that of rice dwarf.

The virus has a polyhedral head with a diameter of around 61 nm. The contractile tail is 100 nm long, and is attached to the head via a tail capital. The neck region bears fibres. The mature virus particle contains at least 19 proteins.

In molecular biology the Bacterial Microcompartment (BMC) domain is a protein domain found in a variety of shell proteins, including CsoS1A, CsoS1B and CsoS1C of Thiobacillus neapolitanus (Halothiobacillus neapolitanus) and their orthologs from other bacteria. These shell proteins form the polyhedral structure of the carboxysome and related structures that plays a metabolic role in bacteria. The BMC domain consists of about 90 amino acid residues, characterized by β-α-β motif connected by a β-hairpin. The majority of the shell proteins consist of a single BMC domain in each subunit, forming a hexameric structure that assembles to form the flat facets of the polyhedral shell.

A linear, amorphous polynorbornene (Norsorex, developed by CdF Chemie/Nippon Zeon) or organic- inorganic hybrid polymers consisting of polynorbornene units that are partially substituted by polyhedral oligosilsesquioxane (POSS) also have shape-memory effect. 700px Another example reported in the literature is a copolymer consisting of polycyclooctene (PCOE) and poly(5-norbornene- exo,exo-2,3-dicarboxylic anhydride) (PNBEDCA), which was synthesized through ring-opening metathesis polymerization (ROMP). Then the obtained copolymer P(COE-co-NBEDCA) was readily modified by grafting reaction of NBEDCA units with polyhedral oligomeric silsesquioxanes (POSS) to afford a functionalized copolymer P(COE-co-NBEDCA-g-POSS). It exhibits shape-memory effect.

Studies in the 1970s formed the early foundations for many of the computer vision algorithms that exist today, including extraction of edges from images, labeling of lines, non-polyhedral and polyhedral modeling, representation of objects as interconnections of smaller structures, optical flow, and motion estimation. The next decade saw studies based on more rigorous mathematical analysis and quantitative aspects of computer vision. These include the concept of scale-space, the inference of shape from various cues such as shading, texture and focus, and contour models known as snakes. Researchers also realized that many of these mathematical concepts could be treated within the same optimization framework as regularization and Markov random fields.

The triakis icosahedron, a polyhedron in which every edge has endpoints with total degree at least 13 In graph theory and polyhedral combinatorics, areas of mathematics, Kotzig's theorem is the statement that every polyhedral graph has an edge whose two endpoints have total degree at most 13. An extreme case is the triakis icosahedron, where no edge has smaller total degree. The result is named after Anton Kotzig, who published it in 1955 in the dual form that every convex polyhedron has two adjacent faces with a total of at most 13 sides. It was named and popularized in the west in the 1970s by Branko Grünbaum.

Every Halin graph is 3-connected, meaning that it is not possible to delete two vertices from it and disconnect the remaining vertices. It is edge-minimal 3-connected, meaning that if any one of its edges is removed, the remaining graph will no longer be 3-connected. By Steinitz's theorem, as a 3-connected planar graph, it can be represented as the set of vertices and edges of a convex polyhedron; that is, it is a polyhedral graph. And, as with every polyhedral graph, its planar embedding is unique up to the choice of which of its faces is to be the outer face.

The polyhedral model (also called the polytope method) is a mathematical framework for programs that perform large numbers of operations -- too large to be explicitly enumerated -- thereby requiring a compact representation. Nested loop programs are the typical, but not the only example, and the most common use of the model is for loop nest optimization in program optimization. The polyhedral method treats each loop iteration within nested loops as lattice points inside mathematical objects called polyhedra, performs affine transformations or more general non-affine transformations such as tiling on the polytopes, and then converts the transformed polytopes into equivalent, but optimized (depending on targeted optimization goal), loop nests through polyhedra scanning.

If a polyhedron is not regular, the edge midpoints surrounding a vertex may not be coplanar. However, a form of rectification is still possible in this case: every polyhedron has a polyhedral graph as its 1-skeleton, and from that graph one may form the medial graph by placing a vertex at each edge midpoint of the original graph, and connecting two of these new vertices by an edge whenever they belong to consecutive edges along a common face. The resulting medial graph remains polyhedral, so by Steinitz's theorem it can be represented as a polyhedron. The Conway polyhedron notation equivalent to rectification is ambo, represented by a.

Knotenschiefer containing andalusite Knotenschiefer from Głuchołazy, Oppa Mountains, Poland Knotenschiefer is a variety of spotted slate characterized by conspicuous subspherical or polyhedral clots that are often individual minerals such as cordierite, biotite, chlorite, andalusite and others.Bucksch, Herbert (1997). Dictionary of Geotechnical Engineering, Vol. 2., Springer- Verlag, Berlin. .

Microscopically, they appear identical to seminomas and very close to embryonic primordial germ cells, having large, polyhedral, rounded clear cells. The nuclei are uniform and round or square with prominent nucleoli and the cytoplasm has high levels of glycogen. Inflammation is another prominent histologic feature of dysgerminomas.

193–200, New York, NY, USA, 1988. proposed an O(log4 n)-time algorithm for the hidden-surface problem, using O((n + v)/log n) CREW PRAM processors for a restricted model of polyhedral terrains, where v is the output size. In 2011 Devai publishedF. Devai.

Decidualization succeeds predecidualization if pregnancy occurs. This is an expansion of it, further developing the uterine glands, the zona compacta and the epithelium of decidual cells lining it. The decidual cells become filled with lipids and glycogen and take the polyhedral shape characteristic for decidual cells.

The topological class of a polyhedron is defined by its Euler characteristic and orientability. From this perspective, any polyhedral surface may be classed as certain kind of topological manifold. For example, the surface of a convex or indeed any simply connected polyhedron is a topological sphere.

Typically, a fundamental domain is required to be a connected subset with some restrictions on its boundary, for example, smooth or polyhedral. The images of a chosen fundamental domain under the group action then tile the space. One general construction of fundamental domains uses Voronoi cells.

The fruit is a woody, cup-shaped, cylindrical or hemispherical capsule, long and wide. The fruit generally has two ribs, a thick rim and broad valves with the tips usually just below the rim. The brown seeds are polyhedral and have narrow wings along the main edges.

Magic: The Gathering Cards contain spells that players can cast on their opponents. Game play contains mystical characters such as Hydra, Sorceress and Guardian. Dungeons and Dragons Game play employs polyhedral dice. Players rely on dice rolling to determine the attack and defence strengths of their characters.

Transmission electron micrograph showing a species of the cyanobacteria Synechococcus. The carboxysomes appear as polyhedral dark structures. Synechococcus (from the Greek synechos, in succession, and the Greek kokkos, granule) is a unicellular cyanobacterium that is very widespread in the marine environment. Its size varies from 0.8 to 1.5 µm.

The graph of an n-gonal prism has 2n vertices and 3n edges. They are regular, cubic graphs. Since the prism has symmetries taking each vertex to each other vertex, the prism graphs are vertex-transitive graphs. As polyhedral graphs, they are also 3-vertex-connected planar graphs.

Leydig cells, also known as interstitial cells of Leydig, are found adjacent to the seminiferous tubules in the testicle. They produce testosterone in the presence of luteinizing hormone (LH). Leydig cells are polyhedral in shape, and have a large prominent nucleus, an eosinophilic cytoplasm and numerous lipid-filled vesicles.

A chopper has an edge on one side. It is unifacial if the edge was created by flaking on one face of the core, or bifacial if on two. Discoid tools are roughly circular with a peripheral edge. Polyhedral tools are edged in the shape of a polyhedron.

In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the (simple) 3-vertex- connected planar graphs (with at least four vertices). That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs. Branko Grünbaum has called this theorem “the most important and deepest known result on 3-polytopes.” The theorem appears in a 1922 paper of Ernst Steinitz, after whom it is named.

Barnette's conjecture is an unsolved problem in graph theory, a branch of mathematics, concerning Hamiltonian cycles in graphs. It is named after David W. Barnette, a professor emeritus at the University of California, Davis; it states that every bipartite polyhedral graph with three edges per vertex has a Hamiltonian cycle.

In geometry, Kalai's 3d conjecture is a conjecture on the polyhedral combinatorics of centrally symmetric polytopes, made by Gil Kalai in 1989.. It states that every d-dimensional centrally symmetric polytope has at least 3d nonempty faces (including the polytope itself as a face but not including the empty set).

As the foam is formed, it changes in structure. As the liquid foams up into the gas, the foam bubbles begin as packed uniform spheres. This phase is the wet phase. The farther up the column the foam travels, the air bubbles distort to form polyhedral shapes, the dry phase.

Her main research fields include hyperbolic geometry, geometric group theory, geometry of discrete groups (especially reflection groups, Coxeter groups), convex and polyhedral geometry, volumes of hyperbolic polytopes, manifolds and polylogarithms. She does historical research into the works and life of Ludwig Schläfli, a Swiss geometer.Der Mathematiker Ludwig Schläfli (15.01.1814 – 20.03.

Lee conformal tetrahedric projection of the world centered on the south pole. Tissot's indicatrix of deformation. Lee conformal tetrahedric projection tessellated several times in the plane. The Lee conformal world in a tetrahedron is a polyhedral, conformal map projection that projects the globe onto a tetrahedron using Dixon's elliptic functions.

There has been much research on Hamiltonicity of cubic graphs. In 1880, P.G. Tait conjectured that every cubic polyhedral graph has a Hamiltonian circuit. William Thomas Tutte provided a counter-example to Tait's conjecture, the 46-vertex Tutte graph, in 1946. In 1971, Tutte conjectured that all bicubic graphs are Hamiltonian.

Glue the face with vertices 0,1,2 to the face with vertices 3,1,0 in that order. Glue the face 0,2,3 to the face 3,2,1 in that order. In the hyperbolic structure of the Gieseking manifold, this ideal tetrahedron is the canonical polyhedral decomposition of David B. A. Epstein and Robert C. Penner.

D&D; uses polyhedral dice to resolve in-game events. These are abbreviated by a 'd' followed by the number of sides. Shown counter-clockwise from the bottom are: d4, d6, d8, d10, d12 and d20 dice. A pair of d10 can be used together to represent percentile dice, or d100.

Brown fat cells or plurivacuolar cells are polyhedral in shape. Unlike white fat cells, these cells have considerable cytoplasm, with lipid droplets scattered throughout. The nucleus is round and, although eccentrically located, it is not in the periphery of the cell. The brown color comes from the large quantity of mitochondria.

Isabella Novik (born 1971)Birth year from ISNI authority control file, retrieved 2018-11-30. is a mathematician who works at the University of Washington as the Robert R. & Elaine F. Phelps Professor in Mathematics. Her research concerns algebraic combinatorics and polyhedral combinatorics.Faculty profile, University of Washington, retrieved 2016-11-06.

Histologically, the tumor is described by large, uniformly shaped polyhedral nevus cells that are pigmented and closely packed Typically, it lacks signs of malignancy such as high mitotic rate, necroses or infiltrative growth. Like the malignant melanoma, it shows an immunohistological profile with S-100 protein-, vimentin- and HMB-45-positive tumor cells.

In collaboration with William D. Phillips he exploited NMR for study of dynamic processes in inorganic fluoride compounds.Muetterties, E. L.; Phillips, W. D. Fluoroarsenites. Journal of the American Chemical Society (1957), 79 3686-7. Muetterties's work on boron hydride clusters led to the work on several polyhedral borane anions such as B12H122−.

Exceptions are mostly limited to organolead compounds. Like the lighter members of the group, lead tends to bond with itself; it can form chains and polyhedral structures. Lead is easily extracted from its ores; prehistoric people in Western Asia knew of it. Galena is a principal ore of lead which often bears silver.

For example in the complex (SP-4-3)-(acetonitrile)dichlorido(pyridine)platinum(II) the (SP-4-3) at the beginning of the name describes a square planar geometry, 4 coordinate with a configuration index of 3 indicating the position of the ligands around the central atom. For more detail see polyhedral symbol.

The pores are roughly spherical or polyhedral at first, becoming angular to pentagonal in age, and almost gill-like near the attachment to the stem. Pores are about 1–2 mm in diameter. The stem is solid (i.e., not hollow) long, thick in the upper part, expanding to at the bulbous base.

Organosilicon compounds are widely encountered in commercial products. Most common are sealants, caulks (sealant), adhesives, and coatings made from silicones. Other important uses include synthesis of polyhedral oligomeric silsesquioxanes, agricultural and plant control adjuvants commonly used in conjunction with herbicides and fungicides. Silicone caulk, commercial sealants, are mainly composed of organosilicon compounds.

In a maximal planar graph (or more generally a polyhedral graph) the peripheral cycles are the faces, so maximal planar graphs are strangulated. The strangulated graphs include also the chordal graphs, and are exactly the graphs that can be formed by clique-sums (without deleting edges) of complete graphs and maximal planar graphs..

Typical role-playing dice, showing a variety of colors and styles. Note the older hand-inked green 12-sided die (showing an 11), manufactured before pre-inked dice were common. Many players collect or acquire a large number of mixed and unmatching dice. Polyhedral dice are commonly used in role-playing games.

A blossom is a factor-critical subgraph of a larger graph. Blossoms play a key role in Jack Edmonds' algorithms for maximum matching and minimum weight perfect matching in non-bipartite graphs.. In polyhedral combinatorics, factor-critical graphs play an important role in describing facets of the matching polytope of a given graph.

Stars is a wood engraving print created by the Dutch artist M. C. Escher in 1948, depicting two chameleons in a polyhedral cage floating through space. Although the compound of three octahedra used for the central cage in Stars had been studied before in mathematics, it was most likely invented independently for this image by Escher without reference to those studies. Escher used similar compound polyhedral forms in several other works, including Crystal (1947), Study for Stars (1948), Double Planetoid (1949), and Waterfall (1961). The design for Stars was likely influenced by Escher's own interest in both geometry and astronomy, by a long history of using geometric forms to model the heavens, and by a drawing style used by Leonardo da Vinci.

For these graphs, a convex (but not necessarily strictly convex) drawing can be found within a grid whose length on each side is linear in the number of vertices of the graph, in linear time. However, strictly convex drawings may require larger grids; for instance, for any polyhedron such as a pyramid in which one face has a linear number of vertices, a strictly convex drawing of its graph requires a grid of cubic area. A linear-time algorithm can find strictly convex drawings of polyhedral graphs in a grid whose length on each side is quadratic. Convex but not strictly convex drawing of the complete bipartite graph K_{2,3} Other graphs that are not polyhedral can also have convex drawings, or strictly convex drawings.

Adventures Among the Toroids extends the investigation of polyhedra with regular faces to non-convex polyhedra, and in particular to polyhedra of higher genus than the sphere. Many of these polyhedra can be formed by gluing together smaller polyhedral pieces, carving polyhedral tunnels through them, or piling them into elaborate towers. The toroidal polyhedra described in this book, formed from regular polygons with no self-intersections or flat angles, have come to be called Stewart toroids. A ring of octahedra discussed in the second edition of the book The second edition is rewritten in a different page format, letter sized in landscape mode compared to the tall and narrow by page size of the first edition, with two columns per page.

The first condensed polyhedral borane, , is formed by sharing four vertices between two icosahedra. According to Wade's n + 1 rule for n-vertex closo structures, should have a charge of +2 (n + 1 = 20 + 1 = 21 pairs required; 16 BH units provide 16 pairs; four shared boron atoms provide 6 pairs; thus 22 pairs are available). To account for the existence of as a neutral species, and to understand the electronic requirement of condensed polyhedral clusters, a new variable, m, was introduced and corresponds to the number of polyhedra (sub- clusters). In Wade's n + 1 rule, the 1 corresponds to the core bonding molecular orbital (BMO) and the n corresponds to the number of vertices, which in turn is equal to the number of tangential surface BMOs.

The salivary gland oncocytoma is a well-circumscribed, benign neoplastic growth also called an oxyphilic adenoma. It comprises about 1% of all salivary gland tumors. The histopathology is marked by sheets of large swollen polyhedral epithelial oncocytes, which are granular acidophilic parotid cells with centrally located nuclei. The granules are created by the mitochondria.

A polyhedral torus can be constructed to approximate a torus surface, from a net of quadrilateral faces, like this 6x4 example. In geometry, a toroidal polyhedron is a polyhedron which is also a toroid (a g-holed torus), having a topological genus of 1 or greater. Notable examples include the Császár and Szilassi polyhedra.

A self-dual graph. A plane graph is said to be self-dual if it is isomorphic to its dual graph. The wheel graphs provide an infinite family of self-dual graphs coming from self-dual polyhedra (the pyramids). However, there also exist self-dual graphs that are not polyhedral, such as the one shown.

According to the painter Giorgio de Chirico, Aizenberg admired architecture and the idea of its construction, especially the architecture of the Renaissance. His work is permanently influenced by this fascination. His work shows isolated towers, empty towns, mysterious and uninhabited buildings, and rare polyhedral constructions. He used slow drying oils to perfect his finishes.

Jessen's icosahedron In Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, the 12 isosceles faces are arranged differently so that the figure is non-convex and has right dihedral angles. It is scissors congruent to a cube, meaning that it can be sliced into smaller polyhedral pieces that can be rearranged to form a solid cube.

Lectures in Geometric Combinatorics is a textbook on polyhedral combinatorics. It was written by Rekha R. Thomas, based on a course given by Thomas at the 2004 Park City Mathematics Institute, and published by the American Mathematical Society and Institute for Advanced Study in 2006, as volume 33 of their Student Mathematical Library book series.

The Little Detroit Speedster was a cyclecar built in Detroit, Michigan by the Detroit Cyclecar Company from 1913 to 1914. The cyclecar was a rather small cyclecar that came equipped with a four-cylinder water-cooled engine and a two-speed selective transmission and shaft drive. The bonnet front had a peculiar polyhedral design.

In 2007, a Stella4D version was added, allowing the generation and display of four-dimensional polytopes (polychora), including a library of all convex uniform polychora, and all currently known nonconvex star polychora, as well as the uniform duals. They can be selected from a library or generated from user created polyhedral vertex figure files.

Crystal structure of clinohumite in polyhedral representation, a-axis projection, b-horizontal. H atom are blue spheres. The structure is monoclinic with space group P21/b (a-unique). The unit cell has a = 4.7488 Å; b = 10.2875 Å; c = 13.6967 Å; and alpha = 100.63°; V = 667.65 Å3; Z = 2 for pure Mg hydroxyl-clinohumite.

Usually the cells are polygonal or polyhedral and form a mesh that partitions the domain. Important classes of two-dimensional elements include triangles (simplices) and quadrilaterals (topological squares). In three-dimensions the most-common cells are tetrahedra (simplices) and hexahedra (topological cubes). Simplicial meshes may be of any dimension and include triangles (2D) and tetrahedra (3D) as important instances.

However, the planar graph drawings produced by Tutte's method do not necessarily lift to convex polyhedra. Instead, Barnette and Grünbaum prove this result using an inductive method. It is also always possible, given a polyhedral graph G and an arbitrary cycle C, to find a realization such that C forms the silhouette of the realization under parallel projection..

They are large polyhedral cells, with six surfaces, three of which have a relevant function. The three relevant type of surfaces are sinusoidal, canalicular and intercellular. These surfaces are involved in the exchange of substances between the hepatocyte, the vessels and the biliar canaliculi. The sinusoidal surfaces are separated from the sinusoids because of the perisinusoidal space.

Corderoite has crankshaft chains that are crosswise linked by additional Hg²+. The bond distance between the cation Hg and anion S is 2.422 Angstroms. It has two angles, Hg-S-HG= 94.1º and S-Hg-S= 165.1º. Various sulfide halides of Hg share the feature of being face-sharing [HgS2X4] −6 polyhedral, as corderoite's polyhedra X=Cl.

The compound of cube and octahedron in the upper left was used earlier by Escher, in Crystal (1947). Escher's later work Four Regular Solids (Stereometric Figure) returned to the theme of polyhedral compounds, depicting a more explicitly Keplerian form in which the compound of the cube and octahedron is nested within the compound of the dodecahedron and icosahedron.

In the 19th century, it was enlarged and castellated, serpentine bays added to the canal and an unusual polyhedral sundial given pride of place on a sunken lawn. Other additions were a gothic porch bearing the Aylward crest and a conservatory. The stable-yard and the castellated entrance to the demesne are attributed to Daniel Robertson.

A polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram. The outer vertices of a compound can be connected to form a convex polyhedron called its convex hull. A compound is a facetting of its convex hull.

Polyhedral libraries such as PolyLib and PPL exploit the double description of polyhedra and therefore naturally support vertex enumeration on (non-parametric) polytopes. The Omega Library internally performs vertex enumeration during the computation of the convex hull. PolyLib and isl provide vertex enumeration on parametric polytopes, which is essential for applying Barvinok's algorithm to parametric polytopes.

Einar Thorsteinn (1942–April 28, 2015) was an Icelandic architect with an interest in geometrical structures. Thorsteinn was a follower of Buckminster Fuller and he worked with polyhedral and spherical shapes. He graduated from Technical University of Hannover. He worked with Frei Otto from 1969-1972 helping to design the Munich Olympiapark for the 1972 Summer Olympics.

For low nuclearity clusters, bonding is often described as if it is localized. For this purpose, the eighteen electron rule is used. Thus, 34 electrons in an organometallic complex predicts a dimetallic complex with a metal-metal bond. For higher nuclearity clusters, more elaborate rules are invoked including Jemmis mno rules and Polyhedral skeletal electron pair theory.

Boolarra virus typically measures around 30 nano-meters and is approximately 21 percent RNA. Boolarra virus is characterized by its polyhedral shape. The virus only shares a significant portion of the same amino acid sequence with the Nodamura virus, which is a similar Nodavirus. Among other Nodoviruses, Boolarra was not very similar concerning amino acid sequences.

Motors and computer automation was banned in the class. Each challenger team was only allowed to build one AC50 for competition and only six boats were built. The class achieved a maximum peak speed of over the water, recorded by ACRM telemetry aboard Magic Blue. The class winner sailed by Team New Zealand featured a distinct polyhedral daggerboard stabilizer.

The aircraft features a cantilever low-wing, a T-tail, a two- seats-in-side-by-side configuration enclosed cockpit, electric flaps, a retractable landing gear, and a single engine in tractor configuration. The Sonata is made from composites. Its polyhedral wing is . Standard engines available are the Rotax 582 two-stroke and the Hirth F34 powerplant.

If a cactus is connected, and each of its vertices belongs to at most two blocks, then it is called a Christmas cactus. Every polyhedral graph has a Christmas cactus subgraph that includes all of its vertices, a fact that plays an essential role in a proof by that every polyhedral graph has a greedy embedding in the Euclidean plane, an assignment of coordinates to the vertices for which greedy forwarding succeeds in routing messages between all pairs of vertices.. In topological graph theory, the graphs whose cellular embeddings are all planar are exactly the subfamily of the cactus graphs with the additional property that each vertex belongs to at most one cycle. These graphs have two forbidden minors, the diamond graph and the five-vertex friendship graph.

It is possible to prove a stronger form of Steinitz's theorem, that any polyhedral graph can be realized by a convex polyhedron for which all of the vertex coordinates are integers. For instance, Steinitz's original induction-based proof can be strengthened in this way. However, the integers that would result from this construction are doubly exponential in the number of vertices of the given polyhedral graph. Writing down numbers of this magnitude in binary notation would require an exponential number of bits.. Subsequent researchers have found lifting-based realization algorithms that use only O(n) bits per vertex... It is also possible to relax the requirement that the coordinates be integers, and assign coordinates in such a way that the x-coordinates of the vertices are distinct integers in the range [0,2n − 4] and the other two coordinates are real numbers in the range [0,1], so that each edge has length at least one while the overall polyhedron has volume O(n).. Some polyhedral graphs are known to be realizable on grids of only polynomial size; in particular this is true for the pyramids (realizations of wheel graphs), prisms (realizations of prism graphs), and stacked polyhedra (realizations of Apollonian networks)..

The Dürer graph is a well-covered graph, meaning that all of its maximal independent sets have the same number of vertices, four. It is one of four well-covered cubic polyhedral graphs and one of seven well-covered 3-connected cubic graphs. The only other three well-covered simple convex polyhedra are the tetrahedron, triangular prism, and pentagonal prism.; .

Petit completed postdoctoral research in Alissa Park's group at Columbia University. She worked on carbon capture using nanoparticle organic hybrid materials (NOHMs). She synthesises them by ionic grafting polymer chains onto polyhedral oligomeric silsesquioxane (POSS). She developed several characterisation techniques to analyse their suitability for carbon capture, including nuclear magnetic resonance, Attenuated total reflectance Fourier-transform infrared spectroscopy and differential scanning calorimetry.

The parabasal cells include the stratum granulousum and the stratum spinosum. In these two layers, cells from the lower basal layer transition from active metabolic activity to death (apoptosis). In these mid-layers of the epithelia, the cells begin to lose their mitochondria and other cell organelles. The multiple layers of parabasal cells are polyhedral in shape with prominent nuclei.

For example, {} is a pentagram; {} is a pentagon. A regular polyhedron that has q regular p-sided polygon faces around each vertex is represented by {p,q}. For example, the cube has 3 squares around each vertex and is represented by {4,3}. A regular 4-dimensional polytope, with r {p,q} regular polyhedral cells around each edge is represented by {p,q,r}.

Dual graphs have several roles in meshing. One can make a polyhedral Voronoi diagram mesh by dualizing a Delaunay triangulation simplicial mesh. One can create a cubical mesh by generating an arrangement of surfaces and dualizing the intersection graph; see spatial twist continuum. Sometimes both the primal mesh and its dual mesh are used in the same simulation; see Hodge star operator.

Jack R. Edmonds (born April 5, 1934) is an American-born and educated computer scientist and mathematician who lived and worked in Canada for much of his life. He has made fundamental contributions to the fields of combinatorial optimization, polyhedral combinatorics, discrete mathematics and the theory of computing. He was the recipient of the 1985 John von Neumann Theory Prize.

Polyhedral 'foam' by truncated octahedra In 1887, Lord Kelvin asked how space could be partitioned into cells of equal volume with the least area of surface between them, i.e., what was the most efficient bubble foam?. This problem has since been referred to as the Kelvin problem. He proposed a foam, based on the bitruncated cubic honeycomb, which is called the Kelvin structure.

Wu et al. apply the isolobal analogy to explore relationships involving structures, energies and magnetic properties between polyhedral boron carbonyls and their hydrocarbon relatives. As determined in this study, although isolobal, these two sets of molecules have significant differences in their strain energy. Goldman and Tyler used the isolobal analogy to determine the most likely mechanism for a deletion reaction.

A pentamer is an entity composed of five sub-units. In chemistry, it applies to molecules made of five monomers. In biochemistry, it applies to macromolecules, in particular to pentameric proteins, made of five proteic sub-units. In microbiology, a pentamer is one of the proteins composing the polyhedral protein shell that encloses the bacterial micro-compartments known as carboxysomes.

Francisco (Paco) Santos Leal (born May 28, 1968) is a Spanish mathematician at the University of Cantabria, known for finding a counterexample to the Hirsch conjecture in polyhedral combinatorics.. In 2015 he won the Fulkerson Prize for this research.2015 Fulkerson Prize citation, retrieved 2015-07-18. Santos francisco Santos was born in Valladolid, Spain.Curriculum vitae , retrieved 2015-07-18.

Martin Grötschel (born 10 September 1948) is a German mathematician known for his research on combinatorial optimization, polyhedral combinatorics, and operations research. From 1991 to 2012 he was Vice President of the Zuse Institute Berlin (ZIB) and served from 2012 to 2015 as ZIB's President. Since October 2015 he has been President of the Berlin-Brandenburg Academy of Sciences and Humanities (BBAW).

A combination puzzle collection A combination puzzle, also known as a sequential move puzzle, is a puzzle which consists of a set of pieces which can be manipulated into different combinations by a group of operations. Many such puzzles are of a polyhedral shape, and consist of multiple layers of pieces along each many axes which can rotate independently of each other.

These provide optimal visibility sets, where there is no image error and no redundancy. They are, however, complex to implement and typically run a lot slower than other PVS based visibility algorithms. Teller computed exact visibility for a scene subdivided into cells and portalsSeth Teller, Visibility Computations in Densely Occluded Polyhedral Environments (Ph.D. dissertation, Berkeley, 1992) (see also portal rendering).

Self-intersecting polyhedral Klein bottle with quadrilateral faces Some polyhedra have two distinct sides to their surface. For example, the inside and outside of a convex polyhedron paper model can each be given a different colour (although the inside colour will be hidden from view). These polyhedra are orientable. The same is true for non-convex polyhedra without self-crossings.

The frame of a large polyhedral dome was loosely woven by a robotic arm out of thin nylon threads, and suspended in an open room. The dome was designed with gaps where it would be warmest. Silkworms were released onto the frame in waves, where they added layers of silk before being removed. This involved engineering, sericulture, and modelling sun in the room.

Teller's parents are Joan Teller and Samuel H. Teller of Bolton, Connecticut; Samuel Teller is a senior judge in the Connecticut Superior Court in Rockville. Teller received his undergraduate degree from the Wesleyan University, and a Ph.D. from the University of California, Berkeley in 1992. His dissertation, Visibility Computations in Densely Occluded Polyhedral Environments, was supervised by Carlo H. Séquin.

The fantasy role- playing game Dungeons & Dragons (D&D;) is largely credited with popularizing dice in such games. Some games use only one type, like Exalted which uses only ten-sided dice. Others use numerous types for different game purposes, such as D&D;, which makes use of all common polyhedral dice. Dice are usually used to determine the outcome of events.

If m polyhedra condense to form a macropolyhedron, m core BMOs will be formed. Thus the skeletal electron pair (SEP) requirement of closo-condensed polyhedral clusters is m + n. Single-vertex sharing is a special case where each subcluster needs to satisfy Wade's rule separately. Let a and b be the number of vertices in the subclusters including the shared atom.

As such, the generalized Jemmis rule can be stated as follows: :The SEP requirement of condensed polyhedral clusters is m + n + o + p − q, where m is the number of subclusters, n is the number of vertices, o is the number of single-vertex shared condensations, p is the number of missing vertices and q is the number of caps.

The Barnette–Bosák–Lederberg graph has a similar construction to the Tutte graph but is composed of two Tutte fragments, connected through a pentagonal prism, instead of three connected through a tetrahedron. Without the constraint of having exactly three edges at every vertex, much smaller non-Hamiltonian polyhedral graphs are possible, including the Goldner–Harary graph and the Herschel graph.

Many polyhedra are also coloured such that no same-coloured faces touch each other along an edge or at a vertex. :For example, a 20-face icosahedron can use twenty colours, one colour, ten colours, or five colours, respectively. An alternative way for polyhedral compound models is to use a different colour for each polyhedron component. Net templates are then made.

Terminology about prisms and cylinders is identical. Thus, for example, since a truncated prism is a prism whose bases do not lie in parallel planes, a solid cylinder whose bases do not lie in parallel planes would be called a truncated cylinder. From a polyhedral viewpoint, a cylinder can also be seen as a dual of a bicone as an infinite-sided bipyramid.

FluoroPOSS (Fluorinated Polyhedral Oligomeric Silsesquioxanes) is a synthetic microfiber with very low surface energy, which makes it oil-repellent. Mixed with an ordinary polymer, it forms a material which can be applied to other materials such as metal, glass, plastic, plant fibers or leaves. The application process is called electrospinning. FluoroPOSS has been developed at the Massachusetts Institute of Technology (MIT).

For a proof, see . For a more general characterization of chordal planar graphs, and an efficient recognition algorithm for these graphs, see . The observation that every chordal polyhedral graph is maximal planar was stated explicitly by . In an Apollonian network, every maximal clique is a complete graph on four vertices, formed by choosing any vertex and its three earlier neighbors.

The Song is made from composites. Its polyhedral wing comes in two optional spans: (with flaperons) and (with ailerons and either spoilers or flaps). Standard engines available are the Bailey V5 four-stroke and the Verner JCV 360 four-stroke powerplant. Randall Fishman of Electric Aircraft Corporation produces an electric-powered version of the Song, the Electric Aircraft Corporation ElectraFlyer-ULS.

It is the second in an infinite series of uniform antiprismatic prisms. It is one of 18 uniform polyhedral prisms created by using uniform prisms to connect pairs of parallel Platonic solids and Archimedean solids. It is one of four four-dimensional Hanner polytopes; the other three are the tesseract, the 16-cell, and the dual of the octahedral prism (a cubic bipyramid).

A Hamiltonian path (but not cycle) in the Herschel graph As a bipartite graph that has an odd number of vertices, the Herschel graph does not contain a Hamiltonian cycle (a cycle of edges that passes through each vertex exactly once). For, in any bipartite graph, any cycle must alternate between the vertices on either side of the bipartition, and therefore must contain equal numbers of both types of vertex and must have an even length. Thus, a cycle passing once through each of the eleven vertices cannot exist in the Herschel graph. It is the smallest non-Hamiltonian polyhedral graph, whether the size of the graph is measured in terms of its number of vertices, edges, or faces.. There exist other polyhedral graphs with 11 vertices and no Hamiltonian cycles (notably the Goldner–Harary graph.) but none with fewer edges.

Boettcher cells are a special cell type located in the inner ear. Boettcher cells are polyhedral cells on the basilar membrane of the cochlea, and are located beneath Claudius cells. Boettcher cells are considered supporting cells for the organ of Corti, and are present only in the lower turn of the cochlea. These cells interweave with each other, and project microvilli into the intercellular space.

Parallel transport Polyhedral space a simplicial complex with a metric such that each simplex with induced metric is isometric to a simplex in Euclidean space. Principal curvature is the maximum and minimum normal curvatures at a point on a surface. Principal direction is the direction of the principal curvatures. Path isometry Proper metric space is a metric space in which every closed ball is compact.

The liquid that separates the flat faces between two polyhedral bubbles is called the lamellae; it is a continuous liquid phase. The areas where three lamellae meet are called plateau borders. When the bubbles in the foam are the same size the lamellae in the plateau borders meet at 120 degree angles. Since the lamella is slightly curved, the plateau region is at low pressure.

Commentators have interpreted the cage's compound shape as a reference to double and triple stars in astronomy, or to twinned crystals in crystallography. The image contrasts the celestial order of its polyhedral shapes with the more chaotic forms of biology. Prints of Stars belong to the permanent collections of major museums including the Rijksmuseum, the National Gallery of Art, and the National Gallery of Canada.

It is about 2.5 mm. in diameter and is irregularly oval in shape; several smaller nodules are found around or near the main mass. It consists of irregular masses of round or polyhedral cells epitheloid cells, which are grouped around a dilated sinusoidal capillary vessel. Each cell contains a large round or oval nucleus, the protoplasm surrounding which is clear, and is not stained by chromic salts.

Oregon State University Press, Corvallis. p.418 this complex has a Eurasiatic and Siberian appearance. These authors also note that small blades and polyhedral cores are absent from subsequent Paleoindian fluted-point assemblages in this region, reinforcing the technological distinctiveness of the Miller complex. The adjacent Krajacic Site is located about 10 miles southeast of Meadowcroft, and it is also important in defining the Miller complex.

Clathrin is a protein that plays a major role in the formation of coated vesicles. Clathrin was first isolated and named by Barbara Pearse in 1976. It forms a triskelion shape composed of three clathrin heavy chains and three light chains. When the triskelia interact they form a polyhedral lattice that surrounds the vesicle, hence the protein's name, which is derived from the Latin clathrum meaning lattice.

The primary problem in PVS computation then becomes: Compute the set of polygons that can be visible from anywhere inside each region of a set of polyhedral regions. There are various classifications of PVS algorithms with respect to the type of visibility set they compute.S. Nirenstein, E. Blake, and J. Gain. Exact from-region visibility culling, In Proceedings of the 13th workshop on Rendering, pages 191–202.

Not every planar graph corresponds to a convex polyhedron in this way: the trees do not, for example. Steinitz's theorem says that the polyhedral graphs formed from convex polyhedra are precisely the finite 3-connected simple planar graphs. More generally, Euler's formula applies to any polyhedron whose faces are simple polygons that form a surface topologically equivalent to a sphere, regardless of its convexity.

The aircraft was derived from the Airsport Sonata motor glider and designed to comply with the Fédération Aéronautique Internationale microlight rules. It features a cantilever low-wing, a T-tail, a two-seats-in-side-by-side configuration enclosed cockpit, fixed tricycle landing gear and a single engine in tractor configuration. The Sonet is made from composites. Its polyhedral wing comes in three optional spans: , and .

Dihedral angle is the upward angle from horizontal of the wings of a fixed-wing aircraft, or of any paired nominally-horizontal surfaces on any aircraft. The term can also apply to the wings of a bird. Dihedral angle is also used in some types of kites such as box kites. Wings with more than one angle change along the full span are said to be polyhedral.

The third of Hilbert's list of mathematical problems, presented in 1900, was the first to be solved. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second? Based on earlier writings by Gauss,Carl Friedrich Gauss: Werke, vol. 8, pp.

4-sided dice The Royal Game of Ur, dating from 2600 BC, was played with a set of tetrahedral dice. Especially in roleplaying, this solid is known as a 4-sided die, one of the more common polyhedral dice, with the number rolled appearing around the bottom or on the top vertex. Some Rubik's Cube-like puzzles are tetrahedral, such as the Pyraminx and Pyramorphix.

The corneal epithelium consists of several layers of cells. The cells of the deepest layer are columnar, known as basal cells which are attached by multiprotein complexes known as hemidesmosomes to an underlying basement membrane. Then follow two or three layers of polyhedral cells, commonly known as wing cells. The majority of these are prickle cells, similar to those found in the stratum mucosum of the cuticle.

Obsidian prismatic blade fragment, ChunchucmilThe chipped-stone assemblage of Chunchucmil is dominated by obsidian prismatic blades. The prismatic blade industry was ubiquitous throughout Mesoamerica and primarily used in the production of obsidian tools. Lithic analyses have determined that the majority of the blades at Chunchucmil were likely imported in finished form, as suggested by the general scarcity of polyhedral cores, production debitage, rejuvenation artifacts, and manufacturing errors at the site.Clark 1997.

It is NP-complete to test whether a graph is pancyclic, even for the special case of 3-connected cubic graphs, and it is also NP-complete to test whether a graph is node-pancyclic, even for the special case of polyhedral graphs., Theorems 2.3 and 2.4. It is also NP- complete to test whether the square of a graph is Hamiltonian, and therefore whether it is pancyclic.

Escher's solid. This image does not depict the stellation, because different visible parts of a single hexagonal face of the stellation have different colors. However, the coloring is consistent with a depiction of the polyhedral compound of three flattened octahedra. Escher's solid is topologically equivalent to the disdyakis dodecahedron, a Catalan solid, which can be seen as a rhombic dodecahedron with shorter rhombic pyramids augumented to each face.

In the work of Archimedes it already appears that the length of a circle can be approximated by the length of regular polyhedra inscribed or circumscribed in the circle. In general, for smooth or rectifiable curves their length can be defined as the supremum of the lengths of polygonal curves inscribed in them. The Schwarz lantern shows that surface area cannot be defined as the supremum of inscribed polyhedral surfaces.

They suggested that the "negatively curved" (or non-euclidean) nature of microscopic growth patterns of biological organisms is one of the key reasons why large-scale organisms do not look like crystals or polyhedral shapes but in fact in many cases resemble self-similar fractals. In particular they suggested that such "negatively curved" local structure is manifested in highly folded and highly connected nature of the brain and the lung tissue.

The polyhedral honeycomb associated with the Weaire–Phelan structure (obtained by flattening the faces and straightening the edges) is also referred to loosely as the Weaire–Phelan structure. It was known well before the Weaire–Phelan structure was discovered, but the application to the Kelvin problem was overlooked.A diagram can be found in , as shown on Ken Brakke's page. It is found in two related geometries of crystal structure in chemistry.

The configuration index is a single digit which is the priority number of the ligand trans to the ligand of lowest priority in the plane perpendicular to the 4 fold axis. (If there is more than one choice then the highest numerical value second digit is taken.) NB this procedure gives the same result as SP-4, however in this case the polyhedral symbol specifies that the complex is non- planar.

The smallest possible number of vertices for a non-hamiltonian polyhedral graph is 11. Therefore, the Goldner–Harary graph is a minimal example of graphs of this type. However, the Herschel graph, another non-Hamiltonian polyhedron with 11 vertices, has fewer edges. As a non-Hamiltonian maximal planar graph, the Goldner–Harary graph provides an example of a planar graph with book thickness greater than two.. See in particular Figure 9.

A model of the He P.1078 B variant The Heinkel P.1078 project had three quite different variants. All of them were a single-seat fighters with polyhedral swept wings. The wings were swept back at 40 degrees and included wood in their construction. All of the projected aircraft had the wing tips angled downwards and all of them would be powered by a single Heinkel HeS 011 turbojet.

Singularities of codimension 2 are of major importance; they are characterized by a single number, the conical angle. The singularities can also studied topologically. Then, for example, there are no topological singularities of codimension 2. In a 3-dimensional polyhedral space without a boundary (faces not glued to other faces) any point has a neighborhood homeomorphic either to an open ball or to a cone over the projective plane.

Two polytopes are called combinatorially isomorphic if their face lattices are isomorphic. The polytope graph (polytopal graph, graph of the polytope, 1-skeleton) is the set of vertices and edges of the polytope only, ignoring higher-dimensional faces. For instance, a polyhedral graph is the polytope graph of a three-dimensional polytope. By a result of Whitney the face lattice of a three-dimensional polytope is determined by its graph.

The mammalian Leydig cell is a polyhedral epithelioid cell with a single eccentrically located ovoid nucleus. The nucleus contains one to three prominent nucleoli and large amounts of dark- staining peripheral heterochromatin. The acidophilic cytoplasm usually contains numerous membrane-bound lipid droplets and large amounts of smooth endoplasmic reticulum (SER). Besides the obvious abundance of SER with scattered patches of rough endoplasmic reticulum, several mitochondria are also prominent within the cytoplasm.

St. Wenceslas’ Church is located on a rock terrace above the level of street. The construction is made of rubble stone with remains of Romanesque walls. The building includes a polyhedral chancel, a rectangular nave, a fléche and a Baroque sacristy in the south. The main nave is 14 meters high, the smaller presbytery is 13 meters high with the ending shaped as five sides of an octagon.

Like other role-playing games, Lejendary Adventure is played using polyhedral dice, pencils, paper, and sometimes miniatures. Unlike Dungeons & Dragons, Lejendary Adventure has a player character creation system that is skill-based, resulting in very flexible character creation to allow role-playing of almost any kind of character. Lejendary Adventures still provide Archetypes using "Orders." Orders are guild-like organizations in the world that provide benefits to the characters.

The age of metasediments is attributed to the Cambrian or Superior Precambrian. There are basic intrusive rocks that have been identified as dolerites and which occur in the form of veins 0.5 to 3m thick, with an orientation predominantly N-S and vertical inclination. They are dark grey, fine grain and micro-porphyritic, being altered in the contact with the mineralized veins. They present irregular fractures and polyhedral disjunction.

A 2-dimensional pyramid is a triangle, formed by a base edge connected to a noncolinear point called an apex. A 4-dimensional pyramid is called a polyhedral pyramid, constructed by a polyhedron in a 3-space hyperplane of 4-space with another point off that hyperplane. Higher-dimensional pyramids are constructed similarly. The family of simplices represent pyramids in any dimension, increasing from triangle, tetrahedron, 5-cell, 5-simplex, etc.

Electron- counting rules are used to predict the preferred electron count for molecules. The octet rule, the 18-electron rule, and Hückel's 4n + 2 pi-electron rule are proven to be useful in predicting the molecular stability. Wade's rules were formulated to explain the electronic requirement of monopolyhedral borane clusters. The Jemmis mno rules are an extension of Wade's rules, generalized to include condensed polyhedral boranes as well.

An extended version of the 3c–2e bond model features heavily in cluster compounds described by the polyhedral skeletal electron pair theory, such as boranes and carboranes. These molecules derive their stability from having a completely filled set of bonding molecular orbitals as outlined by Wade's rules. Resonance structures of 3c-2e bond in diborane. The monomer BH3 is unstable since the boron atom has an empty p-orbital.

Pulleyblank obtained his Ph.D. from the University of Waterloo in 1973; his thesis, supervised by Jack Edmonds, concerned perfect matching theory from the point of view of polyhedral combinatorics. He worked at IBM beginning in the late 1960s and then returned to academia in 1974,. as a professor at the University of Calgary, before moving back to Waterloo in 1982.William Pulleyblank, Waterloo Combinatorics & Optimization, retrieved 2014-12-10.

Lastly, once the structures were known it became clear that new theories of chemical bonding were needed to explain them. Lipscomb was awarded the Nobel prize in Chemistry in 1976 for his achievements in this field. The correct structure of diborane was predicted by H. Christopher Longuet-Higgins 5 years before its determination. Polyhedral skeletal electron pair theory (Wade's rules) can be used to predict the structures of boranes.

The virus polyhedrals comprise 12% of Gypchek with larvae body parts and other inert solids making up the remaining 88%. Gypchek's toxicological and pathogenicity testing revealed no effects on laboratory animals, wild mammals, birds, and fish at field doses. While it is non-toxic to warm-blooded animals, impurities may cause eye irritation. The appearance is listed as, "dried insect body parts and virus polyhedral" and has a musty odor.

In convex geometry, Gordan's lemma states that the semigroup of integral points in the dual cone of a rational convex polyhedral cone is finitely generated. In algebraic geometry, the prime spectrum of the semigroup algebra of such a semigroup is, by definition, an affine toric variety; thus, the lemma says an affine toric variety is indeed an algebraic variety. The lemma is named after the German mathematician Paul Gordan (1837–1912).

Of these graphs, the first four are planar graphs. They are the only four cubic polyhedral graphs (graphs of simple convex polyhedra) that are well-covered. Four of the graphs (the two prisms, the Dürer graph, and ) are generalized Petersen graphs. The 1- and 2-connected cubic well-covered graphs are all formed by replacing the nodes of a path or cycle by three fragments of graphs which labels , , and .

The circle packing theorem implies that every polyhedral graph can be represented as the graph of a polyhedron that has a midsphere. A stronger form of the circle packing theorem asserts that any polyhedral graph and its dual graph can be represented by two circle packings, such that the two tangent circles representing a primal graph edge and the two tangent circles representing the dual of the same edge always have their tangencies at right angles to each other at the same point of the plane. A packing of this type can be used to construct a convex polyhedron that represents the given graph and that has a midsphere, a sphere tangent to all of the edges of the polyhedron. Conversely, if a polyhedron has a midsphere, then the circles formed by the intersections of the sphere with the polyhedron faces and the circles formed by the horizons on the sphere as viewed from each polyhedron vertex form a dual packing of this type.

Coxeter explained in 1937 how he and Petrie began to expand the classical subject of regular polyhedra: :One day in 1926, J. F. Petrie told me with much excitement that he had discovered two new regular polyhedral; infinite but free of false vertices. When my incredulity had begun to subside, he described them to me: one consisting of squares, six at each vertex, and one consisting of hexagons, four at each vertex.H.S.M. Coxeter (1937) "Regular skew polyhedral in three and four dimensions and their topological analogues", Proceedings of the London Mathematical Society (2) 43: 33 to 62 In 1938 Petrie collaborated with Coxeter, Patrick du Val, and H.T. Flather to produce The Fifty-Nine Icosahedra for publication.H. S. M. Coxeter, Patrick du Val, H.T. Flather, J.F. Petrie (1938) The Fifty-nine Icosahedra, University of Toronto studies, mathematical series 6: 1–26 Realizing the geometric facility of the skew polygons used by Petrie, Coxeter named them after his friend when he wrote Regular Polytopes.

In reality, there are only dependencies from the first N/2 iterations into the last N/2, so we can execute this loop as a sequence of two fully parallel loops (from 0...N/2 and from N/2+1...N). The characterization of this dependence, the analysis of parallelism, and the transformation of the code can be done in terms of the instance-wise information provided by any polyhedral framework. Instance-wise analysis and transformation allows the polyhedral model to unify additional transformations (such as index set splitting, loop peeling, tiling, loop fusion or fission, and transformation of imperfectly nested loops) with those already unified by the unimodular framework (such as loop interchange, skewing, and reversal of perfectly nested loops). It has also stimulated the development of new transformations, such as Pugh and Rosser's iteration-space slicing (an instance-wise version of program slicing; note that the code was never released with the Omega Library).

The Cartan–Hadamard theorem provides an example of a local-to-global correspondence in Riemannian and metric geometry: namely, a local condition (non-positive curvature) and a global condition (simple-connectedness) together imply a strong global property (contractibility); or in the Riemannian case, diffeomorphism with Rn. The metric form of the theorem demonstrates that a non-positively curved polyhedral cell complex is aspherical. This fact is of crucial importance for modern geometric group theory.

Gil Kalai received his Ph.D. from Hebrew University in 1983, under the supervision of Micha Perles,. and joined the Hebrew University faculty in 1985 after a postdoctoral fellowship at the Massachusetts Institute of Technology.Profile at the Technical University of Eindhoven as an instructor of a minicourse on polyhedral combinatorics. He was the recipient of the Pólya Prize in 1992, the Erdős Prize of the Israel Mathematical Society in 1993, and the Fulkerson Prize in 1994.

The bacteria Enterobacter aerogenes, Bacillus thuringiensis, Pseudomonas aeruginosa and Serratia marcescens are identified as causing mortality to the teak defoliator. A synnematous fungus of the genus Hirsutella is found to be pathogenic to this pest. An absolutely specific virus with refractile polyhedral inclusion bodies, staining blue in Giemsa and thick blue in Buffalo Black, named as Hyblaea puera nucleopolyhedrovirus (HpNPV) is found to be very effective in the biological control of this pest.

Such lines of intersection are not part of the polyhedral structure and are sometimes called false edges. Likewise where three such lines intersect at a point that is not a corner of any face, these points are false vertices. The images below show spheres at the true vertices, and blue rods along the true edges. For example, the small stellated dodecahedron has 12 pentagram faces with the central pentagonal part hidden inside the solid.

Naturally occurring (crystal) formations of tetrahexahedra are observed in copper and fluorite systems. Polyhedral dice shaped like the tetrakis hexahedron are occasionally used by gamers. A 24-cell viewed under a vertex-first perspective projection has a surface topology of a tetrakis hexahedron and the geometric proportions of the rhombic dodecahedron, with the rhombic faces divided into two triangles. The tetrakis hexahedron appears as one of the simplest examples in building theory.

In the mathematical field of graph theory, an antiprism graph is a graph that has one of the antiprisms as its skeleton. An n-sided antiprism has 2n vertices and 4n edges. They are regular, polyhedral (and therefore by necessity also 3-vertex-connected, vertex-transitive, and planar graphs), and also Hamiltonian graphs.Read, R. C. and Wilson, R. J. An Atlas of Graphs, Oxford, England: Oxford University Press, 2004 reprint, Chapter 6 special graphs pp.

Polymake is software for the algorithmic treatment of convex polyhedra.Official Website Albeit primarily a tool to study the combinatorics and the geometry of convex polytopes and polyhedra, it is by now also capable of dealing with simplicial complexes, matroids, polyhedral fans, graphs, tropical objects, toric varieties and other objects. Polymake has been cited in over 100 recent articles indexed by Zentralblatt MATH as can be seen from its entry in the swMATH database.

Professor Alexander Stewart Herschel, DCL, FRS (5 February 1836 – 18 June 1907) was a British astronomer. Although much less well known than his grandfather William Herschel or his father John Herschel, he did pioneering work in meteor spectroscopy. He also worked on identifying comets as the source of meteor showers. The Herschel graph, the smallest non-Hamiltonian polyhedral graph, is named after Herschel due to his pioneering work on Hamilton's Icosian game.

In the late 1960s several investigations studied polyhedral magnetic fields as a possibility to confine a fusion plasma. The first proposal to combine this configuration with an electrostatic potential well in order to improve electron confinement was made by Oleg Lavrentiev in 1975. The idea was picked up by Robert Bussard in 1983. His 1989 patent application cited Lavrentiev, although in 2006 he appears to claim to have (re)discovered the idea independently.

In mathematics, the upper bound theorem states that cyclic polytopes have the largest possible number of faces among all convex polytopes with a given dimension and number of vertices. It is one of the central results of polyhedral combinatorics. Originally known as the upper bound conjecture, this statement was formulated by Theodore Motzkin, proved in 1970 by Peter McMullen, and strengthened from polytopes to subdivisions of a sphere in 1975 by Richard P. Stanley.

The sodium atom is located within the P-O and Al-O polyhedral in an irregular cavity. The coordination of the sodium is best described as the uncommon seven- coordination. The presence of a hydrogen ion in the same cavity where a sodium ion is causes a repulsion between the two, forcing sodium to one side of the cavity so that is it more coordinated with oxygen than its other side. Gatehouse et al.

The Fighter's Player Pack is an AD&D; game accessory which comes in a case with everything a newcomer needs to get his fighter PC ready for a campaign: a pad of character sheets, a stand-up reference screen, a brief but informative player's guide, seven polyhedral dice, three pewter miniatures, and a shiny red pencil. The case is designed so that a copy of the Player's Handbook can fit snugly inside the lid.

The Thief's Player Pack is an AD&D; game accessory which comes in a case with everything a newcomer needs to get his thief PC ready for a campaign: a pad of character sheets, a stand-up reference screen, a brief but informative player's guide, seven polyhedral dice, three pewter miniatures, and a shiny red pencil. The case is designed so that a copy of the Player's Handbook can fit snugly inside the lid.

In the mathematical field of graph theory, the Tutte graph is a 3-regular graph with 46 vertices and 69 edges named after W. T. Tutte. It has chromatic number 3, chromatic index 3, girth 4 and diameter 8. The Tutte graph is a cubic polyhedral graph, but is non-hamiltonian. Therefore, it is a counterexample to Tait's conjecture that every 3-regular polyhedron has a Hamiltonian cycle.. Reprinted in Scientific Papers, Vol.

They suggested that the "negatively curved" (or non-euclidean) nature of microscopic growth patterns of biological organisms is one of the key reasons why large-scale organisms do not look like crystals or polyhedral shapes but in fact in many cases resemble self-similar fractals. In particular they suggested (see section 3.4 of ) that such "negatively curved" local structure is manifested in highly folded and highly connected nature of the brain and the lung tissue.

In terms of group theory, if G is the symmetry group of a polyhedral compound, and the group acts transitively on the polyhedra (so that each polyhedron can be sent to any of the others, as in uniform compounds), then if H is the stabilizer of a single chosen polyhedron, the polyhedra can be identified with the orbit space G/H – the coset gH corresponds to which polyhedron g sends the chosen polyhedron to.

It has a single hall divided into three naves, with seven pairs of polyhedral pillars. Animals rested in the narrow aisles to the left and right of the main hall. Between the pillars were stone troughs for the animals, and in the corner of one of the halls was a pool of water. Travelers slept in a separate room built at the end of the narrow aisles on the western side of the caravanserai.

The most recent versions of the game's rules are detailed in three core rulebooks: The Player's Handbook, the Dungeon Master's Guide and the Monster Manual. The only items required to play the game are the rulebooks, a character sheet for each player, and a number of polyhedral dice. Many players also use miniature figures on a grid map as a visual aid, particularly during combat. Some editions of the game presume such usage.

In geometry, a skew apeirohedron is an infinite skew polyhedron consisting of nonplanar faces or nonplanar vertex figures, allowing the figure to extend indefinitely without folding round to form a closed surface. Skew apeirohedra have also been called polyhedral sponges. Many are directly related to a convex uniform honeycomb, being the polygonal surface of a honeycomb with some of the cells removed. Characteristically, an infinite skew polyhedron divides 3-dimensional space into two halves.

Supplemental materials appearing in the boxed set included geomorphs, monster and treasure lists, and a set of polyhedral dice. For a period in 1979, TSR experienced a dice shortage. Basic sets published during this time frame came with two sheets of numbered cutout cardstock chits that functioned in lieu of dice, along with a coupon for ordering dice from TSR. The rulebook also included a brief sample dungeon with a full-page map.

In the mathematical field of graph theory, the Frucht graph is a 3-regular graph with 12 vertices, 18 edges, and no nontrivial symmetries. It was first described by Robert Frucht in 1939. The Frucht graph is a pancyclic Halin graph with chromatic number 3, chromatic index 3, radius 3, and diameter 4. As with every Halin graph, the Frucht graph is polyhedral (planar and 3-vertex- connected) and Hamiltonian, with girth 3.

One example is the pentaphosphide Rh4(CO)5(PPh2)5− where all Rh---Rh edges are bridged by PPh2. A carbide-containing butterfly cluster is [Fe4C(CO)12]2− where the carbide is bonded to all four Fe centers. Bonding in such clusters is often discussed in the context of polyhedral skeletal electron pair theory. This theory predicts that tetrametallic clusters with 60 valence electrons will adopt tetrahedral geometry with six M-M bonds.

The Herschel graph is the smallest possible polyhedral graph that does not have a Hamiltonian cycle. A possible Hamiltonian path is shown. Any Hamiltonian cycle can be converted to a Hamiltonian path by removing one of its edges, but a Hamiltonian path can be extended to Hamiltonian cycle only if its endpoints are adjacent. All Hamiltonian graphs are biconnected, but a biconnected graph need not be Hamiltonian (see, for example, the Petersen graph).

Suppose that N is a finite-rank free abelian group. A strongly convex rational polyhedral cone in N is a convex cone (of the real vector space of N) with apex at the origin, generated by a finite number of vectors of N, that contains no line through the origin. These will be called "cones" for short. For each cone σ its affine toric variety Uσ is the spectrum of the semigroup algebra of the dual cone.

After Israyelian's death, the construction works were taken over and continued by the co-author, architect Ardzroun Galikian. From 1971-76, the interior look of the church was significantly improved. On the eastern part of the church, a gallery was added for the church choir. As a result of such additions, it was found necessary to remove the old dome and the old drum and to replace them by a much higher dome with polyhedral fan-shaped spire.

In blocky structure, the structural units are blocklike or polyhedral. They are bounded by flat or slightly rounded surfaces that are casts of the faces of surrounding peds. Typically, blocky structural units are nearly equidimensional but grade to prisms and to plates. The structure is described as angular blocky if the faces intersect at relatively sharp angles; as subangular blocky if the faces are a mixture of rounded and plane faces and the corners are mostly rounded.

In mathematics, the Schwarz lantern (also known as Schwarz's boot, after mathematician Hermann Schwarz) is a pathological example of the difficulty of defining the area of a smooth (curved) surface as the limit of the areas of polyhedra. The curved surface in question is a portion of a right circular cylinder. The discrete polyhedral approximation considered has 2n axial "slices". 2m vertices are placed radially along each slice at a circumferential distance of \pi / m from each other.

This site yielded a great variety of the distinctive Meadowcroft-style blade implements and several small, cylindrical polyhedral cores. At Cactus Hill in Virginia, similar points have been found, where they are dubbed as the Early Triangular type. Some similar finds were made at the Page-Ladson site in Florida as well. Because of the very long occupational sequence at Meadowcroft, it became a very important site, and is seen as quite valuable for comparative analysis.

A “cuboct” cubic lattice of vertex connected octahedrons, similar to the perovskite mineral structure provides a regular polyhedral unit cell that satisfies Maxwell’s rigidity criterion and has a coordination number z of eight. The dependence of the relative density on the coordination number is small relative to the dependence on strut diameter. Winding the reinforcing fibers around the connection holes optimizes their load bearing capacity, while coupling them to struts which themselves retain uniaxial fiber orientation.

Physically, the Bit was represented within the movie by a blue polyhedral shape that alternated between the compound of dodecahedron and icosahedron and the small triambic icosahedron (the first stellation of the icosahedron). When the Bit says the answer "yes", it briefly changes into a yellow octahedron and when it says "no" it changes into a red 35th stellation of an icosahedron; these resemble prismatic forms or "3-D versions" of the Latin letters 'O' and 'X', respectively.

A critical aspect of this work was the development of methods of brain flattening. The first fully accurate method of cortical flattening was developed by Schwartz in 1986, based on the computation of exact minimal geodesic distances on a polyhedral mesh representing the cortical surface , together with metric multidimensional scaling . Variants of this algorithm, especially the recent improvements contributed in the thesis work of Mukund Balasubramanian (see ) underlie most current quantitatively accurate approaches to cortical flattening.

Furthermore it follows that any TU matrix has only 0, +1 or −1 entries. The opposite is not true, i.e., a matrix with only 0, +1 or −1 entries is not necessarily unimodular. A matrix A is TU if and only if AT is TU. Totally unimodular matrices are extremely important in polyhedral combinatorics and combinatorial optimization since they give a quick way to verify that a linear program is integral (has an integral optimum, when any optimum exists).

These involve the use of dice, usually as randomisers. Most dice used in games are the standard cubical dice numbered from 1 to 6, though games with polyhedral dice or those marked with symbols other than numbers exist. The most common use of dice is to randomly determine the outcome of an interaction in a game. An example is a player rolling a die or dice to determine how many board spaces to move a game token.

A Halin graph is a graph formed from an undirected plane tree (with no degree- two nodes) by connecting its leaves into a cycle, in the order given by the plane embedding of the tree. Equivalently, it is a polyhedral graph in which one face is adjacent to all the others. Every Halin graph is planar. Like outerplanar graphs, Halin graphs have low treewidth, making many algorithmic problems on them more easily solved than in unrestricted planar graphs..

The most obvious family of prismatic 4-polytopes is the polyhedral prisms, i.e. products of a polyhedron with a line segment. The cells of such a 4-polytopes are two identical uniform polyhedra lying in parallel hyperplanes (the base cells) and a layer of prisms joining them (the lateral cells). This family includes prisms for the 75 nonprismatic uniform polyhedra (of which 18 are convex; one of these, the cube-prism, is listed above as the tesseract).

The Borromean rings are a hyperbolic link: the complement of the Borromean rings in the 3-sphere admits a complete hyperbolic metric of finite volume. The canonical (Epstein–Penner) polyhedral decomposition of the complement consists of two ideal regular octahedra. The volume is 16\Lambda(\pi/4) \approx 7.32772\dots where \Lambda is the Lobachevsky function. This was a central example in the video Not Knot about knot complements, produced in 1991 by the Geometry Center.

The Bidiakis cube is a cubic Hamiltonian graph and can be defined by the LCF notation [-6,4,-4]4. The Bidiakis cube can also be constructed from a cube by adding edges across the top and bottom faces which connect the centres of opposite sides of the faces. The two additional edges need to be perpendicular to each other. With this construction, the Bidiakis cube is a polyhedral graph, and can be realized as a convex polyhedron.

The definition of a cone may be extended to higher dimensions (see convex cones). In this case, one says that a convex set C in the real vector space Rn is a cone (with apex at the origin) if for every vector x in C and every nonnegative real number a, the vector ax is in C. In this context, the analogues of circular cones are not usually special; in fact one is often interested in polyhedral cones.

This scheme is essentially a list of spatial cells occupied by the solid. The cells, also called voxels are cubes of a fixed size and are arranged in a fixed spatial grid (other polyhedral arrangements are also possible but cubes are the simplest). Each cell may be represented by the coordinates of a single point, such as the cell's centroid. Usually a specific scanning order is imposed and the corresponding ordered set of coordinates is called a spatial array.

The most obvious family of prismatic 4-polytopes is the polyhedral prisms, i.e. products of a polyhedron with a line segment. The cells of such a 4-polytope are two identical uniform polyhedra lying in parallel hyperplanes (the base cells) and a layer of prisms joining them (the lateral cells). This family includes prisms for the 75 nonprismatic uniform polyhedra (of which 18 are convex; one of these, the cube-prism, is listed above as the tesseract).

In topology, a surface is generally defined as a manifold of dimension two. This means that a topological surface is a topological space such that every point has a neighborhood that is homeomorphic to an open subset of a Euclidean plane. Every topological surface is homeomorphic to a polyhedral surface such that all facets are triangles. The combinatorial study of such arrangements of triangles (or, more generally, of higher-dimensional simplexes) is the starting object of algebraic topology.

In the 19th century, it was enlarged and castellated, serpentine bays added to the canal, and an unusual polyhedral sundial given a place of pride on a sunken lawn. Other additions were a gothic porch bearing the Aylward crest and a conservatory. The stable-yard and the castellated entrance to the demesne are attributed to Daniel Robertson. The interior preserves much of its 18th-century character and features including a Georgian staircase, Gothic plasterwork, and a Victorian drawing-room.

A Δ-Y transform can be performed by removing a triangular face from a polyhedron and extending its neighboring faces until the point where they meet, but only when that triple intersection point of the three neighboring faces is on the correct side of the polyhedron; when the triple intersection point is not on the correct side, a projective transformation of the polyhedron suffices to move it to the correct side. Therefore, by induction on the number of Δ-Y and Y-Δ transforms needed to reduce a given graph to K4, every polyhedral graph can be realized as a polyhedron. A later work by Epifanov strengthened Steinitz's proof that every polyhedral graph can be reduced to K4 by Δ-Y and Y-Δ transforms. Epifanov proved that if two vertices are specified in a planar graph, then the graph can be reduced to a single edge between those terminals by combining Δ-Y and Y-Δ transforms with series-parallel reductions.. Epifanov's proof was complicated and non-constructive, but it was simplified by Truemper using methods based on graph minors.

Truemper observed that every grid graph is reducible by Δ-Y and Y-Δ transforms in this way, that this reducibility is preserved by graph minors, and that every planar graph is a minor of a grid graph.. This idea can be used to replace Steinitz's lemma that a reduction sequence exists, in a proof of Steinitz's theorem using induction in the same way. However, there exist graphs that require a nonlinear number of steps in any sequence of Δ-Y and Y-Δ transforms. More precisely, Ω(n3/2) steps are sometimes necessary, and the best known upper bound on the number of steps is even worse, O(n2).. An alternative form of induction proof is based on removing edges (and compressing out the degree-two vertices that might be performed by this removal) or contracting edges and forming a minor of the given planar graph. Any polyhedral graph can be reduced to K4 by a linear number of these operations, and again the operations can be reversed and the reversed operations performed geometrically, giving a polyhedral realization of the graph.

The Szilassi polyhedron, a non-convex polyhedral realization of the Heawood graph with the topology of a torus In any dimension higher than three, the algorithmic Steinitz problem (given a lattice, determine whether it is the face lattice of a convex polytope) is complete for the existential theory of the reals by Richter-Gebert's universality theorem. However, because a given graph may correspond to more than one face lattice, it is difficult to extend this completeness result to the problem of recognizing the graphs of 4-polytopes, and this problem's complexity remains open. Researchers have also found graph-theoretic characterizations of the graphs of certain special classes of three-dimensional non-convex polyhedra.. and four-dimensional convex polytopes.... However, in both cases, the general problem remains unsolved. Indeed, even the problem of determining which complete graphs are the graphs of non-convex polyhedra (other than K4 for the tetrahedron and K7 for the Császár polyhedron) remains unsolved.. László Lovász has shown a correspondence between polyhedral representations of graphs and matrices realizing the Colin de Verdière graph invariants of the same graphs..

In the mathematical fields of numerical analysis and approximation theory, box splines are piecewise polynomial functions of several variables. Box splines are considered as a multivariate generalization of basis splines (B-splines) and are generally used for multivariate approximation/interpolation. Geometrically, a box spline is the shadow (X-ray) of a hypercube projected down to a lower-dimensional space. Box splines and simplex splines are well studied special cases of polyhedral splines which are defined as shadows of general polytopes.

In the granular structure, the structural units are approximately spherical or polyhedral and are bounded by curved or very irregular faces that are not casts of adjoining peds. In other words, they look like cookie crumbs. Granular structure is common in the surface soils of rich grasslands and highly amended garden soils with high organic matter content. Soil mineral particles are both separated and bridged by organic matter breakdown products, and soil biota exudates, making the soil easy to work.

Afwillite is composed of double chains that consist of calcium and silicon polyhedral connected to each other by sharing corners and edges. This causes continuous sheets to form parallel to its miller index [-101] faces. The sheets are bonded together by hydrogen bonds and are all connected by Ca-Si-O bonds (Malik and Jeffery, 1976). Each calcium atom is in 6-fold octahedral coordination with the oxygen, and the silicon is in 4-fold tetrahedral coordination around the oxygen.

In geometry, the first stellation of the rhombic dodecahedron is a self- intersecting polyhedron with 12 faces, each of which is a non-convex hexagon. It is a stellation of the rhombic dodecahedron and has the same outer shell and the same visual appearance as two other shapes: a solid, Escher's solid, with 48 triangular faces, and a polyhedral compound of three flattened octahedra with 24 overlapping triangular faces. Escher's solid can tessellate space to form the stellated rhombic dodecahedral honeycomb.

Stamp of the former German Democratic Republic honoring Euler on the 200th anniversary of his death. Across the centre it shows his polyhedral formula, in English written as "v − e + f = 2". Concerned about the continuing turmoil in Russia, Euler left St. Petersburg on 19 June 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Prussia. He lived for 25 years in Berlin, where he wrote over 380 articles.

E Marchand, P. Bouthemy, F Chaumette, and V. Moreau. Robust visual tracking by coupling 2d and 3d pose estimation. In Proceedings of IEEE International Conference on Image Processing, 1999. used an affine motion model from the image motion in addition to a rough polyhedral CAD model to extract the object pose with respect to the camera to be able to servo onto the object (on the lines of PBVS). 2-1/2-D visual servoing developed by Malis et al.

Specific examples of this are ferritin, capsid, and the tobacco mosaic virus, which are formed by the self-assembly of protein subunits into a polyhedral symmetry. Nonbiological polyhedra formed with metal ions and organic linkers are metal based macromolecular cages that have nanocavities with multiple openings or pores that allow small molecules to permeate and pass through. MOPs have been used to encapsulate a number of guests through various host-guest interactions (e.g. electrostatic interactions, hydrogen bonding, and steric interactions).

He is known for his work in polyhedral combinatorics and discrete geometry, and in particular for proving what was then called the upper bound conjecture and now is the upper bound theorem. This result states that cyclic polytopes have the maximum possible number of faces among all polytopes with a given dimension and number of vertices. McMullen also formulated the g-conjecture, later the g-theorem of Louis Billera, Carl W. Lee, and Richard P. Stanley, characterizing the f-vectors of simplicial spheres..

This is indicated by a lack of production debitage, including polyhedral cores, decortical flakes, and large percussion flakes, among rural occupations. Obsidian was generally transported, where applicable, along coastal trade routes. Of primary importance is the circum- peninsular trade route that linked the southeast Maya area to the Gulf coast of Mexico. Examples of evidence of this include the higher quantities of obsidian found among coastal sites, such as small island occupations off the coast of Belize, then at sites located in-land.

When applying right rudder in an aircraft with dihedral the left hand wing will have increased angle of attack and the right hand wing will have decreased angle of attack which will result in a roll to the right. An aircraft with anhedral will show the opposite effect. This effect of the rudder is commonly used in model aircraft where if sufficient dihedral or polyhedral is included in the wing design, primary roll control such as ailerons may be omitted altogether.

In Hilbert's third problem, he posed the question of whether two polyhedra of equal volumes can always be cut into polyhedral pieces and reassembled into each other. Hilbert's student Max Dehn, in his 1900 habilitation thesis, invented the Dehn invariant in order to prove that this is not always possible, providing a negative solution to Hilbert's problem. Although Dehn formulated his invariant differently, the modern approach is to describe it as a value in a tensor product, following .. See in particular p. 61..

Because very small changes in its edge lengths can cause much bigger changes in its angles, physical models of the polyhedron appear to be flexible. As with the simpler Schönhardt polyhedron, the interior of Jessen's icosahedron cannot be triangulated into tetrahedra without adding new vertices. However, because it has Dehn invariant equal to zero, it is scissors-congruent to a cube, meaning that it can be sliced into smaller polyhedral pieces that can be rearranged to form a solid cube.

It follows from this fact that (up to combinatorial equivalence, the choice of the outer face, and the orientation of the plane) every polyhedral graph has a unique planar embedding.See the remarks following Theorem 2.8 in . As Tutte observes, this was already known to . In planar graphs, the cycle space is generated by the faces, but in non-planar graphs peripheral cycles play a similar role: for every 3-vertex-connected finite graph, the cycle space is generated by the peripheral cycles.

In graph theory, the Golomb graph is a polyhedral graph with 10 vertices and 18 edges. It is named after Solomon W. Golomb, who constructed it (with a non- planar embedding) as a unit distance graph that requires four colors in any graph coloring. Thus, like the simpler Moser spindle, it provides a lower bound for the Hadwiger–Nelson problem: coloring the points of the Euclidean plane so that each unit line segment has differently-colored endpoints requires at least four colors.

The Kennedy group also focuses on the novel structures obtained through the introduction of metals and heteroatoms into polyhedral boranes: these "disobedient skeletons", some described as "iso-nido" or "iso-closo", do not follow Wade's rules. These boron clusters may act as flexible scaffolds for catalytically active metals. Professor Kennedy is a leading authority on the reaction chemistry of metallaboranes. He has also been involved with the synthesis of relatively inert monocarborane derivatives as non-coordinating counter-anions for many industrial applications.

The [B12H12]2− anion's B12 core is a regular icosahedron. The [B12H12]2− as a whole also has icosahedral molecular symmetry, and it belongs to the molecular point group Ih. Its icosahedral shape is consistent with the classification of this cage as "closo" in polyhedral skeletal electron pair theory. Crystals of Cs2B12H12 feature Cs+ ions in contact with twelve hydrides provided by four B12H122−. The B-B bond distances are 178 pm, and the B-H distances are 112 pm.

The triakis tetrahedron, a polyhedral realization of an 8-vertex Apollonian network Apollonian networks are planar 3-connected graphs and therefore, by Steinitz's theorem, can always be represented as the graphs of convex polyhedra. The convex polyhedron representing an Apollonian network is a 3-dimensional stacked polytope. Such a polytope can be obtained from a tetrahedron by repeatedly gluing additional tetrahedra one at a time onto its triangular faces. Therefore, Apollonian networks may also be defined as the graphs of stacked 3d polytopes.

The Hoabinhian level (later hunter–gatherers) contains split pebble tools and abundant faunal remains that dates between 11.000 and 5000 years BP. The team discovered "a large stone featuring what appear to be etchings in the shape of an arrow, dyed with a redocher color...It could be the first case of art in Cambodia"[sic]. The team uncovered rudimentary stone tools (chert flakes and polyhedral, multiplatform cores) in the deepest Palaeolithic levels from as far back as 71.000 years BP.

Convex hulls have wide applications in many fields. Within mathematics, convex hulls are used to study polynomials, matrix eigenvalues, and unitary elements, and several theorems in discrete geometry involve convex hulls. They are used in robust statistics as the outermost contour of Tukey depth, are part of the bagplot visualization of two-dimensional data, and define risk sets of randomized decision rules. Convex hulls of indicator vectors of solutions to combinatorial problems are central to combinatorial optimization and polyhedral combinatorics.

In combinatorial optimization and polyhedral combinatorics, central objects of study are the convex hulls of indicator vectors of solutions to a combinatorial problem. If the facets of these polytopes can be found, describing the polytopes as intersections of halfspaces, then algorithms based on linear programming can be used to find optimal solutions.; see especially remarks following Theorem 2.9. In multi- objective optimization, a different type of convex hull is also used, the convex hull of the weight vectors of solutions.

Geoffrey Colin Shephard is a mathematician who works on convex geometry and reflection groups. He asked Shephard's problem on the volumes of projected convex bodies, posed another problem on polyhedral nets, proved the Shephard–Todd theorem in invariant theory of finite groups, began the study of complex polytopes, and classified the complex reflection groups. Shephard earned his Ph.D. in 1954 from Queen's College, Cambridge, under the supervision of J. A. Todd. He was a professor of mathematics at the University of East Anglia until his retirement.

A blade is defined as a flake with parallel or subparallel margins that is usually at least twice as long as it is wide. There are numerous specialized types of blade flakes. Channel flakes are characteristic flakes caused by the fluting of certain Paleo-Indian projectile points; such fluting produced grooves in the projectile points which may have facilitated hafting. Prismatic blades are long, narrow specialized blades with parallel margins which may be removed from polyhedral blade cores, another common lithic feature of Paleo-Indian lithic culture.

In combinatorial mathematics, an Eulerian poset is a graded poset in which every nontrivial interval has the same number of elements of even rank as of odd rank. An Eulerian poset which is a lattice is an Eulerian lattice. These objects are named after Leonhard Euler. Eulerian lattices generalize face lattices of convex polytopes and much recent research has been devoted to extending known results from polyhedral combinatorics, such as various restrictions on f-vectors of convex simplicial polytopes, to this more general setting.

A cycle double cover of the Petersen graph, corresponding to its embedding on the projective plane as a hemi-dodecahedron. In graph-theoretic mathematics, a cycle double cover is a collection of cycles in an undirected graph that together include each edge of the graph exactly twice. For instance, for any polyhedral graph, the faces of a convex polyhedron that represents the graph provide a double cover of the graph: each edge belongs to exactly two faces. It is an unsolved problem, posed by George Szekeres.

Rigid unit modes (RUMs) represent a class of lattice vibrations or phonons that exist in network materials such as quartz, cristobalite or zirconium tungstate. Network materials can be described as three-dimensional networks of polyhedral groups of atoms such as SiO4 tetrahedra or TiO6 octahedra. A RUM is a lattice vibration in which the polyhedra are able to move, by translation and/or rotation, without distorting. RUMs in crystalline materials are the counterparts of floppy modes in glasses, as introduced by Jim Phillips and Mike Thorpe.

For an unbounded polytope (sometimes called: polyhedron), the H-desciption is still valid, but the V-description should be extended. Theodore Motzkin (1936) proved that any unbounded polytope can be represented as a sum of a bounded polytope and a convex polyhedral cone. In other words, every vector in an unbounded polytope is a convex sum of its vertices (its "defining points"), plus a conical sum of the Euclidean vectors of its infinite edges (its "defining rays"). This is called the finite basis theorem.

The success of the drug therapy of the virus and Parkinson's diseases using abiotic polyhedral adamantane molecules suggests the pharmaceutical potential of other xenobiotics with rigid three-dimensional molecular structure. For example, the functionalized fullerenes have been used for drug therapy of the HIV and other virus diseases: their hydrophobic ball-like molecules “block” a virus active site. This site can also be a target for other xenobiotics, which are geometrically similar to those of the functionalized fullerenes and are complementary to the HIV protease active site,.

Kawai and K. Fujita, Ed.), World Scientific, 2002, pp. 215-237. for a mathematical aspect of topological density which is closely related to the large deviation property of simple random walks. Another invariant arises from the relationship between tessellations and Euclidean graphs. If we regard a tessellation as an assembly of (possibly polyhedral) solid regions, (possibly polygonal) faces, (possibly linear) curves, and vertices – that is, as a CW-complex – then the curves and vertices form a Euclidean graph (or 1-skeleton) of the tessellation.

Brückner's photo of the final stellation of the icosahedron, a stellated polyhedron first studied by Brückner Photo of polyhedra models by Brückner. Johannes Max Brückner (5 August 1860 – 1 November 1934) was a German geometer, known for his collection of polyhedral models. Brückner was born on August 5, 1860 in Hartau, in the Kingdom of Saxony, a town that is now part of Zittau, Germany. He completed a Ph.D. at Leipzig University in 1886, supervised by Felix Klein and Wilhelm Scheibner, with a dissertation concerning conformal maps.

The graph of an Icosidodecahedron, an example for which the conjecture is true. In mathematical programming and polyhedral combinatorics, the Hirsch conjecture is the statement that the edge-vertex graph of an n-facet polytope in d-dimensional Euclidean space has diameter no more than n − d. That is, any two vertices of the polytope must be connected to each other by a path of length at most n − d. The conjecture was first put forth in a letter by to George B. Dantzig in 1957, pp.

Portrait of Nicolaus Kratzer is a 1528 half-length oil on canvas portrait by Hans Holbein the Younger. It is now in the Louvre, whilst a copy after it hangs in the National Portrait Gallery.Copy of Holbein's portrait of Kratzer from the NPG It shows the astronomer Nikolaus Kratzer, a friend of Thomas More and Holbein himself. In his hand he holds a half-finished polyhedral sundial, whilst on the shelves behind him are a semi-circular star quadrant, a shepherd's dial and other instruments.

Blossite is part of the copper vanadates class, the V5+ form a tetrahedral coordination surrounded by oxygen atoms. The VO4 tetrahedra is closely related to thortvetite-group compounds by the formation of [V2O7]4−. The [V2O7] planes lie along [100], the divanadate units are staggered orienting the V-OB-V vector parallel to [120] in one plane and parallel to [120] in the adjacent plane. The independent copper cation in blossite forms as a polyhedral structure coordinated by five oxygen atoms forming an apically elongate square pyramid.

In the mathematical field of graph theory, the 26-fullerene graph is a polyhedral graph with V = 26 vertices and E = 39 edges. Its planar embedding has three hexagonal faces (including the one shown as the external face of the illustration) and twelve pentagonal faces. As a planar graph with only pentagonal and hexagonal faces, meeting in three faces per vertex, this graph is a fullerene. The existence of this fullerene has been known since at least 1968.. See line 19 of table, p.

Many of the boranes readily oxidise on contact with air, some violently. The parent member BH3 is called borane, but it is known only in the gaseous state, and dimerises to form diborane, B2H6. The larger boranes all consist of boron clusters that are polyhedral, some of which exist as isomers. For example, isomers of B20H26 are based on the fusion of two 10-atom clusters. The most important boranes are diborane B2H6 and two of its pyrolysis products, pentaborane B5H9 and decaborane B10H14.

A traditional die is a cube with each of its six faces marked with a different number of dots (pips) from one to six. When thrown or rolled, the die comes to rest showing a random integer from one to six on its upper surface, with each value being equally likely. Dice may also have polyhedral or irregular shapes and may have faces marked with numerals or symbols instead of pips. Loaded dice are designed to favor some results over others for cheating or entertainment.

Combinatorial commutative algebra is a relatively new, rapidly developing mathematical discipline. As the name implies, it lies at the intersection of two more established fields, commutative algebra and combinatorics, and frequently uses methods of one to address problems arising in the other. Less obviously, polyhedral geometry plays a significant role. One of the milestones in the development of the subject was Richard Stanley's 1975 proof of the Upper Bound Conjecture for simplicial spheres, which was based on earlier work of Melvin Hochster and Gerald Reisner.

For the Berlekamp–van Lint–Seidel and Games graphs, see In enumerative combinatorics, there are signed subsets of a set of elements. In polyhedral combinatorics, the hypercube and all other Hanner polytopes have a number of faces (not counting the empty set as a face) that is a power of three. For example, a 2-cube, or square, has 4 vertices, 4 edges and 1 face, and . Kalai's conjecture states that this is the minimum possible number of faces for a centrally symmetric polytope.

Tunnels & Trolls was also the first system to publish a series of fantasy- themed gamebooks - adventures which are designed to be played by one person, without the need for a referee. At least twenty such adventures were published by Flying Buffalo. The Fighting Fantasy series achieved great popularity using this format. Both T&T;'s simplicity and its reliance on use of six-sided dice (as compared to the various polyhedral dice used by Dungeons and Dragons) contributed to its success in this format.

The third distinctive feature of self- assembly is that the building blocks are not only atoms and molecules, but span a wide range of nano- and mesoscopic structures, with different chemical compositions, functionalities, and shapes. Research into possible three- dimensional shapes of self-assembling micrites examines Platonic solids (regular polyhedral). The term ‘micrite’ was created by DARPA to refer to sub- millimeter sized microrobots, whose self-organizing abilities may be compared with those of slime mold. Recent examples of novel building blocks include polyhedra and patchy particles.

As well as being defined by recursive subdivision of triangles, the Apollonian networks have several other equivalent mathematical characterizations. They are the chordal maximal planar graphs, the chordal polyhedral graphs, and the planar 3-trees. They are the uniquely 4-colorable planar graphs, and the planar graphs with a unique Schnyder wood decomposition into three trees. They are the maximal planar graphs with treewidth three, a class of graphs that can be characterized by their forbidden minors or by their reducibility under Y-Δ transforms.

For instance, they may be restricted to being the closures of disjoint open sets. The Bolyai–Gerwien theorem states that any polygon may be dissected into any other polygon of the same area, using interior-disjoint polygonal pieces. It is not true, however, that any polyhedron has a dissection into any other polyhedron of the same volume using polyhedral pieces. This process is possible, however, for any two honeycombs (such as cube) in three dimension and any two zonohedra of equal volume (in any dimension).

These columns, colored pink below and gradually deepening in color to red near the top, have a corrugated surface texture. The columns often fork near the top into additional branches that support a lattice-like, or clathrate dome. The meshes of the fertile net are roughly polyhedral and there is an abrupt transition from columns to lattice. The olive-green gleba is held on the bottom of and in between the meshes of the clathrate dome, and the inner side of the upper arms.

The first BMCs were observed in the 1950s in electron micrographs of cyanobacteria, and were later named carboxysomes after their role in carbon fixation was established. Until the 1990s, carboxysomes were thought to be an oddity confined to certain autotrophic bacteria. But then genes coding for proteins homologous to those of the carboxysome shell were identified in the pdu (propanediol utilization) and eut (ethanolamine utilization) operons. Subsequently, transmission electron micrographs of Salmonella cells grown on propanediol or ethanolamine showed the presence of polyhedral bodies similar to carboxysomes.

257 is a prime number of the form 2^{2^n}+1, specifically with n = 3, and therefore a Fermat prime. Thus a regular polygon with 257 sides is constructible with compass and unmarked straightedge. It is currently the second largest known Fermat prime.. It is also a balanced prime, an irregular prime, a prime that is one more than a square, and a Jacobsthal–Lucas number. There are exactly 257 combinatorially distinct convex polyhedra with eight vertices (or polyhedral graphs with eight nodes).

In organic chemistry, spherical aromaticity is formally used to describe an unusually stable nature of some spherical compounds such as fullerenes, polyhedral boranes. In 2000, Andreas Hirsch and coworkers in Erlangen, Germany, formulated a rule to determine when a fullerene would be aromatic. They found that if there were 2(n+1)2 π-electrons, then the fullerene would display aromatic properties. This follows from the fact that an aromatic fullerene must have full icosahedral (or other appropriate) symmetry, so the molecular orbitals must be entirely filled.

The internal structure of conifer Tree rings are records of the influence of environmental conditions, their anatomical characteristics record growth rate changes produced by these changing conditions. The microscopic structure of conifer wood consists of two types of cells: parenchyma, which have an oval or polyhedral shape with approximately identical dimensions in three directions, and strongly elongated tracheids. Tracheids make up more than 90% of timber volume. The tracheids of earlywood formed at the beginning of a growing season have large radial sizes and smaller, thinner cell walls.

A planar graph is an undirected graph that can be embedded into the Euclidean plane without any crossings. A planar graph is called polyhedral if and only if it is 3-vertex-connected, that is, if there do not exist two vertices the removal of which would disconnect the rest of the graph. A graph is bipartite if its vertices can be colored with two different colors such that each edge has one endpoint of each color. A graph is cubic (or 3-regular) if each vertex is the endpoint of exactly three edges.

The polyhedral capsid from which the virus gets its name is an extremely stable protein crystal that protects the virus in the external environment. It dissolves in the alkaline midgut of moths and butterflies to release the virus particle and infect the larva. An example of an insect that it infects is the fall webworm. NPV was once listed by the International Committee on Taxonomy of Viruses as a subgenus of Eubaculovirinae, but the term now refers to 35 species of the family Baculoviridae—mostly alphabaculoviruses, but also one deltabaculovirus and two gammabaculoviruses.

Two isomers are known of octadecaborane, providing the first example of isomers in a boron-hydride cluster. The clusters are also of interest because the boron centers shared between the two subunits have an unusually high number of B-B interactions. The isomers consists of two B9H11 polyhedral subunits, each having a decaborane-like form, joined at a B–B edge. These two boron atoms are each coordinated to six others; this compound was the first one found to have such a high number of borons coordinated around a single boron center.

Around each silicon there is one OH group and there are three oxygens that neighbor them. The silicon tetrahedra are arranged so that they share an edge with calcium(1), and silicon(2) shares edges with the calcium(2) and calcium(3) polyhedral. The silicon tetrahedra are held together by the OH group and hydrogen bonding occurs between the hydrogen in the OH and the silicon tetrahedra. Hydrogen bonding is caused because the positive ion, hydrogen, is attracted to negatively charge ions which, in this case, are the silicon tetrahedra.

Most commonly, software is used to generate the starting position, as was used in the 2019 World Fischer Random Championship. Ingo Althofer If this is not available, there are several other procedures for generating random starting positions with equal probability. In 1998 Ingo Althöfer proposed in 1998 the "single die method", a method that requires only a single standard die. If a full set of polyhedral dice is available (a tetrahedron (d4), cube (d6), octahedron (d8), dodecahedron (d12), and a icosahedron (d20)), one never needs to reroll any dice.

The visible parts of each face comprise five isosceles triangles which touch at five points around the pentagon. We could treat these triangles as 60 separate faces to obtain a new, irregular polyhedron which looks outwardly identical. Each edge would now be divided into three shorter edges (of two different kinds), and the 20 false vertices would become true ones, so that we have a total of 32 vertices (again of two kinds). The hidden inner pentagons are no longer part of the polyhedral surface, and can disappear.

WEG's game designers created a new rules system for Ghostbusters: A Frightfully Cheerful Roleplaying Game that used only six-sided dice rather that the polyhedral dice favored by other role-playing game companies. WEG would use this D6 System for many of their licensed products. In January 1987, again using funds from Bucci Imports, WEG was able to purchase the games license for Star Wars, and immediately published Star Wars: The Roleplaying Game. Later that year, Greg Costikyan and Eric Goldberg left WEG after a disagreement with Palter.

A closely related woodcut, Study for Stars, completed in August 1948,, p. 99. depicts wireframe versions of several of the same polyhedra and polyhedral compounds, floating in black within a square composition, but without the chameleons. The largest polyhedron shown in Study for Stars, a stellated rhombic dodecahedron, is also one of two polyhedra depicted prominently in Escher's 1961 print Waterfall. The stella octangula, a compound of two tetrahedra that appears in the upper right of Stars, also forms the central shape of another of Escher's astronomical works, Double Planetoid (1949).

Geometric combinatorics is a branch of mathematics in general and combinatorics in particular. It includes a number of subareas such as polyhedral combinatorics (the study of faces of convex polyhedra), convex geometry (the study of convex sets, in particular combinatorics of their intersections), and discrete geometry, which in turn has many applications to computational geometry. Other important areas include metric geometry of polyhedra, such as the Cauchy theorem on rigidity of convex polytopes. The study of regular polytopes, Archimedean solids, and kissing numbers is also a part of geometric combinatorics.

Watermelons suffering from hollow heart are safe to consume. Farmers of the Zentsuji region of Japan found a way to grow cubic watermelons by growing the fruits in metal and glass boxes and making them assume the shape of the receptacle. The cubic shape was originally designed to make the melons easier to stack and store, but these "square watermelons" may be triple the price of normal ones, so appeal mainly to wealthy urban consumers. Pyramid-shaped watermelons have also been developed and any polyhedral shape may potentially be used.

Bridged silsesquioxanes have been used for quantum confined nano-size semiconductors. Silsesquioxane resins have also been used for these applications because they have high dielectric strengths, low dielectric constants, high volume resistivities, and low dissipation factors, making them very suitable for electronics applications. These resins have heat and fire resistant properties, which can be used to make fiber- reinforced composites for electrical laminates. Polyhedral oligomeric silsesquioxanes have been examined as a means to give improved mechanical properties and stability, with an organic matrix for good optical and electrical properties.

Cortex Plus uses polyhedral dice common to many roleplaying games and utilizes standard dice notation, ranging from d4 (a 4 sided tetrahedral die) to d12 (a 12-sided dodecahedral die). The cubed d6 is the "default" die used in the game. Cortex Plus uses dice pools ranging from d4 (terrible) to d12 (the best possible). Every die in your pool that rolls a natural 1 (called an 'Opportunity') not only doesn't count toward your total, but also causes some form of negative consequence for the characters to overcome.

The Miller complex is further defined by surveys done in the Cross Creek watershed, where other lanceolate points, small prismatic blades, and small polyhedral blade cores have been recovered. According to Adovasio et al.,Adovasio, J. M., D. Pedler, J. Donahue, and R. Stuckenrath (1999), No Vestige of a Beginning nor Prospect for an End: Two Decades of Debate on Meadowcroft Rockshelter. In Ice Age Peoples of North America: Environments, Origins, and Adaptations of the First Americans, edited by R. Bonnichsen and K. L. Turmire, pp. 416–31.

In the geometry of crystal nets, one can treat edges as line segments. For example, in a crystal net, it is presumed that edges do not “collide” in the sense that when treating them as line segments, they do not intersect. Several polyhedral constructions can be derived from crystal nets. For example, a vertex figure can be obtained by subdividing each edge (treated as a line segment) by the insertion of subdividing points, and then the vertex figure of a given vertex is the convex hull of the adjacent subdividing points (i.e.

German stamp featuring Jamnitzer's work A later work on perspective, Artes Excelençias de la Perspectiba (1688) by P. Gómez de Alcuña, was heavily influenced by Jamnitzer. A 2008 German postage stamp, issued to commemorate the 500th anniversary of Jamnitzer's birth, included a reproduction of one of the pages of the book, depicting two polyhedral cones tilted towards each other. The full sheet of ten stamps also includes another figure from the book, a skeletal icosahedron. A French edition of Perspectiva corporum regularium, edited by Albert Flocon, was published by Brieux in 1964.

The 14 chapters of the book can be grouped into two parts, with the first 2/3 of the book concerning the combinatorial properties of convex polytopes and the remainder connecting these topics to abstract algebra. The topics covered include Schlegel diagrams and Gale diagrams, irrational polytopes, point set triangulations, regular triangulations and their polyhedral representation by secondary polytopes, the permutohedron as an example of a secondary polytope, Gröbner bases, toric ideals, and toric varieties, and the connections between Gröbner bases of toric ideals and regular triangulations of points.

Weaire–Phelan structure The Weaire–Phelan structure has Pmn (223) symmetry. It has 3 orientations of stacked tetradecahedrons with pyritohedral cells in the gaps. It is found as a crystal structure in chemistry where it is usually known as the "Type I clathrate structure". Gas hydrates formed by methane, propane, and carbon dioxide at low temperatures have a structure in which water molecules lie at the nodes of the Weaire–Phelan structure and are hydrogen bonded together, and the larger gas molecules are trapped in the polyhedral cages.

The typical framework building blocks are polyhedral units, with 4-, 5-, 6- or 7-coordinate metal centres. These units share edges and/or vertices, or, less commonly, faces (such as in the ion , which has face-shared octahedra with Mo atoms at the vertices of an icosahedron). The most common unit for polymolybdates is the octahedral {MoO6} unit, often distorted by the Mo atom being off-centre to give one shorter Mo–O bond. Some polymolybdates contain pentagonal bipyramidal units; these are the key building blocks in the molybdenum blues.

The effusive dedication, to powerful patrons as well as to the men who controlled his position in Graz, also provided a crucial doorway into the patronage system.Caspar. Kepler, pp. 65–71. Though the details would be modified in light of his later work, Kepler never relinquished the Platonist polyhedral-spherist cosmology of Mysterium Cosmographicum. His subsequent main astronomical works were in some sense only further developments of it, concerned with finding more precise inner and outer dimensions for the spheres by calculating the eccentricities of the planetary orbits within it.

While the cubical six-sided die became the most common type in many parts of the world, other shapes were always known, like 20-sided dice in Ptolemaic and Roman times. The modern tradition of using sets of polyhedral dice started around the end of the 1960s when non- cubical dice became popular among players of wargames, and since have been employed extensively in role-playing games and trading card games. Dice using both the numerals 6 and 9, which are reciprocally symmetric through rotation, typically distinguish them with a dot or underline.

Snap rounding is a method of approximating line segment locations by creating a grid and placing each point in the centre of a cell (pixel) of the grid. The method preserves certain topological properties of the arrangement of line segments. Drawbacks include the potential interpolation of additional vertices in line segments (lines become polylines), the arbitrary closeness of a point to a non-incident edge, and arbitrary numbers of intersections between input line-segments. The 3 dimensional case is worse, with a polyhedral subdivision of complexity becoming complexity O(n4).

Turda Museum Turda Museum Franziska Tesaurus is the richest Gepid royal tomb found in Romania. It was found while searching the Potaissa Roman castrum at Turda in 1996, by Mihai Bărbulescu, between the secondary sewer and the frigidarium. The inventory of the tomb was composed of: polyhedral golden rings with almandine, hemicyclical gold plated brooch, gold-plated silver belt with gold garments and almandine, amber necklace, embroidery decorations, bone comb, nomadic mirror, silver shoe belts, and small fragments of clothing. It was put on display on 3 April 2007 in Turda History Museum.

Benton, p. 463 A primary weakness of shell was that it typically produced only a few large fragments, the count increasing with caliber of the shell. A Confederate mid-war innovation perhaps influenced by British ordnance/munition imports was the "polygonal cavity" or "segmented" shell which used a polyhedral cavity core to create lines of weakness in the shell wall that would yield more regular fragmentation patterns—typically 12 similarly sized fragments. While segmented designs were most common in spherical shell, it was applied to specific rifled projectiles as well.

The main focus of the book is on the specification of geometric data that will determine uniquely the shape of a three- dimensional convex polyhedron, up to some class of geometric transformations such as congruence or similarity. It considers both bounded polyhedra (convex hulls of finite sets of points) and unbounded polyhedra (intersections of finitely many half-spaces). The 1950 Russian edition of the book included 11 chapters. The first chapter covers the basic topological properties of polyhedra, including their topological equivalence to spheres (in the bounded case) and Euler's polyhedral formula.

Halothiobacillus neapolitanus: (A) arranged within the cell, and (B) intact upon isolation. Scale bars indicate 100 nm. Carboxysomes are bacterial microcompartments (BMCs) consisting of polyhedral protein shells filled with the enzymes ribulose-1,5-bisphosphate carboxylase/oxygenase (RuBisCO)--the predominant enzyme in carbon fixation and the rate limiting enzyme in the Calvin cycle--and carbonic anhydrase. Carboxysomes are thought to have evolved as a consequence of the increase in oxygen concentration in the ancient atmosphere; this is because oxygen is a competing substrate to carbon dioxide in the RuBisCO reaction.

Polyhedra in Great Stella's library include the Platonic solids, the Archimedean solids, the Kepler-Poinsot solids, the Johnson solids, some Johnson Solid near-misses, numerous compounds including the uniform polyhedra, and other polyhedra too numerous to list here. Operations which can be performed on these polyhedra include stellation, faceting, augmentation, dualization (also called "reciprocation"), creating convex hulls, and others. All versions of the program enable users to print nets for polyhedra. These nets may then be assembled into actual three- dimensional polyhedral models of great beauty and complexity.

In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube. While mathematical literature refers to any such polyhedron as a cuboid, other sources use "cuboid" to refer to a shape of this type in which each of the faces is a rectangle (and so each pair of adjacent faces meets in a right angle); this more restrictive type of cuboid is also known as a rectangular cuboid, right cuboid, rectangular box, rectangular hexahedron, right rectangular prism, or rectangular parallelepiped.

It was launched from Musudan-ri using a Paektusan rocket, at 03:07 GMT on 31 August 1998, a few days before the 50th anniversary of North Korea's independence from Japan. On 4 September, the Korean Central News Agency announced that the satellite had successfully been placed into low Earth orbit. The China National Space Administration was involved in the development of Kwangmyŏngsŏng-1, which had a 72-faced polyhedral shape, similar to Dong Fang Hong I, the first Chinese satellite. The mass of the satellite is unclear, with estimates ranging from to .

Since periodicity and twins were ruled out, Blech, unaware of the two- dimensional tiling work, was looking for another possibility: a completely new structure containing cells connected to each other by defined angles and distances but without translational periodicity. Blech decided to use a computer simulation to calculate the diffraction intensity from a cluster of such a material without long-range translational order but still not random. He termed this new structure multiple polyhedral. The idea of a new structure was the necessary paradigm shift to break the impasse.

The first Basic Set was available as a 48-page standalone rulebook featuring artwork by David C. Sutherland III, or as part of a boxed set, which was packaged in a larger, more visually appealing box than the original boxed set, allowing the game to be stocked on retail shelves and targeted at the general public via toy stores. The boxed set included a set of polyhedral dice and supplemental materials. In that same year, Games Workshop (U.K.) published their own version of the rulebook, with a cover by John Blanche, and illustrations by Fangorn.

In chemistry the polyhedral skeletal electron pair theory (PSEPT) provides electron counting rules useful for predicting the structures of clusters such as borane and carborane clusters. The electron counting rules were originally formulated by Kenneth Wade and were further developed by Michael Mingos and others; they are sometimes known as Wade's rules or the Wade–Mingos rules. The rules are based on a molecular orbital treatment of the bonding. These notes contained original material that served as the basis of the sections on the 4n, 5n, and 6n rules.

The wattle bagworm has many natural enemies. They include parasitic wasps, flies and beetles, and various predators, such as spiders and birds, not to mention fungal diseases such as Entomophthora and Isaria species, bacterial diseases such as Bacillus thuringiensis, and polyhedral virus diseases. Attempts to use such a virus for bagworm control during the 1950s gave results too inconsistent to be satisfactory at the time. In the wild probably the most important insect enemy of Kotochalia junodi is an interesting parasitoid wasp, a member of the Ichneumonidae, Sericopimpla sericata.

The digon is an important construct in the topological theory of networks such as graphs and polyhedral surfaces. Topological equivalences may be established using a process of reduction to a minimal set of polygons, without affecting the global topological characteristics such as the Euler value. The digon represents a stage in the simplification where it can be simply removed and substituted by a line segment, without affecting the overall characteristics. The cyclic groups may be obtained as rotation symmetries of polygons: the rotational symmetries of the digon provide the group C2.

Domokos and Várkonyi are interested to find a polyhedral solution with the surface consisting of a minimal number of flat planes. Therefore, they offer a prize to anyone who finds such solution, which amounts to $10,000 divided by the number of planes in the solution. Obviously, one can approximate a curvilinear gömböc with a finite number of discrete surfaces; however, their estimate is that it would take thousands of planes to achieve that. They hope, by offering this prize, to stimulate finding a radically different solution from their own.

Gilman theorizes that the structure can enable travel from one plane or dimension to another. Gilman begins experiencing bizarre dreams in which he seems to float without physical form through an otherworldly space of unearthly geometry and indescribable colors and sounds. Among the elements, both organic and inorganic, he perceives shapes that he innately recognizes as entities which appear and disappear instantaneously and at random. Several times, his dreaming-self encounters bizarre clusters of "iridescent, prolately spheroidal bubbles", as well as a rapidly changing polyhedral-figure, both of which appear sapient.

More than one comic (or cartoon) appeared in Polymancer magazine during its publication run. There was a serial comic called “SideQuest” about a group of rolepayers that appeared in issues 1 through 8. As well as a one panel cartoon entitled "Quotable Quotes", that started in issue #8; this one panel cartoon was based on humorous quotes from games that were supplied the magazine's readership. There was also "Dice Quest" which began in issue #10, a 4 panel cartoon about the dice used by roleplayers; the characters being anthropomorphized polyhedral dice.

The Nuraghe is a single small tower, is 4.70 meters tall at the highest point and is built with large polyhedral stones of basaltic and trachyte rock. The main facade overlooks the stretch of mountainous coast above the beach of Cala Fuili. The structure of the tower is very modest compared to the great extension of the Nuragic village that lies at its feet. The village consists of various circular huts made of unpolished stones and of rectangular buildings constructed with stones placed on top of each other without any kind of adhesive.

Projected dynamical systems have evolved out of the desire to dynamically model the behaviour of nonstatic solutions in equilibrium problems over some parameter, typically take to be time. This dynamics differs from that of ordinary differential equations in that solutions are still restricted to whatever constraint set the underlying equilibrium problem was working on, e.g. nonnegativity of investments in financial modeling, convex polyhedral sets in operations research, etc. One particularly important class of equilibrium problems which has aided in the rise of projected dynamical systems has been that of variational inequalities.

Mysterium was published late in 1596, and Kepler received his copies and began sending them to prominent astronomers and patrons early in 1597; it was not widely read, but it established Kepler's reputation as a highly skilled astronomer. The effusive dedication, to powerful patrons as well as to the men who controlled his position in Graz, also provided a crucial doorway into the patronage system.Caspar. Kepler, pp. 65–71. Though the details would be modified in light of his later work, Kepler never relinquished the Platonist polyhedral- spherist cosmology of Mysterium Cosmographicum.

A polyhedron realized from a circle packing. The circles representing the vertices of the polyhedron are their horizons on the sphere, and the circles representing the faces (dual vertices) are the intersections of the sphere with the face planes. According to one variant of the circle packing theorem, for every polyhedral graph and its dual graph, there exists a system of circles in the plane or on any sphere, representing the vertices of both graphs, so that two adjacent vertices in the same graph are represented by tangent circles, a primal and dual vertex that represent a vertex and face that touch each other are represented by orthogonal circles, and all remaining pairs of circles are disjoint.. From such a representation on a sphere, one can find a polyhedral realization of the given graph as the intersection of a collection of halfspaces, one for each circle that represents a dual vertex, with the boundary of the halfspace containing the circle. Alternatively and equivalently, one can find the same polyhedron as the convex hull of a collection of points (its vertices), such that the horizon seen when viewing the sphere from any vertex equals the circle that corresponds to that vertex.

In mathematics, a cyclic polytope, denoted C(n,d), is a convex polytope formed as a convex hull of n distinct points on a rational normal curve in Rd, where n is greater than d. These polytopes were studied by Constantin Carathéodory, David Gale, Theodore Motzkin, Victor Klee, and others. They play an important role in polyhedral combinatorics: according to the upper bound theorem, proved by Peter McMullen and Richard Stanley, the boundary Δ(n,d) of the cyclic polytope C(n,d) maximizes the number fi of i-dimensional faces among all simplicial spheres of dimension d − 1 with n vertices.

This phase is thermodynamically unstable at room temperature with respect to the binary oxides ZrO2 and WO3, but may be synthesised by heating stoichiometric quantities of these oxides together and then quenching the material by rapidly cooling it from approximately 900 °C to room temperature. The structure of cubic zirconium tungstate consists of corner-sharing ZrO6 octahedral and WO4 tetrahedral structural units. Its unusual expansion properties are thought to be due to vibrational modes known as Rigid Unit Modes (RUMs), which involve the coupled rotation of the polyhedral units that make up the structure, and lead to contraction.

It is so called because such an embedding can be found as the equilibrium position for a system of springs representing the edges of the graph. Steinitz's theorem states that every 3-connected planar graph can be represented as the edges of a convex polyhedron in three- dimensional space. A straight-line embedding of G, of the type described by Tutte's theorem, may be formed by projecting such a polyhedral representation onto the plane. The Circle packing theorem states that every planar graph may be represented as the intersection graph of a collection of non-crossing circles in the plane.

By providing a provably polynomial-time ε-approximation algorithm, they resolved a long-standing open problem in optimal control. Their first paper considered time-optimal control ("fastest path") of a point mass under Newtonian dynamics, amidst polygonal (2D) or polyhedral (3D) obstacles, subject to state bounds on position, velocity, and acceleration. Later they extended the technique to many other cases, for example, to 3D open-chain kinematic robots under full Lagrangian dynamics. More recently, many practical heuristic algorithms based on stochastic optimization and iterative sampling were developed, by a wide range of authors, to address the kinodynamic planning problem.

It can be proven by mathematical induction (as Steinitz did), by finding the minimum-energy state of a two-dimensional spring system and lifting the result into three dimensions, or by using the circle packing theorem. Several extensions of the theorem are known, in which the polyhedron that realizes a given graph has additional constraints; for instance, every polyhedral graph is the graph of a convex polyhedron with integer coordinates, or the graph of a convex polyhedron all of whose edges are tangent to a common midsphere. In higher dimensions, the problem of characterizing the graphs of convex polytopes remains open.

Under such conditions, the polyhedral crystal form will be unstable, it will sprout protrusions at its corners and edges where the degree of supersaturation is at its highest level. The tips of these protrusions will clearly be the points of highest supersaturation. It is generally believed that the protrusion will become longer (and thinner at the tip) until the effect of interfacial free energy in raising the chemical potential slows the tip growth and maintains a constant value for the tip thickness. In the subsequent tip-thickening process, there should be a corresponding instability of shape.

The visualization of a crystal structure can be reduced to the arrangement of atoms, ions, or molecules in the unit cell, with or without cell outlines. Structure elements extending beyond single unit cells, such as isolated molecular or polyhedral units as well as chain, net, or framework structures, can often be better understood by extending the structure representation into adjacent cells. The space group of a crystal is a mathematical description of the symmetry inherent in the structure. The motif of the crystal structure is given by the asymmetric unit, a minimal subset of the unit cell contents.

In the more abstract setting of incidence geometry, which is a set having a symmetric and reflexive relation called incidence defined on its elements, a flag is a set of elements that are mutually incident. This level of abstraction generalizes both the polyhedral concept given above as well as the related flag concept from linear algebra. A flag is maximal if it is not contained in a larger flag. An incidence geometry (Ω, ) has rank if Ω can be partitioned into sets Ω1, Ω2, ..., Ω, such that each maximal flag of the geometry intersects each of these sets in exactly one element.

Around 80% of the oxygen atoms are shared among three or more Al-O polyhedra, and the majority of inter-polyhedral connections are corner- sharing, with the remaining 10–20% being edge-sharing. The breakdown of octahedra upon melting is accompanied by a relatively large volume increase (~20%), the density of the liquid close to its melting point is 2.93 g/cm3. The structure of molten alumina is temperature dependent and the fraction of 5- and 6-fold aluminium increases during cooling (and supercooling), at the expense of tetrahedral AlO4 units, approaching the local structural arrangements found in amorphous alumina.

They are polyhedral graphs, meaning that every Halin graph can be used to form the vertices and edges of a convex polyhedron, and the polyhedra formed from them have been called roofless polyhedra or domes. Every Halin graph has a Hamiltonian cycle through all its vertices, as well as cycles of almost all lengths up to their number of vertices. The Halin graphs can be recognized in linear time. Because Halin graphs have low treewidth, many computational problems that are hard on other kinds of planar graphs, such as finding Hamiltonian cycles, can also be solved quickly on Halin graphs.

Gerard Caris based his entire artistic oeuvre on the regular dodecahedron and the pentagon, which is presented as a new art movement coined as Pentagonism. A climbing wall consisting of three dodecahedral pieces In modern role-playing games, the regular dodecahedron is often used as a twelve-sided die, one of the more common polyhedral dice. Immersive Media, a camera manufacturing company, has made the Dodeca 2360 camera, the world's first 360° full-motion camera which captures high-resolution video from every direction simultaneously at more than 100 million pixels per second or 30 frames per second. It is based on regular dodecahedron.

Most notably, these papers demonstrated how a good characterization of the polyhedron associated with a combinatorial optimization problem could lead, via the duality theory of linear programming, to the construction of an efficient algorithm for the solution of that problem. Additional landmark work of Edmonds is in the area of matroids. He found a polyhedral description for all spanning trees of a graph, and more generally for all independent sets of a matroid. Building on this, as a novel application of linear programming to discrete mathematics, he proved the matroid intersection theorem, a very general combinatorial min-max theorem.

In the late 1970s Connelly and D. Sullivan formulated the bellows conjecture stating that the volume of a flexible polyhedron is invariant under flexing. This conjecture was proved for polyhedra homeomorphic to a sphere by using elimination theory, and then proved for general orientable 2-dimensional polyhedral surfaces by . The proof extends Piero della Francesca's formula for the volume of a tetrahedron to a formula for the volume of any polyhedron. The extended formula shows that the volume must be a root of a polynomial whose coefficients depend only on the lengths of the polyhedron's edges.

Ancient philosophy posited a set of classical elements to explain observed patterns in nature. These elements originally referred to earth, water, air and fire rather than the chemical elements of modern science. The term 'elements' (stoicheia) was first used by the Greek philosopher Plato in about 360 BCE in his dialogue Timaeus, which includes a discussion of the composition of inorganic and organic bodies and is a speculative treatise on chemistry. Plato believed the elements introduced a century earlier by Empedocles were composed of small polyhedral forms: tetrahedron (fire), octahedron (air), icosahedron (water), and cube (earth).

In April 2019, J. Yuhara and others reported the deposition of a single atom thickness by molecular beam epitaxy with a segregation method upon a palladium surface in a crystal lattice with Miller indices (111). The structure was confirmed with scanning tunneling microscopy (STM) revealing a nearly flat honeycomb structure. There is no evidence of any three-dimensional islands, but one notices a unique nanostructured tessellation all over the terraces looking like a space-filling polyhedral foam reduced to dimension 2. Their appearance reminds you of the famous Weaire-Phelan bubble structure of the envelope of the Beijing Olympics’ “WaterCube”.

Fred did two years of postdoctoral work with Professor Fred Hawthorne at the University of California, Riverside. During that time, he greatly expanded the range of known carboranes to include polyhedral B9C2H11, B8C2H10, B7C2H9, and B6C2H8 carboranes, the B7C2H13 system, and their derivatives.Thermal isomerization of C-phenyldicarbaundecaborate(12), Garrett, Philip M.; Tebbe, Fred N.; Hawthorne, M. Frederick, Journal of the American Chemical Society (1964), 86(22), 5016-17The B6C2H8, B7C2H9, and B8C2H10 carborane systems, Tebbe, Fred N.; Garrett, Philip M.; Young, Donald C.; Hawthorne, M. Frederick, Journal of the American Chemical Society (1966), 88(3), 609-10.

The family Entolomataceae was first defined by the Czech mycologists František Kotlaba and Zdenek Pouzar in 1972. Although the family as a whole is quite well defined, many different internal classifications of the Entolomataceae have been proposed. Please see List of Entolomataceae genera for a table of the main genera which have been placed in this family at one time or another. The current view is that Entolomataceae with angular (polyhedral) spores should be classified in genus Entoloma, those with bumpy spores should be in Rhodocybe, and those with longitudinally ridged spores should be put in Clitopilus.

Miniatures wargamers began using dice in the shape of Platonic solids in the late 1960s and early ’70s, to obtain results that could not easily be produced on a conventional six-sided die. Dungeons & Dragons emerged in this milieu, and was the first game with widespread commercial availability to use such dice. In its earliest edition, D&D; had no standardized way to call for polyhedral die rolls or to refer to the results of such rolls. In some places the text gives a verbal instruction; in others, it only implies the roll to be made by describing the range of its results.

Sample dice products Q-Workshop booth at Gen Con Indy 2008 Q WORKSHOP is a Polish company located in Poznań that specializes in design and production of polyhedral dice and dice accessories for use in various games (role-playing games, board games and tabletop wargames). They also run an online retail store and maintain an active social media community. Q WORKSHOP was established in 2001 by Patryk Strzelewicz. Initially, the company sold its products via online auction services, but in 2005 a website and online store were also established. Currently there are over 120 designs of dice in stock.

Having developed a suitable design for such an aircraft, development activity on the Lampyridae programme proceeded to the construction of a single three-quarter scale piloted model of the aircraft. This model was initially used for a series of wind tunnel tests. Commenced during 1985, these tests are known to have involved at least two models, a 1:3.5-scale low-speed model and a 1:20-scale transonic model. According to Lobert, the results produced by the wind tunnel tests demonstrated the Lampyridae's design to have possessed favourable high-quality aerodynamic properties, despite the initial disadvantages presented by the polyhedral airframe design.

Majumdar did her undergraduate studies at the University of Bristol. As a graduate student at Bristol, she also worked with Hewlett Packard Laboratories. She was awarded a Ph.D. in applied mathematics at the University of Bristol in 2006; her dissertation, Liquid crystals and tangent unit-vector fields in polyhedral geometries, was jointly supervised by Jonathan Robbins and Maxim Zyskin. After working as a Royal Commission of the Exhibition of 1851 Research Fellow at the University of Oxford, she moved to the University of Bath in 2012, having been awarded a 5-year EPSRC Career Acceleration Fellowship in 2011.

In China, Taiwan and Thailand, sky lanterns are traditionally made from oiled rice paper on a bamboo frame. The source of hot air may be a small candle or fuel cell composed of a waxy flammable material. In Brazil and Mexico sky lanterns were traditionally made of several patches of thin translucent paper (locally called "silk paper"), in various bright colors, glued together to make a multicolored polyhedral shell. A design that was fairly common was two pyramids joined by the base (a bipyramid, such as the octahedron) sometimes with a cube or prism inserted in the middle.

Typically, while there may be many species of constituents, there are two main classes: somewhat compact and often polyhedral secondary building units (SBUs), and linking or bridging building units. A popular class of examples are the Metal-Organic Frameworks (MOFs), in which (classically) the secondary building units are metal ions or clusters of ions and the linking building units are organic ligands. These SBUs and ligands are relatively controllable, and some new crystals have been synthesized using designs of novel nets. An organic variant are the Covalent Organic Frameworks (COFs), in which the SBUs might (but not necessarily) be themselves organic.

Airflushed intake filter called TAPIS Since the late 1990s, Taprogge offers another filter system which retains fouling already at the intake into the cooling water system – in this way the entire system and the long cooling water pipes can be protected. The system called TAPIS (Taprogge Air Powered Intake System) is installed in the water at the cooling water pipe inlet in the form of a polyhedral housing with plain filter surfaces. It is cleaned by pressurized air blast. In contrast to submarine rakes for seaborne matter, the stainless steel filter has no moving parts and masters biggest water flows.

After spending two years as a post-doctoral student at the University of Cambridge and two further years lecturing successively at Cornell University and Derby College of Technology, in 1961 Wade became a Lecturer at Durham University. In 1971, he was appointed Senior Lecturer and was promoted to Reader in 1977. Between 1983 and 1998, he was Professor of Chemistry at the university and served, between 1986 and 1989, as Chairman of its Department of Chemistry. Wade's Rules, also known as Polyhedral skeletal electron pair theory, are a set of electron counting rules to predict the shapes of borane clusters.

The skeleton of any convex polyhedron is a planar graph, and the skeleton of any k-dimensional convex polytope is a k-connected graph. Conversely, Steinitz's theorem states that any 3-connected planar graph is the skeleton of a convex polyhedron; for this reason, this class of graphs is also known as the polyhedral graphs. A Euclidean graph is a graph in which the vertices represent points in the plane, and the edges are assigned lengths equal to the Euclidean distance between those points. The Euclidean minimum spanning tree is the minimum spanning tree of a Euclidean complete graph.

The first set of necessary and sufficient conditions for the embeddability of a semigroup in a group were given in . Though theoretically important, the conditions are countably infinite in number and no finite subset will suffice, as shown in . (Accessed on 11 May 2009) A different (but also countably infinite) set of necessary and sufficient conditions were given in , where it was shown that a semigroup can be embedded in a group if and only if it is cancellative and satisfies a so-called "polyhedral condition". The two embedding theorems by Malcev and Lambek were later compared in .

In 1978, Mingos, Stephen G. Davies and Malcolm Green compiled a set of rules that summarise where nucleophilic additions will occur on pi ligands. Mingos' 1984 paper on the polyhedral skeletal electron pair theory develops Wade's electron counting rules for predicting the molecular geometry of cluster compounds. In 1990 he was appointed Reader in Inorganic Chemistry and for the academic year 1991/92 he served as Assessor. From 1992 until 1999 he worked at Imperial College London as Sir Edward Frankland British Petroleum Professor of Inorganic Chemistry (1992–99) and Dean of the Royal College of Science (1996–99).

As observed, the Dehn invariant is an invariant for the dissection of polyhedra, in the sense that cutting up a polyhedron into smaller polyhedral pieces and then reassembling them into a different polyhedron does not change the Dehn invariant of the result. Another such invariant is the volume of the polyhedron. Therefore, if it is possible to dissect one polyhedron into a different polyhedron , then both and must have the same Dehn invariant as well as the same volume. extended this result by proving that the volume and the Dehn invariant are the only invariants for this problem.

Naturally occurring ice spikes, often in the form of circular ice candles or polyhedral ice towers (usually triangular), are occasionally found in containers of frozen rainwater or tapwater. Water expands by 9% as it freezes into ice and the simplest shape of an ice crystal that reflects its internal structure is a hexagonal prism. The top and bottom faces of the crystal are hexagonal planes called basal planes and the direction that is perpendicular to the basal planes is called the c-axis. The process begins when surface water nucleates around irregularities where it meets the container wall and freezes inward.

One stronger form of the circle packing theorem, on representing planar graphs by systems of tangent circles, states that every polyhedral graph can be represented by a polyhedron with a midsphere. The horizon circles of a canonical polyhedron can be transformed, by stereographic projection, into a collection of circles in the Euclidean plane that do not cross each other and are tangent to each other exactly when the vertices they correspond to are adjacent.; . Schramm states that the existence of an equivalent polyhedron with a midsphere was claimed by , but that Koebe only proved this result for polyhedra with triangular faces.

Code_Saturne is a general-purpose computational fluid dynamics free computer software package. Developed since 1997 at Électricité de France R&D;, Code_Saturne is distributed under the GNU GPL licence. It is based on a co- located finite-volume approach that accepts meshes with any type of cell (tetrahedral, hexahedral, prismatic, pyramidal, polyhedral...) and any type of grid structure (unstructured, block structured, hybrid, conforming or with hanging nodes...). Its basic capabilities enable the handling of either incompressible or expandable flows with or without heat transfer and turbulence (mixing length, 2-equation models, v2f, Reynolds stress models, Large eddy simulation...).

The EPH must be approximated in the IDM technique before the decision maps are displayed. Methods for approximating the EPH depend on the convexity properties of the EPH. Approximation methods are typically based either on approximation of the EPH by a convex polyhedral set or on approximation of the EPH by a large but finite number of domination cones in objective space with vertices that are close to the Pareto front. The first form can be applied only in the convex problems, while the second form is universal and can be used in general nonlinear problems.

A Moravian star hung outside a church A polyhedron which does not cross itself, such that all of the interior can be seen from one interior point, is an example of a star domain. The visible exterior portions of many self-intersecting star polyhedra form the boundaries of star domains, but despite their similar appearance, as abstract polyhedra these are different structures. For instance, the small stellated dodecahedron has 12 pentagram faces, but the corresponding star domain has 60 isosceles triangle faces, and correspondingly different numbers of vertices and edges. Polyhedral star domains appear in various types of architecture, usually religious in nature.

The effect is largest for cations with high charge and low C.N. (especially when r+/r- approaches the lower limit of the polyhedral stability). As one example, Pauling considered the three mineral forms of titanium dioxide, each with a coordination number of 6 for the Ti4+ cations. The most stable (and most abundant) form is rutile, in which the coordination octahedra are arranged so that each one shares only two edges (and no faces) with adjoining octahedra. The other two, less stable, forms are brookite and anatase, in which each octahedron shares three and four edges respectively with adjoining octahedra.

In polyhedral combinatorics, the Gale transform turns the vertices of any convex polytopes into a set of vectors or points in a space of a different dimension, the Gale diagram of the polytope. It can be used to describe high- dimensional polytopes with few vertices, by transforming them into sets of points in a space of a much lower dimension. The process can also be reversed, to construct polytopes with desired properties from their Gale diagrams. The Gale transform and Gale diagram are named after David Gale, who introduced these methods in a 1956 paper on neighborly polytopes.

In mathematics, the vertex enumeration problem for a polytope, a polyhedral cell complex, a hyperplane arrangement, or some other object of discrete geometry, is the problem of determination of the object's vertices given some formal representation of the object. A classical example is the problem of enumeration of the vertices of a convex polytope specified by a set of linear inequalities:Eric W. Weisstein CRC Concise Encyclopedia of Mathematics, 2002, , p. 3154, article "vertex enumeration" :Ax \leq b where A is an m×n matrix, x is an n×1 column vector of variables, and b is an m×1 column vector of constants.

The following characterization describes tropical varieties intrinsically without reference to algebraic varieties and tropicalization. A set V in \R^n is an irreducible tropical variety if it is the support of a weighted polyhedral complex of pure dimension d that satisfies the zero-tension condition and is connected in codimension one. When d is one, the zero-tension condition means that around each vertex, the weighted-sum of the out-going directions of edges equals zero. For higher dimension, sums are taken instead around each cell of dimension d-1 after quotienting out the affine span of the cell.

The effect of comorbid pathologies on clinical implications, diagnosis, prognosis and therapy of many diseases is polyhedral and patient-specific. The interrelation of the disease, age and drug pathomorphism greatly affect the clinical presentation and progress of the primary nosology, character and severity of the complications, worsens the patient's life quality and limit or make difficult the remedial-diagnostic process. Comorbidity affects life prognosis and increases the chances of fatality. The presence of comorbid disorders increases bed days, disability, hinders rehabilitation, increases the number of complications after surgical procedures, and increases the chances of decline in aged people.

The carboxysome shell is thought to be only sparingly permeable to carbon dioxide, which results in an effective increase in carbon dioxide concentration around RuBisCO, thus enhancing carbon fixation. Mutants that lack genes coding for the carboxysome shell display a high carbon requiring phenotype due to the loss of the concentration of carbon dioxide, resulting in increased oxygen fixation by RuBisCO. The shells have also been proposed to restrict the diffusion of oxygen, thus preventing the oxygenase reaction, reducing wasteful photorespiration. Electron micrograph of Synechococcus elongatus PCC 7942 cell showing the carboxysomes as polyhedral dark structures.

Deadlands features a unique way of creating playing characters for the game. Instead of spending character points, or randomly rolling dice, a character's abilities are determined by drawing cards from a standard 54-card poker deck (jokers included), which determine the character's Traits (their basic attributes). The game also uses polyhedral dice (d4, d6, d8, d10, d12, and d20) which are referred to as the "Bones", and a set of white, red, and blue poker chips called "Fate Chips".The Classic Deadlands mini-chips set used a 4:2:1 ratio of 40 white, 20 red and 10 blue chips.

Example of a alt= In 3D computer graphics and solid modeling, a polygon mesh' is a collection of ', s and s that defines the shape of a polyhedral object. The faces usually consist of triangles (triangle mesh), quadrilaterals (quads), or other simple convex polygons (n-gons), since this simplifies rendering, but may also be more generally composed of concave polygons, or even polygons with holes. The study of polygon meshes is a large sub-field of computer graphics (specifically 3D computer graphics) and geometric modeling. Different representations of polygon meshes are used for different applications and goals.

In the same manner as Holbein, Kratzer's talents obtained him a court position as astronomer and clock maker to the king. Sundial by Kratzer, formerly in the garden of Corpus Christi College, Oxford.Kratzer also collaborated with Holbein on producing maps, and in return the artist produced a portrait of Kratzer in 1528 that now hangs in the Louvre; it depicts the craftsman surrounded by the tools of his trade, and with an unfinished polyhedral sundial. His close relationship with Holbein and More also may be observed in his annotations of Holbein's draft for his portrait of the More family.

Cauchy's theorem is a theorem in geometry, named after Augustin Cauchy. It states that convex polytopes in three dimensions with congruent corresponding faces must be congruent to each other. That is, any polyhedral net formed by unfolding the faces of the polyhedron onto a flat surface, together with gluing instructions describing which faces should be connected to each other, uniquely determines the shape of the original polyhedron. For instance, if six squares are connected in the pattern of a cube, then they must form a cube: there is no convex polyhedron with six square faces connected in the same way that does not have the same shape.

He made original contributions to Max Dehn's theory of the equivalence of polyhedra under polyhedral dissection and reassembly (scissors-congruence), extending and generalizing the theory with an entire class of new relations. Nicoletti collaborated in the writing of Enciclopedia Hoepli delle Matematiche elementari e complementi (published from 1930 to 1951) with the contribution of two monographic articles: Forme razionali di una o più variabili (Rational forms of one or more variables) and Proprietà generali delle funzioni algebriche (General properties of algebraic functions). A leading expert in mathematics education, he edited with Roberto Marcolongo a series of successful editions for secondary schools. Nicoletti was an Invited Speaker of the ICM in Rome.

Face diagram of a square pyramid showing one of its flags In (polyhedral) geometry, a flag is a sequence of faces of a polytope, each contained in the next, with exactly one face from each dimension. More formally, a flag ψ of an n-polytope is a set {F−1, F0, ..., Fn} such that Fi ≤ Fi+1 (−1 ≤ i ≤ n − 1) and there is precisely one Fi in ψ for each i, (−1 ≤ i ≤ n). Since, however, the minimal face F−1 and the maximal face Fn must be in every flag, they are often omitted from the list of faces, as a shorthand. These latter two are called improper faces.

Stars is a wood engraving print; that is, it was produced by carving the artwork into the end grain of a block of wood (unlike a woodcut which uses the side grain), and then using this block to print the image. It was created by Escher in October 1948. Although most published copies of Stars are monochromatic, with white artwork against a black background, the copy in the National Gallery of Canada is tinted in different shades of turquoise, yellow, green, and pale pink. The print depicts a hollowed-out compound of three octahedra, a polyhedral compound composed of three interlocking regular octahedra, floating in space.

From the beginning of his geometry research, Williams considered polyhedral geometry as the basis of a Form Language comprising three levels: the Formative (geometry), the Purportive (psychology), and the Symbolic. With respect to the symbolic Level, he followed the lead of symbologist and mythographer Robert Lawlor.Lawlor, R. Sacred Geometry: Philosophy and practice, London: Thames & Hudson, 1989 (1st edition 1979, 1980, or 1982), In The Integration of Universal Constants Williams presented relationships among numerous diverse subjects: geometric form, color spectrum, the music octave, the periodic table, astronomy, astrology, psychology, tarot, chakras, gender, seasons of the year, among others. The relationships are depicted in six integrated cosmology charts.

By Steinitz's theorem, these graphs are exactly the polyhedral graphs, the graphs of convex polyhedra. A planar graph is 3-vertex-connected if and only if its dual graph is 3-vertex-connected. More generally, a planar graph has a unique embedding, and therefore also a unique dual, if and only if it is a subdivision of a 3-vertex-connected planar graph (a graph formed from a 3-vertex-connected planar graph by replacing some of its edges by paths). For some planar graphs that are not 3-vertex- connected, such as the complete bipartite graph , the embedding is not unique, but all embeddings are isomorphic.

Its first introduces the subject and gives an overview of some important result, after which the second chapter provides background material on non-Archimedean ordered field, algebraic varieties, convex polytopes, and Gröbner bases. Chapter three concerns tropical varieties, defined in several different ways, correspondences between classical varieties and their tropicalizations, the "Fundamental Theorem of Tropical Geometry" proving that these definitions are equivalent, and tropical intersection theory. Chapter four studies tropical connections to the Grassmannian, neighbor joining in the space of metric trees, and matroids. chapter five considers tropical analogues of some of the important concepts in linear algebra, and chapter six connects tropical varieties to toric varieties and polyhedral geometry.

Subgroups of the projective orthogonal group correspond to subgroups of the orthogonal group that contain -I (that have central symmetry). As always with a quotient map (by the lattice theorem), there is a Galois connection between subgroups of O and PO, where the adjunction on O (given by taking the image in PO and then the preimage in O) simply adds -I if absent. Of particular interest are discrete subgroups, which can be realized as symmetries of projective polytopes – these correspond to the (discrete) point groups that include central symmetry. Compare with discrete subgroups of the Spin group, particularly the 3-dimensional case of binary polyhedral groups.

Kepler's Platonic solid model of the Solar System from Mysterium Cosmographicum (1596) The Platonic solids have been known since antiquity. It has been suggested that certain carved stone balls created by the late Neolithic people of Scotland represent these shapes; however, these balls have rounded knobs rather than being polyhedral, the numbers of knobs frequently differed from the numbers of vertices of the Platonic solids, there is no ball whose knobs match the 20 vertices of the dodecahedron, and the arrangement of the knobs was not always symmetric. The ancient Greeks studied the Platonic solids extensively. Some sources (such as Proclus) credit Pythagoras with their discovery.

Polyhedral Frameworks support dependence analysis in a variety of ways, helping to capture the impact of symbolic terms, identify conditional dependences, and separating out the effects of memory aliasing. The effects of memory aliasing, in particular, have been described in two ways: many authors distinguish between "true" data dependences (corresponding to actual flow of information) from false dependences arising from memory aliasing or limited precision of dependence analysis. The Omega Project publications use specific terms to identify specific effects on analysis. They maintain the traditional distinction of flow-, output-, and anti-dependences, based on the types of array access (write to read, write to write, or read to write, respectively).

He has brought synthesis and design to the field of biopolymers and the methodology of nucleic acids to the field of molecular recognition. His pioneering research at the interface of chemistry and biology has contributed greatly to a set of general chemical principles for sequence specific recognition at single sites in the human genome." # M. Frederick Hawthorne 1994 "For outstanding contributions to the fields of inorganic chemistry and organometallic chemistry through his seminal discoveries n the rapidly expanding area of borane clusters. Inparticular, his work has provided pioneering insights into the syntheses, structures, bonding, and reactivity patterns of polyhedral borane anions, carboranes, and metallocarboranes.

She has described the emergence of shape and patterns in membranes and in multicomponent complex mixtures. She and her students and postdocs discovered that electrostatics leads to spontaneous symmetry breaking in ionic membranes such as viral capsids (for which they were awarded the 2007 Cozzarelli Prize) and in fibers. They also demonstrated the spontaneous emergence of various regular and irregular polyhedral geometries in closed membranes with non-homogeneous elastic properties such as bacterial microcompartments, including carboxysomes, via a mechanism that explains observed shapes in crystalline shells formed by more than one component such as archaea and organelle wall envelopes as well as in ionic vesicles.

The Sylvester–Gallai theorem has been proved in many different ways. Gallai's 1944 proof switches back and forth between Euclidean and projective geometry, in order to transform the points into an equivalent configuration in which an ordinary line can be found as a line of slope closest to zero; for details, see . The 1941 proof by Melchior uses projective duality to convert the problem into an equivalent question about arrangements of lines, which can be answered using Euler's polyhedral formula. Another proof by Leroy Milton Kelly shows by contradiction that the connecting line with the smallest nonzero distance to another point must be ordinary.

The Protoclassic is growing in acceptance as a distinct period in Maya history, but is generally referred to as the Terminal Preclassic (0 – 250 AD). Increases in obsidian production technology, procurement, and distribution can be used as lines of evidence in this debate. In Copan and its hinterland regions the pattern of large flakes spalls and small nodules continued until the late Protoclassic when the population increased and a subsequent rise in production technology (Aoyama 2001). Polyhedral cores and blade production debitage are noted in assemblages related to principle urban group residences suggesting political control by a ruler over obsidian trade and distribution (Aoyama 2001).

Obsidian found at Wild Cane Cay is primarily from Highland Guatemala sources but there is some from Central Mexico, with an 80% increase in overall densities mainly in the form of cores during the Postclassic (Mckillop 1989). An estimated 21,686 cores overwhelmingly linked to production areas is recorded for the Postclassic at this site which far exceeds household needs (Mckillop 1989). This positions Wild Cane Cay as an important port of trade in the Postclassic exchange system of obsidian. Ambergis Key on the coast of Belize shows procurement of already reduced polyhedral cores primarily from Ixtepeque obsidian, but other Guatemalan sources are noted (Stemp et al. 2011).

A sculpture of the small stellated dodecahedron in M. C. Escher's Gravitation, near the Mesa+ Institute of Universiteit Twente A polyhedron model is a physical construction of a polyhedron, constructed from cardboard, plastic board, wood board or other panel material, or, less commonly, solid material. Since there are 75 uniform polyhedra, including the five regular convex polyhedra, five polyhedral compounds, four Kepler-Poinsot polyhedra, and thirteen Archimedean solids, constructing or collecting polyhedron models has become a common mathematical recreation. Polyhedron models are found in mathematics classrooms much as globes in geography classrooms. Polyhedron models are notable as three-dimensional proof-of-concepts of geometric theories.

Description of Acton Court with a picture of the stone sundial Peter Drinkwater has presented a critical evaluation of the sundials attributed to Kratzer, in particular the one in the Holbein portrait. He comments that "Kratzer triumphed, not through genius or creativity, but through having learned what others had discovered and invented, and by being the first to apply that learning in England". John North concurs: "Kratzer doubtless had nothing new to offer of a fundamental kind. Many of his dials were unusual, but his favorite polyhedral dial was perhaps more useful as a repository of verses [...] than for actually announcing the time with any accuracy".

Model of an icosahedron made with metallic spheres and magnetic connectors The 12 edges of a regular octahedron can be subdivided in the golden ratio so that the resulting vertices define a regular icosahedron. This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly subdividing each edge into the golden mean along the direction of its vector. The five octahedra defining any given icosahedron form a regular polyhedral compound, while the two icosahedra that can be defined in this way from any given octahedron form a uniform polyhedron compound. Regular icosahedron and its circumscribed sphere.

See for proofs showed that Barnette's conjecture is equivalent to a superficially stronger statement, that for every two edges e and f on the same face of a bipartite cubic polyhedron, there exists a Hamiltonian cycle that contains e but does not contain f. Clearly, if this statement is true, then every bipartite cubic polyhedron contains a Hamiltonian cycle: just choose e and f arbitrarily. In the other directions, Kelmans showed that a counterexample could be transformed into a counterexample to the original Barnette conjecture. Barnette's conjecture is also equivalent to the statement that the vertices of the dual of every cubic bipartite polyhedral graph can be partitioned into two subsets whose induced subgraphs are trees.

She became interested in clathrin, a protein that is key to both inward and outward membrane trafficking in cells, shortly after its isolation by Barbara Pearse. She used monoclonal antibodies to map the structure of clathrin and to probe its assembly into its characteristic polyhedral structures. Clathrin-mediated endocytosis is involved in the uptake of antigens from outside the cell that are eventually presented on the surface of the cell by the major histocompatibility complex class II (MHC class II). Brodsky discovered that the pathway of MHC class II export meets the antigen import pathway in a specialized endocytic compartment where antigens can be processed into peptides and loaded onto the MHC class II molecule for presentation.

As the original site is now under water it is unlikely that the dispute will ever be resolved. It is acknowledged that the square shape results solely from interpretations by José Pires Gonçalves, a doctor and an amateur archaeologist from Reguengos de Monsaraz, previously responsible for identifying the nearby Menhir of Outeiro, who identified the stones as a cromlech in 1969, having been alerted to the existence of the central phallic stone by two local residents, José Cruz e Leonel Franco. The 55 menhirs are of different types of granite of local origin. These are mainly between 0.37 and 2.10 meters in height, with varying shapes (ovoid, slightly flattened, cylindrical, sub-square, conical or polyhedral).

Where the components of the crystal lie at the centres of the polyhedra it forms one of the Frank–Kasper phases.. . Where the components of the crystal lie at the corners of the polyhedra, it is known as the "Type I clathrate structure". Gas hydrates formed by methane, propane and carbon dioxide at low temperatures have a structure in which water molecules lie at the nodes of the Weaire–Phelan structure and are hydrogen bonded together, and the larger gas molecules are trapped in the polyhedral cages. Some alkali metal silicides and germanides also form this structure (Si/Ge at nodes, alkali metals in cages), as does the silica mineral melanophlogite (silicon at nodes, linked by oxygen along edges).

Click here for an animated version. The Herschel graph is planar and 3-vertex- connected, so it follows by Steinitz's theorem that it is a polyhedral graph: there exists a convex polyhedron (an enneahedron) having the Herschel graph as its skeleton.. This polyhedron has nine quadrilaterals for faces, which can be chosen to form three rhombi and six kites. Its dual polyhedron is a rectified triangular prism, formed as the convex hull of the midpoints of the edges of a triangular prism. This polyhedron has the property that its faces cannot be numbered in such a way that consecutive numbers appear on adjacent faces, and such that the first and last number are also on adjacent faces.

By Steinitz's theorem, the Goldner–Harary graph is a polyhedral graph: it is planar and 3-connected, so there exists a convex polyhedron having the Goldner–Harary graph as its skeleton. Geometrically, a polyhedron representing the Goldner–Harary graph may be formed by gluing a tetrahedron onto each face of a triangular dipyramid, similarly to the way a triakis octahedron is formed by gluing a tetrahedron onto each face of an octahedron. That is, it is the Kleetope of the triangular dipyramid.. Same page, 2nd ed., Graduate Texts in Mathematics 221, Springer-Verlag, 2003, .. The dual graph of the Goldner–Harary graph is represented geometrically by the truncation of the triangular prism.

Gliders are typically partial to slow flying and have high aspect ratio, as well as very low wing loading (weight to wing area ratio). Two and three-channel gliders which use only rudder control for steering and dihedral or polyhedral wing shape to automatically counteract rolling are popular as training craft, due to their ability to fly very slowly and high tolerance to error. Powered gliders have recently seen an increase in popularity. By combining the efficient wing size and wide speed envelope of a glider airframe with an electric motor, it is possible to achieve long flight times and high carrying capacity, as well as glide in any suitable location regardless of thermals or lift.

In algebraic geometry, a Du Val singularity, also called simple surface singularity, Kleinian singularity, or rational double point, is an isolated singularity of a complex surface which is modeled on a double branched cover of the plane, with minimal resolution obtained by replacing the singular point with a tree of smooth rational curves, with intersection pattern dual to a Dynkin diagram of A-D-E singularity type. They are the canonical singularities (or, equivalently, rational Gorenstein singularities) in dimension 2. They were studied by and Felix Klein. The Du Val singularities also appear as quotients of C2 by a finite subgroup of SL2(C); equivalently, a finite subgroup of SU(2), which are known as binary polyhedral groups.

In geometry, the Dehn invariant of a polyhedron is a value used to determine whether polyhedra can be dissected into each other or whether they can tile space. It is named after Max Dehn, who used it to solve Hilbert's third problem on whether all polyhedra with equal volume could be dissected into each other. Two polyhedra have a dissection into polyhedral pieces that can be reassembled into either one, if and only if their volumes and Dehn invariants are equal. A polyhedron can be cut up and reassembled to tile space if and only if its Dehn invariant is zero, so having Dehn invariant zero is a necessary condition for being a space-filling polyhedron.

The book has 19 chapters. After two chapters introducing background material in linear algebra, topology, and convex geometry, two more chapters provide basic definitions of polyhedra, in their two dual versions (intersections of half-spaces and convex hulls of finite point sets), introduce Schlegel diagrams, and provide some basic examples including the cyclic polytopes. Chapter 5 introduces Gale diagrams, and the next two chapters use them to study polytopes with a number of vertices only slightly higher than their dimension, and neighborly polytopes. Chapters 8 through 11 study the numbers of faces of different dimensions in polytopes through Euler's polyhedral formula, the Dehn–Sommerville equations, and the extremal combinatorics of numbers of faces in polytopes.

The "Eureka moment" came when the computer simulation showed sharp ten-fold diffraction patterns, similar to the observed ones, emanating from the three-dimensional structure devoid of periodicity. The multiple polyhedral structure was termed later by many researchers as icosahedral glass but in effect it embraces any arrangement of polyhedra connected with definite angles and distances (this general definition includes tiling, for example). Shechtman accepted Blech's discovery of a new type of material and it gave him the courage to publish his experimental observation. Shechtman and Blech jointly wrote a paper entitled "The Microstructure of Rapidly Solidified Al6Mn" and sent it for publication around June 1984 to the Journal of Applied Physics (JAP).

After the release of the AD&D; game, the Basic Set saw a major revision in 1981 by editor Tom Moldvay. The game was not brought in line with AD&D; but instead further away from that ruleset, and thus the basic D&D; game became a separate and distinct product line from AD&D.; The former was promoted as a continuation of the tone of original D&D;, while AD&D; was an advancement of the mechanics. The revised version of the set included a larger, sixty-four page rule book with a red border and a color cover by Erol Otus, the module B2 The Keep on the Borderlands, six polyhedral dice, and a marking crayon.

When the two curves are embedded in a metric space other than Euclidean space, such as a polyhedral terrain or some Euclidean space with obstacles, the distance between two points on the curves is most naturally defined as the length of the shortest path between them. The leash is required to be a geodesic joining its endpoints. The resulting metric between curves is called the geodesic Fréchet distance... Cook and Wenk describe a polynomial-time algorithm to compute the geodesic Fréchet distance between two polygonal curves in a simple polygon. If we further require that the leash must move continuously in the ambient metric space, then we obtain the notion of the homotopic Fréchet distance.

In 1952-53 he again returned to Europe and went to China in 1954. Between 1961 and 1974 Sérgio de Camargo remained in Paris, where he became a member of the Groupe de Recherche d’Art Visuel (GRAV) in 1963. During that period he concentrated on structuring monochrome white surfaces some in "Polyhedral Volumes of Mutable Readings" using parallelepiped shapes and others with cylindrical wooden reliefs, in both cases proposing the play of lights and shadows alternating between order and chaos, fullness and emptiness. As retold by Guy Brett, a curator and friend: “Cutting an apple to eat it, he sliced off nearly half and then made another cut at a different angle to take a piece out.

Large-scale deaths of teak defoliator larvae characterized by cessation of feeding, flaccidity and subsequent liquefaction of body tissues have been reported by Stebbing as early as 1903. During 1985, an investigation of microbial pathogens of H. puera was undertaken in the plantations of Nilambur Forest Divisions of Kerala, India, by KFRI detected several dead insects with the characteristic symptoms as observed by Stebbing. Microscopic observation of tissues revealed the presence of refractile polyhedral inclusions bodies, which stained blue in Giemsa, measured 0.9–2.4 micrometers in diameter in the scanning electron micrograph taken by Jean Adams at USDA, confirmed its identity as NPV. The NPVs come under the family of baculoviridae and its virions are enveloped rod shaped nucleocapsids containing circular, supercoiled, double stranded DNA.

There are two traditional methods for making polyhedra out of paper: polyhedral nets and modular origami. In the net method, the faces of the polyhedron are placed to form an irregular shape on a flat sheet of paper, with some of these faces connected to each other within this shape; it is cut out and folded into the shape of the polyhedron, and the remaining pairs of faces are attached together. In the modular origami method, many similarly-shaped "modules" are each folded from a single sheet of origami paper, and then assembled to form a polyhedron, with pairs of modules connected by the insertion of a flap from one module into a slot in another module. This book does neither of those two things.

Descartes' theorem on total angular defect of a polyhedron is the polyhedral analog: it states that the sum of the defect at all the vertices of a polyhedron which is homeomorphic to the sphere is 4π. More generally, if the polyhedron has Euler characteristic \chi=2-2g (where g is the genus, meaning "number of holes"), then the sum of the defect is 2\pi \chi. This is the special case of Gauss–Bonnet, where the curvature is concentrated at discrete points (the vertices). Thinking of curvature as a measure, rather than as a function, Descartes' theorem is Gauss–Bonnet where the curvature is a discrete measure, and Gauss–Bonnet for measures generalizes both Gauss–Bonnet for smooth manifolds and Descartes' theorem.

Eluvathingal Devassy Jemmis or E. D. Jemmis (born 31 October 1951) is a Professor of theoretical chemistry at the Indian Institute of Science, Bangalore, India. He was the founding Director of Indian Institute of Science Education and Research, Thiruvananthapuram (IISER-TVM). His primary area of research is applied theoretical chemistry with emphasis on structure, bonding and reactivity, across the periodic table of the elements. Apart from many of his contributions to applied theoretical chemistry, an equivalent of the structural chemistry of carbon, as exemplified by the Huckel 4n+2 Rule, benzenoid aromatics and graphite, and tetrahedral carbon and diamond, is brought in the structural chemistry of boron by the Jemmis mno rules which relates polyhedral and macropolyhedral boranes to allotropes of boron and boron-rich solids.

His publications covers a wide range of topics in graph theory and combinatorics: convex polyhedra, quasigroups, special decompositions into Hamiltonian paths, Latin squares, decompositions of complete graphs, perfect systems of difference sets, additive sequences of permutations, tournaments and combinatorial games theory. The triakis icosahedron, a polyhedron in which every edge has endpoints with total degree at least 13 One of his results, known as Kotzig's theorem, is the statement that every polyhedral graph has an edge whose two endpoints have total degree at most 13. An extreme case is the triakis icosahedron, where no edge has smaller total degree. Kotzig published the result in Slovakia in 1955, and it was named and popularized in the west by Branko Grünbaum in the mid-1970s.

The small cubicuboctahedron is a polyhedral immersion of the tiling of the Klein quartic by 56 triangles, meeting at 24 vertices. Note each face in the polyhedron consist of multiple faces in the tiling – two triangular faces constitute a square face and so forth, as per this explanatory image. The smallest Hurwitz group is the projective special linear group PSL(2,7), of order 168, and the corresponding curve is the Klein quartic curve. This group is also isomorphic to PSL(3,2). Next is the Macbeath curve, with automorphism group PSL(2,8) of order 504. Many more finite simple groups are Hurwitz groups; for instance all but 64 of the alternating groups are Hurwitz groups, the largest non-Hurwitz example being of degree 167.

"Plato postulated that the basic atomic components of the five Elements each had the form of one of the five regular solids." Similar Platonic geometric symbolismColumn: The crucible, by Philip Ball, 26 June 2009 survive on the contemporary monument to Sir Anthony Ashley (1551-1627/8) (Clerk to the Privy Council) at nearby Wimborne St Giles, erected by his son-in-law Anthony Ashley-Cooper, 1st Earl of Shaftesbury (1621–1683). "All the known examples of these polyhedral sculptures originate within a period of about 30 years, during which England and the rest of Europe saw a resurgence of interest in quasi-mystical geometric symbolism". T Tarnai and J Krähling, Proceedings of the IASS-SLTE 2008 Symposium, 27-31 October 2008, Acapulco, Mexico (ed.

While the examples of perspective in Underweysung der Messung are underdeveloped and contain inaccuracies, there is a detailed discussion of polyhedra. Dürer is also the first to introduce in text the idea of polyhedral nets, polyhedra unfolded to lie flat for printing. Dürer published another influential book on human proportions called Vier Bücher von Menschlicher Proportion (Four Books on Human Proportion) in 1528. Salvador Dalí's Crucifixion (Corpus Hypercubus), 1954, depicts Christ upon the mathematical net of a hypercube, (oil on canvas, 194.3 × 123.8 cm, Metropolitan Museum of Art, New York)Rudy Rucker, The Fourth Dimension: Toward a Geometry of Higher Reality, Courier Corporation, 2014, Dürer's well-known engraving Melencolia I depicts a frustrated thinker sitting by a truncated triangular trapezohedron and a magic square.

When code contains a mixture of affine and non-affine terms, polyhedral libraries can, in principle, be used to produce approximate results, for example by simply omitting such terms when it is safe to do so. In addition to providing a way to flag such approximate results, the Omega Library allows restricted uses of "Uninterpreted Function Symbols" to stand in for any nonlinear term, providing a system that slightly improves the result of dependence analysis and (probably more significantly) provides a language for communication about these terms (to drive other analysis or communication with the programmer). Pugh and Wonnacott discussed a slightly less restricted domain than that allowed in the library, but this was never implemented (a description exists in Wonnacott's dissertation).

The book was predrilled for use in a three-ringed binder, and the complete set of polyhedral dice came in a heat-sealed bag with a small wax crayon to use in coloring the numbers on the dice. The revised rulebook was visually distinct from the previous version: the Holmes booklet had a monochrome pale blue cover, while the Moldvay rulebook had a bright red cover. With the revision of the Basic Set, discrete rulesets for higher character levels were introduced as expansions for the basic game. The Moldvay Basic Set was immediately followed by the accompanying release of an Expert Set edited by Dave Cook with Steve Marsh that supported character levels four through fourteen, with the intent that players would continue with the Expert Set.

He studied for his PhD in Agriculture Entomology at the Kansas State University and specialized in Grain Science Technology. He did a year of post-doctoral work in the Department of Entomology, Texas A&M; University, (1969), on problems related to isolation and purification of polyhedral virus (6). He joined University of Minnesota in 1970 as a faculty in the Department of Entomology, Fisheries and Wildlife. He worked on isolation and characterization fungal toxins (10,11). He joined Prof Marion Andrews in the Department of Pharmacology in 1971 and worked on drug metabolizing enzymes (12,13). In 1972 he started working with James G White in the Department of Pediatrics, who was working on the morphology and ultra structure of blood platelets.

The students are attempting to prove the formula for the Euler characteristic in algebraic topology, which is a theorem about the properties of polyhedra, namely that for all polyhedra the number of their Vertices minus the number of their Edges plus the number of their Faces is 2: (V – E + F = 2). The dialogue is meant to represent the actual series of attempted proofs which mathematicians historically offered for the conjecture, only to be repeatedly refuted by counterexamples. Often the students paraphrase famous mathematicians such as Cauchy, as noted in Lakatos's extensive footnotes. Lakatos termed the polyhedral counterexamples to Euler's formula monsters and distinguished three ways of handling these objects: Firstly, monster-barring, by which means the theorem in question could not be applied to such objects.

Thompson analyses the polyhedral forms of Radiolaria from the Challenger expedition drawn by Ernst Haeckel, 1904. D'Arcy Wentworth Thompson's most famous work, On Growth and Form was written in Dundee, mostly in 1915, but publication was put off until 1917 because of the delays of wartime and Thompson's many late alterations to the text. The central theme of the book is that biologists of its author's day overemphasized evolution as the fundamental determinant of the form and structure of living organisms, and underemphasized the roles of physical laws and mechanics. At a time when vitalism was still being considered as a biological theory, he advocated structuralism as an alternative to natural selection in governing the form of species, with the smallest hint of vitalism as the unseen driving force.

Arch of the entrance portico Towards the south, another dependence of similar size is poured that it pours to the courtyard by a portico of great polilobulated arcades. Again there is a tripartite space, and its east and west ends extend perpendicularly with two lateral galleries which are accessed by wide polyhedral lobes and which end at the end of their arms in separate pointed arches also polilobulated whose alfiz is decorated by complex laqueus and reliefs of arabesques. This whole structure seeks an appearance of solemnity and majesty that the scant depth of these stays would not give a spectator to access the king's room. In addition, all the ornamentation of yeserias of the palace was polychrome in shades of blue and red in the back and gold in the arabesques.

The method continues by setting up a system of linear equations in the vertex coordinates, according to which each remaining vertex should be placed at the average of its neighbors. Then as Tutte showed, this system of equations will have a unique solution in which each face of the graph is drawn as a convex polygon.. The result is almost an equilibrium stress: if one assigns weight one to each interior edge, then each interior vertex of the drawing is in equilibrium. However, it is not always possible to assign negative numbers to the exterior edges so that they, too, are in equilibrium. Such an assignment is always possible when the outer face is a triangle, and so this method can be used to realize any polyhedral graph that has a triangular face.

Numerous other polyhedra and polyhedral compounds float in the background; the four largest are, on the upper left, the compound of cube and octahedron; on the upper right, the stella octangula; on the lower left, a compound of two cubes; and on the lower right, a solid version of the same octahedron 3-compound. The smaller polyhedra visible within the print also include all of the five Platonic solids and the rhombic dodecahedron. In order to depict polyhedra accurately, Escher made models of them from cardboard. Two chameleons are contained within the cage-like shape of the central compound; Escher writes that they were chosen as its inhabitants "because they are able to cling by their legs and tails to the beams of their cage as it swirls through space".

The medial graph of the Herschel graph is a 4-regular planar graph with no Hamiltonian decomposition. The shaded regions correspond to the vertices of the underlying Herschel graph. The Herschel graph also provides an example of a polyhedral graph for which the medial graph cannot be decomposed into two edge-disjoint Hamiltonian cycles. The medial graph of the Herschel graph is a 4-regular graph with 18 vertices, one for each edge of the Herschel graph; two vertices are adjacent in the medial graph whenever the corresponding edges of the Herschel graph are consecutive on one of its faces.. It is 4-vertex-connected and essentially 6-edge-connected, meaning that for every partition of the vertices into two subsets of at least two vertices, there are at least six edges crossing the partition.

Hasik 2008, pp. 47-48. According to aerospace publication Flight International, the MRMF programme had been motivated by the concept that a future fighter could be both lighter and cheaper if it could be so superior at mid-range combat that it could completely eliminate the need to perform any close-range dogfighting-style combat. As such, MBB was required to develop an airframe which possessed a suitable configuration to achieve a forward-facing radar cross-section that would be between 20-30dB (in the X band frequencies) below that of what a conventional fighter would typically achieve. Similar to Lockheed's own approach adopted during its development of the Have Blue demonstrated and production F-117 Nighthawk, MBB's design team harnesses the dimensional principles of an airframe externally covered by polyhedral shapes for the Lampyridae.

Fuller took an intuitive approach to his studies, often going into exhaustive empirical detail while at the same time seeking to cast his findings in their most general philosophical context. For example, his sphere packing studies led him to generalize a formula for polyhedral numbers: 2 P F2 \+ 2, where F stands for "frequency" (the number of intervals between balls along an edge) and P for a product of low order primes (some integer). He then related the "multiplicative 2" and "additive 2" in this formula to the convex versus concave aspects of shapes, and to their polar spinnability respectively. These same polyhedra, developed through sphere packing and related by tetrahedral mensuration, he then spun around their various poles to form great circle networks and corresponding triangular tiles on the surface of a sphere.

During the eighteenth century, Kepler, Nicolas Steno, René Just Haüy, and others gradually associated the packing of Boyle-type corpuscular units into arrays with the apparent emergence of polyhedral structures resembling crystals as a result. During the nineteenth century, there was considerably more work done on polyhedra and also of crystal structure, notably in the derivation of the Crystallographic groups based on the assumption that a crystal could be regarded as a regular array of unit cells. During the early twentieth century, the physics and chemistry community largely accepted Boyle's corpuscular theory of matter—by now called the atomic theory—and X-ray crystallography was used to determine the position of the atomic or molecular components within the unit cells (by the early twentieth century, unit cells were regarded as physically meaningful).

Descartes on Polyhedra: A Study of the "De solidorum elementis" is a book in the history of mathematics, concerning the work of René Descartes on polyhedra. Central to the book is the disputed priority for Euler's polyhedral formula between Leonhard Euler, who published an explicit version of the formula, and Descartes, whose De solidorum elementis includes a result from which the formula is easily derived. Descartes on Polyhedra was written by Pasquale Joseph Federico (1902–1982), and published posthumously by Springer- Verlag in 1982, with the assistance of Federico's widow Bianca M. Federico, as volume 4 of their book series Sources in the History of Mathematics and Physical Sciences. The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries.

Apollonian networks are examples of maximal planar graphs, graphs to which no additional edges can be added without destroying planarity, or equivalently graphs that can be drawn in the plane so that every face (including the outer face) is a triangle. They are also chordal graphs, graphs in which every cycle of four or more vertices has a diagonal edge connecting two non-consecutive cycle vertices, and the order in which vertices are added in the subdivision process that forms an Apollonian network is an elimination ordering as a chordal graph. This forms an alternative characterization of the Apollonian networks: they are exactly the chordal maximal planar graphs or equivalently the chordal polyhedral graphs.The equivalence of planar 3-trees and chordal maximal planar graphs was stated without proof by .

The cube and regular octahedron are dual graphs of each other According to Steinitz's theorem, every polyhedral graph (the graph formed by the vertices and edges of a three- dimensional convex polyhedron) must be planar and 3-vertex-connected, and every 3-vertex-connected planar graph comes from a convex polyhedron in this way. Every three-dimensional convex polyhedron has a dual polyhedron; the dual polyhedron has a vertex for every face of the original polyhedron, with two dual vertices adjacent whenever the corresponding two faces share an edge. Whenever two polyhedra are dual, their graphs are also dual. For instance the Platonic solids come in dual pairs, with the octahedron dual to the cube, the dodecahedron dual to the icosahedron, and the tetrahedron dual to itself.. Polyhedron duality can also be extended to duality of higher dimensional polytopes,.

Each of these steps may be performed with simple trigonometric calculations, and as Collins and Stephenson argue, the system of radii converges rapidly to a unique fixed point for which all covering angles are exactly 2π. Once the system has converged, the circles may be placed one at a time, at each step using the positions and radii of two neighboring circles to determine the center of each successive circle. describes a similar iterative technique for finding simultaneous packings of a polyhedral graph and its dual, in which the dual circles are at right angles to the primal circles. He proves that the method takes time polynomial in the number of circles and in log 1/ε, where ε is a bound on the distance of the centers and radii of the computed packing from those in an optimal packing.

Tutte's work in graph theory includes the structure of cycle spaces and cut spaces, the size of maximum matchings and existence of k-factors in graphs, and Hamiltonian and non-Hamiltonian graphs. He disproved Tait's conjecture, on the Hamiltonicity of polyhedral graphs, by using the construction known as Tutte's fragment. The eventual proof of the four colour theorem made use of his earlier work. The graph polynomial he called the "dichromate" has become famous and influential under the name of the Tutte polynomial and serves as the prototype of combinatorial invariants that are universal for all invariants that satisfy a specified reduction law. The first major advances in matroid theory were made by Tutte in his 1948 Cambridge PhD thesis which formed the basis of an important sequence of papers published over the next two decades.

400px The dicyclic group is a binary polyhedral group — it is one of the classes of subgroups of the Pin group Pin−(2), which is a subgroup of the Spin group Spin(3) — and in this context is known as the binary dihedral group. The connection with the binary cyclic group C2n, the cyclic group Cn, and the dihedral group Dihn of order 2n is illustrated in the diagram at right, and parallels the corresponding diagram for the Pin group. Coxeter writes the binary dihedral group as ⟨2,2,n⟩ and binary cyclic group with angle-brackets, ⟨n⟩. There is a superficial resemblance between the dicyclic groups and dihedral groups; both are a sort of "mirroring" of an underlying cyclic group. But the presentation of a dihedral group would have x2 = 1, instead of x2 = an; and this yields a different structure.

Obsidian trade was largely relegated to the coast with the collapse of Classic Maya society in the Northern and Southern Lowland regions that occupied the inland areas the Yucutan and River basins. The period from 900-1500 AD saw 80% of the population in the Yucutan remain within 50 km of the coast (Rathje and Sabloff 1973). Chichen Itza and Cozumel were used as trading bases by invaders in the Early Postclassic but after a mainland collapse the centralized commercial systems in place collapsed as well (Rathje and Sabloff 1973). For Postclassic Copan, an obsidian pattern similar to the Preclassic returned; nonspecialized production utilizing Ixtepeque obsidian used smaller flakes as opposed to polyhedral cores, resulting in fewer prismatic blades and an overall decline in the quantity and quality of utilitarian obsidian found at the site (Aoyama 2001).

Although written at a graduate level, the main prerequisites for reading the book are linear algebra and general topology, both at an undergraduate level. In a review of the first edition of the book, Werner Fenchel calls it "a remarkable achievement", "a wealth of material", "well organized and presented in a lucid style". Over 35 years later, in giving the Steele Prize to Grünbaum for Convex Polytopes, the American Mathematical Society wrote that the book "has served both as a standard reference and as an inspiration", that it was in large part responsible for the vibrant ongoing research in polyhedral combinatorics, and that it remained relevant to this area. Reviewing and welcoming the second edition, Peter McMullen wrote that despite being "immediately rendered obselete" by the research that it sparked, the book was still essential reading for researchers in this area.

The fact that the volume of any pyramid, regardless of the shape of the base, whether circular as in the case of a cone, or square as in the case of the Egyptian pyramids, or any other shape, is (1/3) × base × height, can be established by Cavalieri's principle if one knows only that it is true in one case. One may initially establish it in a single case by partitioning the interior of a triangular prism into three pyramidal components of equal volumes. One may show the equality of those three volumes by means of Cavalieri's principle. In fact, Cavalieri's principle or similar infinitesimal argument is necessary to compute the volume of cones and even pyramids, which is essentially the content of Hilbert's third problem – polyhedral pyramids and cones cannot be cut and rearranged into a standard shape, and instead must be compared by infinite (infinitesimal) means.

As an example, consider the case of building a tetrahedron from a triangle, the latter of whose elements are enumerated by row 3 of Pascal's triangle: 1 face, 3 edges, and 3 vertices (the meaning of the final 1 will be explained shortly). To build a tetrahedron from a triangle, we position a new vertex above the plane of the triangle and connect this vertex to all three vertices of the original triangle. The number of a given dimensional element in the tetrahedron is now the sum of two numbers: first the number of that element found in the original triangle, plus the number of new elements, each of which is built upon elements of one fewer dimension from the original triangle. Thus, in the tetrahedron, the number of cells (polyhedral elements) is ; the number of faces is the number of edges is the number of new vertices is .

Convex and strictly convex grid drawings of the same graph In graph drawing, a convex drawing of a planar graph is a drawing that represents the vertices of the graph as points in the Euclidean plane and the edges as straight line segments, in such a way that all of the faces of the drawing (including the outer face) have a convex boundary. The boundary of a face may pass straight through one of the vertices of the graph without turning; a strictly convex drawing asks in addition that the face boundary turns at each vertex. That is, in a strictly convex drawing, each vertex of the graph is also a vertex of each convex polygon describing the shape of each incident face. Every polyhedral graph has a strictly convex drawing, for instance obtained as the Schlegel diagram of a convex polyhedron representing the graph.

One direction of Steinitz's theorem (the easier direction to prove) states that the graph of every convex polyhedron is planar and 3-connected. As shown in the illustration, planarity can be shown by using a Schlegel diagram: if one places a light source near one face of the polyhedron, and a plane on the other side, the shadows of the polyhedron edges will form a planar graph, embedded in such a way that the edges are straight line segments. The 3-connectivity of a polyhedral graph is a special case of Balinski's theorem that the graph of any k-dimensional convex polytope is k-connected.. The other, more difficult, direction of Steinitz's theorem states that every planar 3-connected graph is the graph of a convex polyhedron. There are three standard approaches for this part: proofs by induction, lifting two-dimensional Tutte embeddings into three dimensions using the Maxwell–Cremona correspondence, and methods using the circle packing theorem to generate a canonical polyhedron.

Force-directed methods in graph drawing date back to the work of , who showed that polyhedral graphs may be drawn in the plane with all faces convex by fixing the vertices of the outer face of a planar embedding of the graph into convex position, placing a spring-like attractive force on each edge, and letting the system settle into an equilibrium.. Because of the simple nature of the forces in this case, the system cannot get stuck in local minima, but rather converges to a unique global optimum configuration. Because of this work, embeddings of planar graphs with convex faces are sometimes called Tutte embeddings. The combination of attractive forces on adjacent vertices, and repulsive forces on all vertices, was first used by ;. additional pioneering work on this type of force-directed layout was done by .. The idea of using only spring forces between all pairs of vertices, with ideal spring lengths equal to the vertices' graph-theoretic distance, is from ..

By Steinitz's theorem, the 3-connected planar graphs to which Tutte's spring theorem applies coincide with the polyhedral graphs, the graphs formed by the vertices and edges of a convex polyhedron. According to the Maxwell–Cremona correspondence, a two- dimensional embedding of a planar graph forms the vertical projection of a three-dimensional convex polyhedron if and only if the embedding has an equilibrium stress, an assignment of forces to each edge (affecting both endpoints in equal and opposite directions parallel to the edge) such that the forces cancel at every vertex. For a Tutte embedding, assigning to each edge an attractive force proportional to its length (like a spring) makes the forces cancel at all interior vertices, but this is not necessarily an equilibrium stress at the vertices of the outer polygon. However, when the outer polygon is a triangle, it is possible to assign repulsive forces to its three edges to make the forces cancel there, too.

The book is organized historically, and reviewer Robert Bradley divides the topics of the book into three parts. The first part discusses the earlier history of polyhedra, including the works of Pythagoras, Thales, Euclid, and Johannes Kepler, and the discovery by René Descartes of a polyhedral version of the Gauss–Bonnet theorem (later seen to be equivalent to Euler's formula). It surveys the life of Euler, his discovery in the early 1750s that the Euler characteristic V-E+F is equal to two for all convex polyhedra, and his flawed attempts at a proof, and concludes with the first rigorous proof of this identity in 1794 by Adrien-Marie Legendre, based on Girard's theorem relating the angular excess of triangles in spherical trigonometry to their area. Although polyhedra are geometric objects, Euler's Gem argues that Euler discovered his formula through being the first to view them topologically (as abstract incidence patterns of vertices, faces, and edges), rather than through their geometric distances and angles.

He is known for his results in combinatorial number theory, and in particular for Behrend's theorem on the logarithmic density of sets of integers in which no member of the set is a multiple of any other, and for his construction of large Salem–Spencer sets of integers with no three- element arithmetic progression. Behrend sequences are sequences of integers whose multiples have density one; they are named for Behrend, who proved in 1948 that the sum of reciprocals of such a sequence must diverge. He wrote one paper in algebraic geometry, on the number of symmetric polynomials needed to construct a system of polynomials without nontrivial real solutions, several short papers on mathematical analysis, and an investigation of the properties of geometric shapes that are invariant under affine transformations. After moving to Melbourne his interests shifted to topology, first in the construction of polyhedral models of manifolds, and later in point-set topology.

The use of pecking, grinding, and carving techniques may also be employed to produce figurines, jewelry, eccentrics, or other types of objects. Prismatic blade production, a technique employing a pressure flaking-like technique that removed blades from a polyhedral core, was ubiquitous throughout Mesoamerica. Modern attempts to redesign production techniques are heavily based on Spanish records and accounts of witnessed obsidian knapping. Motolinia, a 16th-century Spanish observer, left this account of prismatic blade production: > It is in this manner: First they get out a knife stone (obsidian core) which > is black like jet and 20 cm or slightly less in length, and they make it > cylindrical and as thick as the calf of the leg, and they place the stone > between the feet, and with a stick apply force to the edges of the stone, > and at every push they give a little knife springs off with its edges like > those of a razor.

The outcomes of more complex or risky actions are determined by rolling dice. "The Role-Playing Game and the Game of Role-Playing" Different polyhedral dice are used for different actions, such as a twenty-sided die to see whether a hit was made in combat, but an eight- sided die to determine how much damage was dealt. Factors contributing to the outcome include the character's ability scores, skills and the difficulty of the task.Tweet, Cook, Williams; Player's Handbook v3.5, p. 62 In circumstances where a character does not have control of an event, such as when a trap or magical effect is triggered or a spell is cast, a saving throw can be used to determine whether the resulting damage is reduced or avoided.Tweet, Cook, Williams; Player's Handbook v3.5, p. 136"Generally, when you are subject to an unusual or magical attack, you get a saving throw to avoid or reduce the effect." There is identical language in sections titled 'Saving Throws' in (Tweet 2000:119).

After a lemma of Augustin Cauchy on the impossibility of labeling the edges of a polyhedron by positive and negative signs so that each vertex has at least four sign changes, the remainder of chapter 2 outlines the content of the remaining book. Chapters 3 and 4 prove Alexandrov's uniqueness theorem, characterizing the surface geometry of polyhedra as being exactly the metric spaces that are topologically spherical locally like the Euclidean plane except at a finite set of points of positive angular defect, obeying Descartes' theorem on total angular defect that the total angular defect should be 4\pi. Chapter 5 considers the metric spaces defined in the same way that are topologically a disk rather than a sphere, and studies the flexible polyhedral surfaces that result. Chapters 6 through 8 of the book are related to a theorem of Hermann Minkowski that a convex polyhedron is uniquely determined by the areas and directions of its faces, with a new proof based on invariance of domain.

They also suggested that in the polypeptides of a given protein, amino acids occurred in a regular, repeating pattern; for example, they proposed that silk fibroin, known to consist mainly of glycine and alanine, had a sequence of glycine-alanine- glycine-[other], with the glycine/alanine pattern making up three of every four amino acids and other residues falling periodically into the fourth spot. Niemann worked with Bergmann on this theory from 1936–1938; by 1939, they came to reject the theory as other biochemists provided evidence contradicting their proposed formula. After his work at the Rockefeller Institute and at the University College Hospital as a Rockefeller Foundation Fellow, and with strong support from Warren Weaver, Niemann joined Linus Pauling's Crellin Laboratory at Caltech in 1938. In 1939, Niemann and Linus Pauling published a strong critique of Dorothy Wrinch's cyclol hypothesis of protein structure, which held that globular proteins formed inter-linked, cage-like polyhedral structures.

Removing any two vertices (yellow) cannot disconnect a three-dimensional polyhedron: one can choose a third vertex (green), and a nontrivial linear function whose zero set (blue) passes through these three vertices, allowing connections from the chosen vertex to the minimum and maximum of the function, and from any other vertex to the minimum or maximum. In polyhedral combinatorics, a branch of mathematics, Balinski's theorem is a statement about the graph-theoretic structure of three-dimensional polyhedra and higher- dimensional polytopes. It states that, if one forms an undirected graph from the vertices and edges of a convex d-dimensional polyhedron or polytope (its skeleton), then the resulting graph is at least d-vertex-connected: the removal of any d − 1 vertices leaves a connected subgraph. For instance, for a three-dimensional polyhedron, even if two of its vertices (together with their incident edges) are removed, for any pair of vertices there will still exist a path of vertices and edges connecting the pair.. Balinski's theorem is named after mathematician Michel Balinski, who published its proof in 1961,.

One of the more studied manganese oxide-based cathodes is ,a cation ordered member of the spinel structural family (space group Fd3m). In addition to containing inexpensive materials, the three-dimensional structure of lends itself to high rate capability by providing well connected framework for the insertion and de-insertion of ions during discharge and charge of the battery. In particular, the ions occupy the tetrahedral sites within the polyhedral frameworks adjacent to empty octahedral sites.M. M. Thackeray, P.J. Johnson, L.A. de Picciotto, P.G. Bruce, J.B. Goodenough. "Electrochemical extraction of lithium from LiMn2O 4" Materials Research Bulletin 19.2 (1984): 179-187 M. M. Thackeray, Yang Shao‐Horn, Arthur J Kahaian, Keith D Kepler, Eric Skinner, John T Vaughey, Stephen A Hackney "Structural Fatigue in Spinel Electrodes in High Voltage (4 V) Li/LixMn2O4 Cells" Electrochemical and Solid-State Letters 1 (1), 7-9 (1998) As a consequence of this structural arrangement, batteries based on cathodes have demonstrated a higher rate-capability compared to materials with two-dimensional frameworks for diffusion.

The Serenity Role Playing Game was the first game to be produced under the Cortex System. It is a rules light generic roll-over system using polyhedral dice. Each of a character's attributes and skills is assigned one of these dice types, with larger dice representing greater ability, ranging from d4 to d12+d4. When a character attempts an action, such as piloting a spacecraft, shooting a gun, or punching someone, the player rolls the die for the character's applicable attribute and the die for their appropriate skill, adds the results together, and compares the total against a difficulty number based on the difficulty of the task being attempted The Firefly Role-Playing Game was produced under the Cortex Plus system, and the dice pool includes three basic sections; an attribute (Physical/Mental/Social, either one at d6, one at d8, and one at d10 or all three at d8), a skill, and a distinction chosen from a list that may be rolled at d8 or rolled at d4 to gain a plot point.

The book consists of six chapters, the first of which introduces the problem, sets it in the context of the investigation of the mathematical strength of straightedge and compass constructions, and introduces one of the major themes of the book, the relegation of paper folding to recreational mathematics as this sort of investigation fell out of favor among professional mathematicians, and its more recent resurrection as a serious topic of investigation. As a work of history, the book follows Hans-Jörg Rheinberger in making a distinction between epistemic objects, the not-yet-fully-defined subjects of scientific investigation, and technical objects, the tools used in these investigations, and it links the perceived technicality of folding with its fall from mathematical favor. The remaining chapters are organized chronologically, beginning in the 16th century and the second chapter. This chapter includes the work of Albrecht Dürer on polyhedral nets, arrangements of polygons in the plane that can be folded to form a given polyhedron, and of Luca Pacioli on the use of folding to replace the compass and straightedge in geometric constructions; it also discusses the history of paper, and paper folding in the context of bookbinding.

Two planar graphs can have isomorphic medial graphs only if they are dual to each other.. A planar graph with four or more vertices is maximal (no more edges can be added while preserving planarity) if and only if its dual graph is both 3-vertex-connected and 3-regular.. A connected planar graph is Eulerian (has even degree at every vertex) if and only if its dual graph is bipartite. A Hamiltonian cycle in a planar graph corresponds to a partition of the vertices of the dual graph into two subsets (the interior and exterior of the cycle) whose induced subgraphs are both trees. In particular, Barnette's conjecture on the Hamiltonicity of cubic bipartite polyhedral graphs is equivalent to the conjecture that every Eulerian maximal planar graph can be partitioned into two induced trees.. If a planar graph has Tutte polynomial , then the Tutte polynomial of its dual graph is obtained by swapping and . For this reason, if some particular value of the Tutte polynomial provides information about certain types of structures in , then swapping the arguments to the Tutte polynomial will give the corresponding information for the dual structures.

Unlike the Cortex System, Cortex Plus is a roll and keep system in which you roll one die from each category and keep the two highest dice in your dice pool. What goes into your dice pool is whatever is considered important for the game you are playing - but different stories have different things they consider important so the implementation of the system has been different for each game so far. All versions of Cortex Plus use standard polyhedral dice and the normal dice notation ranging from d4 (a 4 sided tetrahedral die) to d12 (a 12-sided dodecahedral die), and narratively notable features are given dice from this list, with d6 being the default. In all cases Cortex Plus uses dice pools ranging from d4 (terrible) to d12 (the best possible), and every die in your pool that rolls a natural 1 (called an "Opportunity") doesn't count toward your total and causes some form of negative consequence which, depending on the game, either creates a complication for the characters to overcome or adds to the Doom Pool that provides the Game Master resources within the scene.

The first printed illustration of a rhombicuboctahedron, by Leonardo da Vinci, published in De Divina Proportione, 1509 The Platonic solids and other polyhedra are a recurring theme in Western art. They are found, for instance, in a marble mosaic featuring the small stellated dodecahedron, attributed to Paolo Uccello, in the floor of the San Marco Basilica in Venice; in Leonardo da Vinci's diagrams of regular polyhedra drawn as illustrations for Luca Pacioli's 1509 book The Divine Proportion; as a glass rhombicuboctahedron in Jacopo de Barbari's portrait of Pacioli, painted in 1495; in the truncated polyhedron (and various other mathematical objects) in Albrecht Dürer's engraving Melencolia I; and in Salvador Dalí's painting The Last Supper in which Christ and his disciples are pictured inside a giant dodecahedron. Albrecht Dürer (1471–1528) was a German Renaissance printmaker who made important contributions to polyhedral literature in his 1525 book, Underweysung der Messung (Education on Measurement), meant to teach the subjects of linear perspective, geometry in architecture, Platonic solids, and regular polygons. Dürer was likely influenced by the works of Luca Pacioli and Piero della Francesca during his trips to Italy.

The 600 vertices of a 120-cell with an edge length of = 3− and a center-to-vertex radius of = 2 include all permutations of: : (0, 0, ±2, ±2) : (±1, ±1, ±1, ±) : (±φ−2, ±φ, ±φ, ±φ) : (±φ−1, ±φ−1, ±φ−1, ±φ2) and all even permutations of : (0, ±φ−2, ±1, ±φ2) : (0, ±φ−1, ±φ, ±) : (±φ−1, ±1, ±φ, ±2) where φ (also called τ) is the golden ratio, . Considering the adjacency matrix of the vertices representing its polyhedral graph, the graph diameter is 15, connecting each vertex to its coordinate-negation, at a Euclidean distance of 4 away (its circumdiameter), and there are 24 different paths to connect them along the polytope edges. From each vertex, there are 4 vertices at distance 1, 12 at distance 2, 24 at distance 3, 36 at distance 4, 52 at distance 5, 68 at distance 6, 76 at distance 7, 78 at distance 8, 72 at distance 9, 64 at distance 10, 56 at distance 11, 40 at distance 12, 12 at distance 13, 4 at distance 14, and 1 at distance 15. The adjacency matrix has 27 distinct eigenvalues ranging from 2−3φ, with a multiplicity of 4, to 4, with a multiplicity of 1.

To compare the constraint-based polyhedral model to prior approaches such as individual loop transformations and the unimodular approach, consider the question of whether we can parallelize (execute simultaneously) the iterations of following contrived but simple loop: for i := 0 to N do A(i) := (A(i) + A(N-i))/2 Approaches that cannot represent symbolic terms (such as the loop-invariant quantity N in the loop bound and subscript) cannot reason about dependencies in this loop. They will either conservatively refuse to run it in parallel, or in some cases speculatively run it completely in parallel, determine that this was invalid, and re-execute it sequentially. Approaches that handle symbolic terms but represent dependencies via direction vectors or distance vectors will determine that the i loop carries a dependence (of unknown distance), since for example when N=10 iteration 0 of the loop writes an array element (A(0)) that will be read in iteration 10 (as A(10-10)) and reads an array element (A(10-0)) that will later be overwritten in iteration 10 (as A(10)). If all we know is that the i loop carries a dependence, we once again cannot safely run it in parallel.