345 Sentences With "dodecahedron" | Random Sentence Generator

Beyoncé is a musical dodecahedron, in that she has many sides.

FAMILY FRIDAY: 'WOOD, MAGNETS, PLAYDOUGH AND SPACE' (Friday) What's a rhombic dodecahedron?

The DodeCal is a dodecahedron-shaped, Scandinavian-looking calendar made out of sycamore wood.

But when it folds in on itself, RAD forms a dodecahedron cage, trapping animals inside.

It's a dodecahedron with a tiny accelerometer inside that can tell which way it's sitting.

The ball is technically a dodecahedron that closes softly around the creature in front of it.

The RAD sampler (short for rotary actuated dodecahedron), is essentially a 3D printed, origami catcher's mitt.

Now he has: the DodeCal is a dodecahedron-shaped, Scandinavian-looking calendar made out of sycamore wood.

It's composed of hundreds of wooden, dodecahedron-shaped structures, each at least the size of a room.

So Ric Bell, DodeCal's creator opted to engrave his calendar on a rhombic-dodecahedron, which comprises 12 four-sided, diamond-shaped faces.

A single motor allows the entire structure to rotate about its joints and fold up into a hollow 12-sided cage, or dodecahedron.

Introducing the rotary actuated dodecahedron (RAD), a 12-faced device developed by researchers at the Wyss Institute for Biologically Inspired Engineering at Harvard University and several other institutions.

For ease of use, the most common words such as "chair" are reserved for densely populated areas, whereas obscure terms such as "dodecahedron," are confined to remote locations like Antarctica.

The device (known as the rotary actuated dodecahedron, or RAD for short) can be attached to the arm of an underwater rover and triggered remotely to capture soft marine life safely.

Zemdegs also goes over the differences between a traditional Rubik's Cube and the likes of a 4x4x4 all the way up to the Megaminx, a Rubik's Cube in the shape of a dodecahedron.

The problem has been that designing, say, a dodecahedron is a tremendously complicated task, and few have the expertise to assemble such a complex molecule, composed of thousands of base pairs, by hand.

This small, spaceship-like dodecahedron hut is made from aluminum panels supported by a metal pole in the center and a metal frame on stilts, which could be embedded into the ground to provide stability.

The rotary actuated dodecahedron, or RAD, has five 3D-printed "petals" with a complex-looking but mechanically simple framework that allows them to close up simultaneously from force applied at a single point near the rear panel.

For example, there are gaps left between the panels of the dodecahedron in order to stop pressure from building up in the interior when the marine rover makes the trip from the ocean floor to the surface.

There are a lot of goofy parodies of the more-ubiquitous-than-it-should-be Flat Earth Theory floating throughout the internet—Twitter accounts dedicated to Dodecahedron Earth Theory, Banana Earth Theory, Taco Earth Theory, and Dinosaur Earth Theory.

The rhombic dodecahedron is one of the favorite three-dimensional shapes of the designer Jane Kostick and the early-childhood educator Sharon Kulik, who will be leading an investigation into the nature of space and patterns, and how shapes can be fitted together so that there are no gaps or overlaps.

Each foglet would measure just 10 microns across (roughly the same size as a human cell), be equipped with a tiny, rudimentary onboard computer to control its actions (which would be controlled externally by an artificially intelligent system), and a dozen telescopic arms that extrude outwards in the shape of a dodecahedron.

For example, Denes's mid-'2548s "Maps Projections" — in which she used isometric projection to render gridded maps of the earth in shapes other than a globe — read as sincere thought experiments in alternative mapping (such as pyramid-, egg-, and dodecahedron-shaped earths) despite the sporadic gag (for example, a hot dog-shaped earth).

A January 26, 1995 article in the Tampa Bay Times describes the intention as "aerobics for the face" with pulsing screws which massaged the muscles and delivered mild electric shocks (essentially Dynatone '95) to each of the twelve "facial zones"–a concept which informs you that your face is a dodecahedron (and newfound awareness that other products may have been neglecting zones this whole time).

3D model of a small stellapentakis dodecahedron The small stellapentakis dodecahedron (or small astropentakis dodecahedron) is a nonconvex isohedral polyhedron. It is the dual of the truncated great dodecahedron. It has 60 intersecting triangular faces.

In geometry, the parabiaugmented dodecahedron is one of the Johnson solids (J59). It can be seen as a dodecahedron with two pentagonal pyramids (J2) attached to opposite faces. When pyramids are attached to a dodecahedron in other ways, they may result in an augmented dodecahedron, a metabiaugmented dodecahedron, a triaugmented dodecahedron, or even a pentakis dodecahedron if the faces are made to be irregular. The dual of this solid is the Gyroelongated pentagonal bifrustum.

In geometry, the augmented dodecahedron is one of the Johnson solids (J58), consisting of a dodecahedron with a pentagonal pyramid (J2) attached to one of the faces. When two or three such pyramids are attached, the result may be a parabiaugmented dodecahedron, a metabiaugmented dodecahedron or a triaugmented dodecahedron.

Great dodecahedron shown solid, surrounding stellated dodecahedron only as wireframe The compound of small stellated dodecahedron and great dodecahedron is a polyhedron compound where the great dodecahedron is interior to its dual, the small stellated dodecahedron. This can be seen as the three-dimensional equivalent of the compound of two pentagrams ({10/4} "decagram"); this series continues into the fourth dimension as compounds of star 4-polytopes.

In geometry, the metabiaugmented dodecahedron is one of the Johnson solids (J60). It can be viewed as a dodecahedron with two pentagonal pyramids (J2) attached to two faces that are separated by one face. (The two faces are not opposite, but not adjacent either.) When pyramids are attached to a dodecahedron in other ways, they may result in an augmented dodecahedron, a parabiaugmented dodecahedron, a triaugmented dodecahedron, or even a pentakis dodecahedron if the faces are made to be irregular.

Five tetrahedra in a dodecahedron. It is a faceting of a dodecahedron, as shown at left.

Ten tetrahedra in a dodecahedron. It is also a facetting of the dodecahedron, as shown at left. Concave pentagrams can be seen on the compound where the pentagonal faces of the dodecahedron are positioned.

3D model of a great stellapentakis dodecahedron In geometry, the great stellapentakis dodecahedron (or great astropentakis dodecahedron) is a nonconvex isohedral polyhedron. It is the dual of the truncated great icosahedron. It has 60 intersecting triangular faces.

3D model of a great stellated truncated dodecahedron In geometry, the great stellated truncated dodecahedron (or quasitruncated great stellated dodecahedron or great stellatruncated dodecahedron) is a nonconvex uniform polyhedron, indexed as U66. It has 32 faces (20 triangles and 12 pentagrams), 90 edges, and 60 vertices. It is given a Schläfli symbol t0,1{5/3,3}.

Animation of a net of a regular (pentagonal) dodecahedron being folded 3D model of a regular dodecahedron A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals (60 face diagonals, 100 space diagonals).. It is represented by the Schläfli symbol {5,3}.

3D model of a small stellated truncated dodecahedron In geometry, the small stellated truncated dodecahedron (or quasitruncated small stellated dodecahedron or small stellatruncated dodecahedron) is a nonconvex uniform polyhedron, indexed as U58. It has 24 faces (12 pentagons and 12 decagrams), 90 edges, and 60 vertices. It is given a Schläfli symbol t{,5}, and Coxeter diagram .

In geometry, the elongated dodecahedron,Coxeter (1973) p.257 extended rhombic dodecahedron, rhombo-hexagonal dodecahedronWilliamson (1979) p169 or hexarhombic dodecahedronFedorov's five parallelohedra in R³ is a convex dodecahedron with 8 rhombic and 4 hexagonal faces. The hexagons can be made equilateral, or regular depending on the shape of the rhombi. It can be seen as constructed from a rhombic dodecahedron elongated by a square prism.

In geometry, the small stellapentakis dodecahedron is a nonconvex isohedral polyhedron. It is the dual of the truncated great dodecahedron. It has 60 intersecting triangular faces.

The regular dodecahedron is the third in an infinite set of truncated trapezohedra which can be constructed by truncating the two axial vertices of a pentagonal trapezohedron. The stellations of the regular dodecahedron make up three of the four Kepler–Poinsot polyhedra. A rectified regular dodecahedron forms an icosidodecahedron. The regular dodecahedron has icosahedral symmetry Ih, Coxeter group [5,3], order 120, with an abstract group structure of A5 × Z2.

Rhombic dodecahedron The rhombic dodecahedron is a zonohedron with twelve rhombic faces and octahedral symmetry. It is dual to the quasiregular cuboctahedron (an Archimedean solid) and occurs in nature as a crystal form. The rhombic dodecahedron packs together to fill space. The rhombic dodecahedron can be seen as a degenerate pyritohedron where the 6 special edges have been reduced to zero length, reducing the pentagons into rhombic faces.

In geometry, the triaugmented dodecahedron is one of the Johnson solids (J61). It can be seen as a dodecahedron with three pentagonal pyramids (J2) attached to nonadjacent faces.

3D model of a great stellated dodecahedron In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedron, with Schläfli symbol {,3}. It is one of four nonconvex regular polyhedra. It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each vertex. It shares its vertex arrangement, although not its vertex figure or vertex configuration, with the regular dodecahedron, as well as being a stellation of a (smaller) dodecahedron.

3D model of a truncated dodecahedron In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.

3D model of a great stellapentakis dodecahedron The great stellapentakis dodecahedron is a nonconvex isohedral polyhedron. It is the dual of the truncated great icosahedron. It has 60 intersecting triangular faces.

The rhombic dodecahedron has several stellations, the first of which is also a parallelohedral spacefiller. Another important rhombic dodecahedron, the Bilinski dodecahedron, has twelve faces congruent to those of the rhombic triacontahedron, i.e. the diagonals are in the ratio of the golden ratio. It is also a zonohedron and was described by Bilinski in 1960.

As a surface geometry, it can be seen as visually similar to a Catalan solid, the disdyakis dodecahedron, with much taller rhombus-based pyramids joined to each face of a rhombic dodecahedron.

The vertices of the first stellation of the rhombic dodecahedron include the 12 vertices of the cuboctahedron, together with eight additional vertices (the degree-3 vertices of the rhombic dodecahedron). Escher's solid has six additional vertices, at the center points of the square faces of the cuboctahedron (the degree-4 vertices of the rhombic dodecahedron). In the first stellation of the rhombic dodecahedron, these six points are not vertices, but are instead the midpoints of pairs of edges that cross at right angles at these points. The first stellation of the rhombic dodecahedron has 12 hexagonal faces, 36 edges, and 20 vertices, yielding an Euler characteristic of 20 − 36 + 12 = −4.

For, in the Bilinski dodecahedron, the long body diagonal is parallel to the short diagonals of two faces, and to the long diagonals of two other faces. In the rhombic dodecahedron, the corresponding body diagonal is parallel to four short face diagonals, and in any affine transformation of the rhombic dodecahedron this body diagonal would remain parallel to four equal-length face diagonals. Another difference between the two dodecahedra is that, in the rhombic dodecahedron, all the body diagonals connecting opposite degree-4 vertices are parallel to face diagonals, while in the Bilinski dodecahedron the shorter body diagonals of this type have no parallel face diagonals.

In the 20th Century, Artist M. C. Escher's interest in geometric forms often led to works based on or including regular solids; Gravitation is based on a small stellated dodecahedron. Norwegian artist Vebjørn Sands sculpture The Kepler Star is displayed near Oslo Airport, Gardermoen. The star spans 14 meters, and consists of an icosahedron and a dodecahedron inside a great stellated dodecahedron.

The pentakis dodecahedron is also a model of some icosahedrally symmetric viruses, such as Adeno-associated virus. These have 60 symmetry related capsid proteins, which combine to make the 60 symmetrical faces of a pentakis dodecahedron.

It can also be projected into 3D-dimensions as --> , a dodecahedron envelope.

3D model of a snub disphenoid In geometry, the snub disphenoid, Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron is a three-dimensional convex polyhedron with twelve equilateral triangles as its faces. It is not a regular polyhedron because some vertices have four faces and others have five. It is a dodecahedron, one of the eight deltahedra (convex polyhedra with equilateral triangle faces) and one of the 92 Johnson solids (non-uniform convex polyhedra with regular faces). It can be thought of as a square antiprism where both squares are replaced with two equilateral triangles.

In a 1962 paper,. Reprinted in (The Beauty of Geometry. Twelve Essays, Dover, 1999, ). H. S. M. Coxeter claimed that the Bilinski dodecahedron could be obtained by an affine transformation from the rhombic dodecahedron, but this is false.

Bicupolae are special in having four faces on every vertex. This means that their dual polyhedra will have all quadrilateral faces. The best known example is the rhombic dodecahedron composed of 12 rhombic faces. The dual of the ortho-form, triangular orthobicupola, is also a dodecahedron, similar to rhombic dodecahedron, but it has 6 trapezoid faces which alternate long and short edges around the circumference.

3D model of a great dodecahedron In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol {5,5/2} and Coxeter–Dynkin diagram of . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagons), with five pentagons meeting at each vertex, intersecting each other making a pentagrammic path. The discovery of the great dodecahedron is sometimes credited to Louis Poinsot in 1810, though there is a drawing of something very similar to a great dodecahedron in the 1568 book Perspectiva Corporum Regularium by Wenzel Jamnitzer.

3D model of a truncated great dodecahedron In geometry, the truncated great dodecahedron is a nonconvex uniform polyhedron, indexed as U37. It has 24 faces (12 pentagrams and 12 decagons), 90 edges, and 60 vertices. It is given a Schläfli symbol t{5,}.

3d model of a rhombic dodecahedron In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of two types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.

The regular dodecahedron is topologically related to a series of tilings by vertex figure n3. The regular dodecahedron can be transformed by a truncation sequence into its dual, the icosahedron: The regular dodecahedron is a member of a sequence of otherwise non-uniform polyhedra and tilings, composed of pentagons with face configurations (V3.3.3.3.n). (For n > 6, the sequence consists of tilings of the hyperbolic plane.) These face-transitive figures have (n32) rotational symmetry.

Asymmetric Exposures brought Hänninen also to geometrical forms like the Platonic solids and especially the fifth- dodecahedron, which she saw through her newest photographs and as result of this she then for the first time made three- dimensional objects containing photographs in the form of dodecahedron.

The great dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the (n-1)-D pentagonal polytope faces of the core nD polytope (pentagons for the great dodecahedron, and line segments for the pentagram) until the figure again closes.

In geometry, a diminished rhombic dodecahedron is a rhombic dodecahedron with one or more vertices removed. This article describes diminishing one 4-valence vertex. This diminishment creates one new square face while 4 rhombic faces are reduced to triangles. It has 13 vertices, 24 edges, and 13 faces.

These four are the only regular star polyhedra, and have come to be known as the Kepler–Poinsot polyhedra. It was not until the mid-19th century, several decades after Poinsot published, that Cayley gave them their modern English names: (Kepler's) small stellated dodecahedron and great stellated dodecahedron, and (Poinsot's) great icosahedron and great dodecahedron. The Kepler–Poinsot polyhedra may be constructed from the Platonic solids by a process called stellation. The reciprocal process to stellation is called facetting (or faceting).

Its dual polyhedron is the great stellated dodecahedron {, 3}, having three regular star pentagonal faces around each vertex.

The first stellation of the rhombic dodecahedron has 12 faces, each of which is a non-convex hexagon. It is a stellation of the rhombic dodecahedron, meaning that each of its faces lies in the same plane as one of the rhombus faces of the rhombic dodecahedron, with each face containing the rhombus in the same plane, and that it has the same symmetries as the rhombic dodecahedron. It is the first stellation, meaning that no other self-intersecting polyhedron with the same face planes and the same symmetries has smaller faces. Extending the faces outwards even farther in the same planes leads to two more stellations, if the faces are required to be simple polygons.

When a regular dodecahedron is inscribed in a sphere, it occupies more of the sphere's volume (66.49%) than an icosahedron inscribed in the same sphere (60.55%). A regular dodecahedron with edge length 1 has more than three and a half times the volume of an icosahedron with the same length edges (7.663... compared with 2.181...), which ratio is approximately , or in exact terms: or . A regular dodecahedron has 12 faces and 20 vertices, whereas a regular icosahedron has 20 faces and 12 vertices. Both have 30 edges.

Alexander's Star A dissection of the great dodecahedron was used for the 1980s puzzle Alexander's Star. Regular star polyhedra first appear in Renaissance art. A small stellated dodecahedron is depicted in a marble tarsia on the floor of St. Mark's Basilica, Venice, Italy, dating from ca. 1430 and sometimes attributed to Paulo Ucello.

The icosidodecahedron has 60 edges, all equivalent. There are four Archimedean solids with 60 vertices: the truncated icosahedron, the rhombicosidodecahedron, the snub dodecahedron, and the truncated dodecahedron. The skeletons of these polyhedra form 60-node vertex-transitive graphs. There are also two Archimedean solids with 60 edges: the snub cube and the icosidodecahedron.

The dihedral angle of a regular dodecahedron is ~116.6°, so it is impossible to fit 4 of them on an edge in Euclidean 3-space. However in hyperbolic space a properly-scaled regular dodecahedron can be scaled so that its dihedral angles are reduced to 90 degrees, and then four fit exactly on every edge.

A rhombic dodecahedron has twelve faces, each of which is a rhombus. Paulingite is a perfect clear rhombic dodecahedron of 0.1 to 1.0 mm in diameter. Their attachment to vesicles causes a hemispherical shape exhibiting 5 to 6 planes of dodecahedral planes. In the vesicular walls, they appear to be dark brown to black.

Let \phi be the golden ratio. The 12 points given by (0, \pm 1, \pm \phi) and cyclic permutations of these coordinates are the vertices of a regular icosahedron. Its dual regular dodecahedron, whose edges intersect those of the icosahedron at right angles, has as vertices the points (\pm 1, \pm 1, \pm 1) together with the points (\pm\phi, \pm 1/\phi, 0) and cyclic permutations of these coordinates. Multiplying all coordinates of this dodecahedron by a factor of (7\phi-1)/11\approx 0.938\,748\,901\,93 gives a slightly smaller dodecahedron.

Three-dimensional simple polyhedra include the prisms (including the cube), the regular tetrahedron and dodecahedron, and, among the Archimedean solids, the truncated tetrahedron, truncated cube, truncated octahedron, truncated cuboctahedron, truncated dodecahedron, truncated icosahedron, and truncated icosidodecahedron. They also include the Goldberg polyhedron and Fullerenes, including the chamfered tetrahedron, chamfered cube, and chamfered dodecahedron. In general, any polyhedron can be made into a simple one by truncating its vertices of valence four or higher. For instance, truncated trapezohedrons are formed by truncating only the high-degree vertices of a trapezohedron; they are also simple.

Stanko Bilinski (22 April 1909 in Našice – 6 April 1998 in Zagreb) was a Croatian mathematician and academician. He was a professor at the University of Zagreb and a fellow of the Croatian Academy of Sciences and Arts In 1960 he discovered a rhombic dodecahedron of the second kind, the Bilinski dodecahedron. Like the standard rhombic dodecahedron, this convex polyhedron has 12 congruent rhombus sides, but they are differently shaped and arranged. Bilinski's discovery corrected a 75-year-old omission in Evgraf Fedorov's classification of convex polyhedra with congruent rhombic faces..

Every complete bipartite graph K_{a,b} is (b,a)-biregular. The rhombic dodecahedron is another example; it is (3,4)-biregular..

3D model of a dodecadodecahedron In geometry, the dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U36. It is the rectification of the great dodecahedron (and that of its dual, the small stellated dodecahedron). It was discovered independently by , and . The edges of this model form 10 central hexagons, and these, projected onto a sphere, become 10 great circles.

Armand Spitz used a dodecahedron as the "globe" equivalent for his Digital Dome planetarium projector. based upon a suggestion from Albert Einstein.

The solution of the puzzle is a cycle containing twenty (in ancient Greek icosa) edges (i.e. a Hamiltonian circuit on the dodecahedron).

Alexander's Star in a solved state. Alexander's Star is a puzzle similar to the Rubik's Cube, in the shape of a great dodecahedron.

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.

Johannes Kepler (1571–1630) used star polygons, typically pentagrams, to build star polyhedra. Some of these figures may have been discovered before Kepler's time, but he was the first to recognize that they could be considered "regular" if one removed the restriction that regular polytopes must be convex. Later, Louis Poinsot realised that star vertex figures (circuits around each corner) can also be used, and discovered the remaining two regular star polyhedra. Cauchy proved Poinsot's list complete, and Cayley gave them their accepted English names: (Kepler's) the small stellated dodecahedron and great stellated dodecahedron, and (Poinsot's) the great icosahedron and great dodecahedron.

It is made of dodecahedron cells {5,3}, and has 3 cells around each edge. There is one regular tessellation of Euclidean 3-space: the cubic honeycomb, with a Schläfli symbol of {4,3,4}, made of cubic cells and 4 cubes around each edge. There are also 4 regular compact hyperbolic tessellations including {5,3,4}, the hyperbolic small dodecahedral honeycomb, which fills space with dodecahedron cells.

The stellation diagram for the regular dodecahedron with the central pentagon highlighted. This diagram represents the dodecahedron face itself. In geometry, a stellation diagram or stellation pattern is a two-dimensional diagram in the plane of some face of a polyhedron, showing lines where other face planes intersect with this one. The lines cause 2D space to be divided up into regions.

The Megaminx twisty puzzle, alongside its larger and smaller order analogues, is in the shape of a regular dodecahedron. In the children's novel The Phantom Tollbooth, the regular dodecahedron appears as a character in the land of Mathematics. Each of his faces wears a different expression – e.g. happy, angry, sad – which he swivels to the front as required to match his mood.

3D model of a small stellated dodecahedron In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol {,5}. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex. It shares the same vertex arrangement as the convex regular icosahedron.

The great grand stellated 120-cell is the final stellation of the 120-cell, and is the only Schläfli- Hess polychoron to have the 120-cell for its convex hull. In this sense it is analogous to the three-dimensional great stellated dodecahedron, which is the final stellation of the dodecahedron and the only Kepler-Poinsot polyhedron to have the dodecahedron for its convex hull. Indeed, the great grand stellated 120-cell is dual to the grand 600-cell, which could be taken as a 4D analogue of the great icosahedron, dual of the great stellated dodecahedron. The edges of the great grand stellated 120-cell are τ6 as long as those of the 120-cell core deep inside the polychoron, and they are τ3 as long as those of the small stellated 120-cell deep within the polychoron.

Lamellae seen optically may indicate twinning. They have a conchoidal fracture. It has a white streak. Rhombic dodecahedron is the dominant crystal form for paulingite.

In geometry a parallelohedron is a polyhedron that can be translated without rotations in 3-dimensional Euclidean space to fill space with a honeycomb in which all copies of the polyhedron meet face-to-face. There are five types of parallelohedron, first identified by Evgraf Fedorov in 1885 in his studies of crystallographic systems: the cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron, and truncated octahedron.

These figures have pentagrams (star pentagons) as faces or vertex figures. The small and great stellated dodecahedron have nonconvex regular pentagram faces. The great dodecahedron and great icosahedron have convex polygonal faces, but pentagrammic vertex figures. In all cases, two faces can intersect along a line that is not an edge of either face, so that part of each face passes through the interior of the figure.

Altamaha Tech's Dodecahedron The Dodecahedron is the official symbol of Altamaha Technical College. The twelve faceted sides are pentagons that represent the multiple facets of technical education, and is a reminder to stay focused on the future and to be multifaceted in the school's programs and services. The inspiration for this choice was the design for the deep space probes in the novel Contact by Carl Sagan.

One shot briefly shows Yamaguchi falling overlaid with words, in the style of the Documentaly album cover artwork. In the final scene, the band can be seen watching themselves in the dodecahedron. They remove their masks, to show the dodecahedron with a brain pattern instead of their faces. The general theme that the band requested for the video was one of sociality and reflecting current society.

It also shares the same edge arrangement with the great icosahedron, with which it forms a degenerate uniform compound figure. It is the second of four stellations of the dodecahedron. The small stellated dodecahedron can be constructed analogously to the pentagram, its two- dimensional analogue, via the extension of the edges (1-faces) of the core polytope until a point is reached where they intersect.

Thus each face diagonal of a cube with side length a is a\sqrt 2.. A regular dodecahedron has 60 face diagonals (and 100 space diagonals)..

If the pentagrammic faces are considered as 5 triangular faces, it shares the same surface topology as the pentakis dodecahedron, but with much taller isosceles triangle faces, with the height of the pentagonal pyramids adjusted so that the five triangles in the pentagram become coplanar. The critical angle is atan(2) above the dodecahedron face. If we regard it as having 12 pentagrams as faces, with these pentagrams meeting at 30 edges and 12 vertices, we can compute its genus using Euler's formula :V - E + F = 2 - 2g and conclude that the small stellated dodecahedron has genus 4. This observation, made by Louis Poinsot, was initially confusing, but Felix Klein showed in 1877 that the small stellated dodecahedron could be seen as a branched covering of the Riemann sphere by a Riemann surface of genus 4, with branch points at the center of each pentagram.

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.

Each of the Platonic solids occurs naturally in one form or another. The tetrahedron, cube, and octahedron all occur as crystals. These by no means exhaust the numbers of possible forms of crystals (Smith, 1982, p212), of which there are 48. Neither the regular icosahedron nor the regular dodecahedron are amongst them, but crystals can have the shape of a pyritohedron, which is visually almost indistinguishable from a regular dodecahedron.

The original motivation for studying zonohedra is that the Voronoi diagram of any lattice forms a convex uniform honeycomb in which the cells are zonohedra. Any zonohedron formed in this way can tessellate 3-dimensional space and is called a primary parallelohedron. Each primary parallelohedron is combinatorially equivalent to one of five types: the rhombohedron (including the cube), hexagonal prism, truncated octahedron, rhombic dodecahedron, and the rhombo-hexagonal dodecahedron.

The original entrance overlooking the Foro Italico is of a monumental neoclassical design. It has been permanently closed, however, so access to the garden is now obtained through the primary entrance on Lincoln Street. That gate area is less developed. The heart of the villa is the dodecahedron fountain, featuring a sculpture of a dodecahedral marble clock created by the mathematician Lorenzo Federici, each face of the dodecahedron featuring a sundial.

Let \phi be the golden ratio. The 12 points given by (0, \pm 1, \pm \phi) and cyclic permutations of these coordinates are the vertices of a regular icosahedron. Its dual regular dodecahedron, whose edges intersect those of the icosahedron at right angles, has as vertices the points (\pm 1, \pm 1, \pm 1) together with the points (\pm\phi, \pm 1/\phi, 0) and cyclic permutations of these coordinates. Multiplying all coordinates of the icosacahedron by a factor of (3\phi+12)/19\approx 0.887\,057\,998\,22 gives a slightly smaller icosahedron. The 12 vertices of this icosahedron, together with the vertices of the dodecahedron, are the vertices of a pentakis dodecahedron centered at the origin.

Matila Ghyka, The Geometry of Art and Life (1977), p.68 The boundary of the 120-cell is composed of 120 dodecahedral cells with 4 meeting at each vertex. It can be thought of as the 4-dimensional analog of the regular dodecahedron. Just as a dodecahedron can be built up as a model with 12 pentagons, 3 around each vertex, the dodecaplex can be built up from 120 dodecahedra, with 3 around each edge.

A hemi-dodecahedron is an abstract regular polyhedron, containing half the faces of a regular dodecahedron. It can be realized as a projective polyhedron (a tessellation of the real projective plane by 6 pentagons), which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and dividing the hemisphere into three equal parts. It has 6 pentagonal faces, 15 edges, and 10 vertices.

Bernal writes that the snub disphenoid is "a very common coordination for the calcium ion in crystallography".. In coordination geometry, it is usually known as the trigonal dodecahedron or simply as the dodecahedron. The snub disphenoid name comes from Norman Johnson's 1966 classification of the Johnson solids, convex polyhedra all of whose faces are regular.. It exists first in a series of polyhedra with axial symmetry, so also can be given the name digonal gyrobianticupola.

3D model of a tetradyakis hexahedron The tetradyakis hexahedron (or great disdyakis dodecahedron) is a nonconvex isohedral polyhedron. It has 48 intersecting scalene triangle faces, 72 edges, and 20 vertices.

Looking in the opposite direction, certain abstract regular polytopes – hemi-cube, hemi- dodecahedron, and hemi-icosahedron – can be constructed as regular figures in the projective plane; see also projective polyhedra.

Along with the rhombic dodecahedron, it is a space-filling polyhedron, one of the five types of parallelohedron identified by Evgraf Fedorov that tile space face-to-face by translations.

200px The pentakis dodecahedron in a model of buckminsterfullerene: each surface segment represents a carbon atom. Equivalently, a truncated icosahedron is a model of buckminsterfullerene, with each vertex representing a carbon atom.

There are generic geometric names for the most common polyhedra. The 5 regular polyhedra are called a tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron with 4, 6, 8, 12, and 20 sides respectively.

3d model of a triakis icosahedron In geometry, the triakis icosahedron (or kisicosahedronConway, Symmetries of things, p.284) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated dodecahedron.

3D model of a great dodecahemidodecahedron In geometry, the great dodecahemidodecahedron is a nonconvex uniform polyhedron, indexed as U70. It has 18 faces (12 pentagrams and 6 decagrams), 60 edges, and 30 vertices. Its vertex figure is a crossed quadrilateral. Aside from the regular small stellated dodecahedron {5/2,5} and great stellated dodecahedron {5/2,3}, it is the only nonconvex uniform polyhedron whose faces are all non-convex regular polygons (star polygons), namely the star polygons {5/2} and {10/3}.

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.

The compound polyhedron known as the stellated octahedron can be represented by a{4,3} (an altered cube), and , 40px. The star polyhedron known as the small ditrigonal icosidodecahedron can be represented by a{5,3} (an altered dodecahedron), and , 40px. Here all the pentagons have been alternated into pentagrams, and triangles have been inserted to take up the resulting free edges. The star polyhedron known as the great ditrigonal icosidodecahedron can be represented by a{5/2,3} (an altered great stellated dodecahedron), and , 40px.

Net of a dodecahedron. All edges in this net have true length. In geometry, true length is any distance between points that is not foreshortened by the view type.Manual of Engineering Drawing 2009, , pp.

The Arloff icosahedron (front right) in the Rhenish State Museum, Bonn A Roman icosahedron was found near Arloff. It was at first misclassified as a dodecahedron. It is on display in the Rheinisches Landesmuseum.

The dihedral angle of a Euclidean regular dodecahedron is ~116.6°, so no more than three of them can fit around an edge in Euclidean 3-space. In hyperbolic space, however, the dihedral angle is smaller than it is in Euclidean space, and depends on the size of the figure; the smallest possible dihedral angle is 60°, for an ideal hyperbolic regular dodecahedron with infinitely long edges. The dodecahedra in this dodecahedral honeycomb are sized so that all of their dihedral angles are exactly 72°.

A simple construction of this space begins with a dodecahedron. Each face of the dodecahedron is identified with its opposite face, using the minimal clockwise twist to line up the faces. Gluing each pair of opposite faces together using this identification yields a closed 3-manifold. (See Seifert–Weber space for a similar construction, using more "twist", that results in a hyperbolic 3-manifold.) Alternatively, the Poincaré homology sphere can be constructed as the quotient space SO(3)/I where I is the icosahedral group (i.e.

Escher would not have been familiar with Brückner's work and H. S. M. Coxeter writes that "It is remarkable that Escher, without any knowledge of algebra or analytic geometry, was able to rediscover this highly symmetrical figure." Earlier in 1948, Escher had made a preliminary woodcut with a similar theme, Study for Stars, but instead of using the compound of three regular octahedra in the study he used a different but related shape, a stellated rhombic dodecahedron (sometimes called Escher's solid), which can be formed as a compound of three flattened octahedra.The compound of three octahedra and a remarkable compound of three square dipyramids, the Escher's solid, Livio Zefiro, University of Genova. This form as a polyhedron is topologically identical to the disdyakis dodecahedron, which can be seen as rhombic dodecahedron with shorter pyramids on the rhombic faces.

It is part of a truncation process between a dodecahedron and icosahedron: This polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry.

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.

Most shape modifications can be adapted to higher-order cubes. In the case of Tony Fisher's Rhombic Dodecahedron, there are 3×3×3, 4×4×4, 5×5×5, and 6×6×6 versions of the puzzle.

A Kepler–Poinsot polyhedron covers its circumscribed sphere more than once, with the centers of faces acting as winding points in the figures which have pentagrammic faces, and the vertices in the others. Because of this, they are not necessarily topologically equivalent to the sphere as Platonic solids are, and in particular the Euler relation :\chi=V-E+F=2\ does not always hold. Schläfli held that all polyhedra must have χ = 2, and he rejected the small stellated dodecahedron and great dodecahedron as proper polyhedra. This view was never widely held.

Irregular dodecahedron The Weaire–Phelan structure differs from Kelvin's in that it uses two kinds of cells, although they have equal volume. From a topological and symmetrical point of view, one is a pyritohedron, an irregular dodecahedron with pentagonal faces, possessing tetrahedral symmetry (Th). Tetrakaidecahedron The second is a form of truncated hexagonal trapezohedron, a species of tetrakaidecahedron with two hexagonal and twelve pentagonal faces, in this case only possessing two mirror planes and a rotoreflection symmetry. Like the hexagons in the Kelvin structure, the pentagons in both types of cells are slightly curved.

If the pentagrammic faces are broken into triangles, it is topologically related to the triakis icosahedron, with the same face connectivity, but much taller isosceles triangle faces. If the triangles are instead made to invert themselves and excavate the central icosahedron, the result is a great dodecahedron. The great stellated dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, by attempting to stellate the n-dimensional pentagonal polytope which has pentagonal polytope faces and simplex vertex figures until it can no longer be stellated; that is, it is its final stellation.

In mathematics he is most known for obtaining the angle in the rhombic dodecahedron shape in 1712, which is still called the Maraldi angle. Craters on the Moon and Mars were named in his and his nephew's honor.

Cartesian coordinates for the vertices of a truncated dodecahedron with edge length 2φ − 2, centered at the origin, are all even permutations of: :(0, ±, ±(2 + φ)) :(±, ±φ, ±2φ) :(±φ, ±2, ±(φ + 1)) where φ = is the golden ratio.

It has C4v symmetry, order 8. Like the rhombic dodecahedron, the long diagonal of each rhombic face is times the length of the short diagonal, so that the acute angles on each face measure arccos(), or approximately 70.53°.

In the solid state the lead(IV) centers are coordinated by four acetate ions, which are bidentate, each coordinating via two oxygen atoms. The lead atom is 8 coordinate and the O atoms form a flattened trigonal dodecahedron.

Two projectors were used to display the images onto the dodecahedron. No trial run was done before the video's shooting, and instead Shiga modelled the set-up as a computer-aided design until he was satisfied with the arrangement.

Wenninger writes "quasitruncated dodecahedron", but this appear to be a mistake. . Coxeter et al. credit its discovery to a paper published in 1881 by Austrian mathematician Johann Pitsch.. According to , the truncated dodecadodecahedron appears as no. XII on p.86.

The rhombic dodecahedral honeycomb can be dissected into a trigonal trapezohedral honeycomb with each rhombic dodecahedron dissected into 4 trigonal trapezohedrons. Each rhombic dodecahedra can also be dissected with a center point into 12 rhombic pyramids of the rhombic pyramidal honeycomb.

It is the only radially equilateral convex polyhedron. Its dual polyhedron is the rhombic dodecahedron. The cuboctahedron was probably known to Plato: Heron's Definitiones quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares.

This polyhedron can be formed from a regular dodecahedron by truncating (cutting off) the corners so the pentagon faces become decagons and the corners become triangles. It is used in the cell-transitive hyperbolic space-filling tessellation, the bitruncated icosahedral honeycomb.

The dodecahedron-first orthographic projection of the dodecahedral prism into 3D space has a dodecahedral envelope. The two dodecahedral cells project onto the entire volume of this envelope, while the 12 decagonal prismic cells project onto its 12 pentagonal faces.

Because the ideal regular tetrahedron, cube, octahedron, and dodecahedron all have dihedral angles that are integer fractions of 2\pi, they can all tile hyperbolic space, forming a regular honeycomb. In this they differ from the Euclidean regular solids, among which only the cube can tile space. The ideal tetrahedron, cube, octahedron, and dodecahedron form respectively the order-6 tetrahedral honeycomb, order-6 cubic honeycomb, order-4 octahedral honeycomb, and order-6 dodecahedral honeycomb; here the order refers to the number of cells meeting at each edge. However, the ideal icosahedron does not tile space in the same way.

A Ho–Mg–Zn quasicrystal in the shape of a dodecahedron. A holmium–magnesium–zinc (Ho–Mg–Zn) quasicrystal is a quasicrystal made of an alloy of the three metals holmium, magnesium and zinc that has the shape of a regular dodecahedron, a Platonic solid with 12 five-sided faces. Unlike the similar pyritohedron shape of some cubic-system crystals such as pyrite, this quasicrystal has faces that are true regular pentagons. The crystal is part of the R-Mg-Zn family of crystals, where R=Y, Gd, Tb, Dy, Ho or Er. They were first discovered in 1994.

Meglos wants to steal the Dodecahedron back from Tigella, as it is a Zolfa-Thuran energy source of immense power. To aid him, Meglos uses an Earthling captured for him by the Gaztaks to occupy and take on humanoid form: and the humanoid form he chooses is the Doctor, whom he has trapped in the bubble. While the hysteresis persists Meglos gets the Gaztaks to take him to Tigella, and infiltrates the city in his new identity. Zastor greets “the Doctor” warmly as an old friend, asking him to examine the Dodecahedron, but others are less sure, especially Lexa.

Thus, in the Seifert–Weber space, each edge is surrounded by five pentagonal faces, and the dihedral angle between these pentagons is 72°. This does not match the 117° dihedral angle of a regular dodecahedron in Euclidean space, but in hyperbolic space there exist regular dodecahedra with any dihedral angle between 60° and 117°, and the hyperbolic dodecahedron with dihedral angle 72° may be used to give the Seifert–Weber space a geometric structure as a hyperbolic manifold. It is a quotient space of the order-5 dodecahedral honeycomb, a regular tessellation of hyperbolic 3-space by dodecahedra with this dihedral angle.

Early conceptual designs of the Pod itself were based, as it existed in the novel, on one of the primary shapes in geometry, a dodecahedron, or a twelve-sided body. Eventually the Pod was modified to a spherical capsule that encases the traveler, with a dodecahedron surrounding the sphere. Zemeckis and the production crew also made several visits to the Kennedy Space Center at Merritt Island adjacent to Cape Canaveral, where officials allowed them access to sites off-limits to most visitors. Filmmakers were also brought onto Launch Complex 39 before the launch of the space shuttle.

Cartesian coordinates for the vertices of a great stellated truncated dodecahedron are all the even permutations of : (0, ±τ, ±(2−1/τ)) : (±τ, ±1/τ, ±2/τ) : (±1/τ2, ±1/τ, ±2) where τ = (1+)/2 is the golden ratio (sometimes written φ).

It is related to the order-4 120-cell honeycomb, {5,3,3,4}, and order-5 120-cell honeycomb, {5,3,3,5}. It is topologically similar to the finite 5-cube, {4,3,3,3}, and 5-simplex, {3,3,3,3}. It is analogous to the 120-cell, {5,3,3}, and dodecahedron, {5,3}.

As a zonohedron, it can be constructed with all but 12 octagons as regular polygons. It has two sets of 48 vertices existing on two distances from its center. It represents the Minkowski sum of a cube, a truncated octahedron, and a rhombic dodecahedron.

Painting of Luca Pacioli, attributed to Jacopo de' Barbari, 1495 (attribution controversial Ritratto Pacioli.). Table is filled with geometrical tools: slate, chalk, compass, a dodecahedron model. A rhombicuboctahedron half-filed with water is suspended from the ceiling. Pacioli is demonstrating a theorem by Euclid.

For example, in 3 dimensions, 4 of the 5 Platonic solids have central symmetry (cube/octahedron, dodecahedron/icosahedron), while the tetrahedron does not – however, the stellated octahedron has central symmetry, though the resulting symmetry group is the same as that of the cube/octahedron.

A polyhedron P is said to have the Rupert property if a polyhedron of the same or larger size and the same shape as P can pass through a hole in P. All five Platonic solids: the cube, the regular tetrahedron, regular octahedron, regular dodecahedron, and regular icosahedron, have the Rupert property. It has been conjectured that all 3-dimensional convex polyhedra have this property. For n greater than 2, the n-dimensional hypercube also has the Rupert property. Of the 13 Archimedean solids, it is known that these nine have the Rupert property: the cuboctahedron, truncated octahedron, truncated cube, rhombicuboctahedron, icosidodecahedron, truncated cuboctahedron, truncated icosahedron, truncated dodecahedron.

Sculpture of a small stellated dodecahedron, as in Escher's 1952 work Gravitation (University of Twente) Escher often incorporated three- dimensional objects such as the Platonic solids such as spheres, tetrahedrons, and cubes into his works, as well as mathematical objects such as cylinders and stellated polyhedra. In the print Reptiles, he combined two- and three- dimensional images. In one of his papers, Escher emphasized the importance of dimensionality: Escher's artwork is especially well-liked by mathematicians such as Doris Schattschneider and scientists such as Roger Penrose, who enjoy his use of polyhedra and geometric distortions. For example, in Gravitation, animals climb around a stellated dodecahedron.

If an odd perfect number is of the form 12k + 1, it has at least twelve distinct prime factors. A twelve-sided polygon is a dodecagon. A twelve-faced polyhedron is a dodecahedron. Regular cubes and octahedrons both have 12 edges, while regular icosahedrons have 12 vertices.

Geometric examples for many values of h, k, and T can be found at List of geodesic polyhedra and Goldberg polyhedra. Many exceptions to this rule exist: For example, the polyomaviruses and papillomaviruses have pentamers instead of hexamers in hexavalent positions on a quasi-T=7 lattice. Members of the double-stranded RNA virus lineage, including reovirus, rotavirus and bacteriophage φ6 have capsids built of 120 copies of capsid protein, corresponding to a "T=2" capsid, or arguably a T=1 capsid with a dimer in the asymmetric unit. Similarly, many small viruses have a pseudo-T=3 (or P=3) capsid, which is organized according to a T=3 lattice, but with distinct polypeptides occupying the three quasi-equivalent positions T-numbers can be represented in different ways, for example T = 1 can only be represented as an icosahedron or a dodecahedron and, depending on the type of quasi-symmetry, T = 3 can be presented as a truncated dodecahedron, an icosidodecahedron, or a truncated icosahedron and their respective duals a triakis icosahedron, a rhombic triacontahedron, or a pentakis dodecahedron.

3D model of a snub dodecadodecahedron In geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U40. It has 84 faces (60 triangles, 12 pentagons, and 12 pentagrams), 150 edges, and 60 vertices. It is given a Schläfli symbol sr{,5}, as a snub great dodecahedron.

'Maha' signifies the Mahogany tree and 'ganitham' Mathematics. Concepts like the Golden ratio and Fibonacci series are used in the design. More than thousand mathematical entities and geometrical shapes are engraved in the sculpture. The five Platonic bodies - tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron - are present in it.

The Wigner–Seitz cell always has the same point symmetry as the underlying Bravais lattice. For example, the cube, truncated octahedron, and rhombic dodecahedron have point symmetry Oh, since the respetive Bravais lattices used to generate them all belong to the cubic lattice system, which has Oh point symmetry.

In this simple example, we observe that the surface inherits the topology of a sphere and so receives a curvature. The final structure, here a pentagonal dodecahedron, allows for a perfect propagation of the pentagonal order. It is called an "ideal" (defect- free) model for the considered structure.

3 or 35. It is the dual of the dodecahedron, which is represented by {5,3}, having three pentagonal faces around each vertex. A regular icosahedron is a strictly convex deltahedron and a gyroelongated pentagonal bipyramid and a biaugmented pentagonal antiprism in any of six orientations. The name comes .

The Basilica of Our Lady (Maastricht), whose enneahedral tower tops form a space-filling polyhedron. Slicing a rhombic dodecahedron in half through the long diagonals of four of its faces results in a self-dual enneahedron, the square diminished trapezohedron, with one large square face, four rhombus faces, and four isosceles triangle faces. Like the rhombic dodecahedron itself, this shape can be used to tessellate three-dimensional space.. An elongated form of this shape that still tiles space can be seen atop the rear side towers of the 12th-century Romanesque Basilica of Our Lady (Maastricht). The towers themselves, with their four pentagonal sides, four roof facets, and square base, form another space-filling enneahedron.

Petersen graph as Kneser graph KG_{5,2} The Petersen graph is the complement of the line graph of K_5. It is also the Kneser graph KG_{5,2}; this means that it has one vertex for each 2-element subset of a 5-element set, and two vertices are connected by an edge if and only if the corresponding 2-element subsets are disjoint from each other. As a Kneser graph of the form KG_{2n-1,n-1} it is an example of an odd graph. Geometrically, the Petersen graph is the graph formed by the vertices and edges of the hemi-dodecahedron, that is, a dodecahedron with opposite points, lines and faces identified together.

The Eureka Ball consists of 12 magnetic design pieces , each a seven-sided, truncated pentagonal pyramid. Together, these Eureka pieces form a beautiful 12-sided dodecahedron. The Eureka Ball also has polarity dots on its pieces to help the user connect them together correctly. The Eureka Ball comes in Red.

The Oregonian, June 13, 1958, p. 35. facility housed in a temporary dome, but in 1967 it was replaced by a larger, 142-seat facility in a distinctive dodecahedron (12-sided) building equipped with a new projector."New OMSI Planetarium Offers Story Of Stars". The Oregonian, December 15, 1967, p. 41.

The hardness of paulingite is 5. The size of the paulingite unit cell is outstanding because it is the largest inorganic compounds exceeding most complex, intermetallic compounds. The measured density is 2.085 g/cm3 and calculated density is 2.10 g/cm3. The figure below shows the dodecahedron shape of paulingite mineral.

Finite reflection groups are the point groups Cnv, Dnh, and the symmetry groups of the five Platonic solids. Dual regular polyhedra (cube and octahedron, as well as dodecahedron and icosahedron) give rise to isomorphic symmetry groups. The classification of finite reflection groups of R3 is an instance of the ADE classification.

Melbourne-based Indigenous artist Reko Rennie was awarded an 200,000 commission to create 'Murri Totems', an installation located at the entrance of the LIMS building. The artwork consists of four vertical structures incorporating the five platonic forms – icosahedron, octahedron, star tetrahedron, hexahedron and dodecahedron – and painted with Rennie’s traditional pattern.

3D model of a pentagonal hexecontahedron In geometry, a pentagonal hexecontahedron is a Catalan solid, dual of the snub dodecahedron. It has two distinct forms, which are mirror images (or "enantiomorphs") of each other. It has 92 vertices that span 60 pentagonal faces. It is the Catalan solid with the most vertices.

However, another highly symmetric class of polyhedra, the Catalan solids, do not all have ideal forms. The Catalan solids are the dual polyhedra to the Archimedean solids, and have symmetries taking any face to any other face. Catalan solids that cannot be ideal include the rhombic dodecahedron and the triakis tetrahedron.; see .

A 6-color Megaminx, solved A 12-color Megaminx, solved A 12-color Megaminx in a star-pattern arrangement The Megaminx or Mégaminx (, ) is a dodecahedron- shaped puzzle similar to the Rubik's Cube. It has a total of 50 movable pieces to rearrange, compared to the 20 movable pieces of the Rubik's Cube.

The Sacrament of the Last Supper (1955): The canvas of this surrealist masterpiece by Salvador Dalí is a golden rectangle. A huge dodecahedron, with edges in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.Hunt, Carla Herndon and Gilkey, Susan Nicodemus. Teaching Mathematics in the Block pp.

In the dialogue Timaeus Plato associated each of the four classical elements (earth, air, water, and fire) with a regular solid (cube, octahedron, icosahedron, and tetrahedron respectively) due to their shape, the so-called Platonic solids. The fifth regular solid, the dodecahedron, was supposed to be the element which made up the heavens.

Floor mosaic by Paolo Uccello, 1430 A small stellated dodecahedron can be seen in a floor mosaic in St Mark's Basilica, Venice by Paolo Uccello circa 1430. See in particular p. 42. The same shape is central to two lithographs by M. C. Escher: Contrast (Order and Chaos) (1950) and Gravitation (1952).

In mathematics, Tait's conjecture states that "Every 3-connected planar cubic graph has a Hamiltonian cycle (along the edges) through all its vertices". It was proposed by and disproved by , who constructed a counterexample with 25 faces, 69 edges and 46 vertices. Several smaller counterexamples, with 21 faces, 57 edges and 38 vertices, were later proved minimal by . The condition that the graph be 3-regular is necessary due to polyhedra such as the rhombic dodecahedron, which forms a bipartite graph with six degree-four vertices on one side and eight degree-three vertices on the other side; because any Hamiltonian cycle would have to alternate between the two sides of the bipartition, but they have unequal numbers of vertices, the rhombic dodecahedron is not Hamiltonian.

3D model of a great icosidodecahedron In geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U54. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 30 vertices. It is given a Schläfli symbol r{3,}. It is the rectification of the great stellated dodecahedron and the great icosahedron.

3D model of a small snub icosicosidodecahedron In geometry, the small snub icosicosidodecahedron or snub disicosidodecahedron is a uniform star polyhedron, indexed as U32. It has 112 faces (100 triangles and 12 pentagrams), 180 edges, and 60 vertices. Its stellation core is a truncated pentakis dodecahedron. It also called a holosnub icosahedron, ß{3,5}.

In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra.Coxeter, Star polytopes and the Schläfli function f(α,β,γ) p. 121 1. The Kepler–Poinsot polyhedra They may be obtained by stellating the regular convex dodecahedron and icosahedron, and differ from these in having regular pentagrammic faces or vertex figures.

The cubic distance-regular graphs have been completely classified. The 13 distinct cubic distance-regular graphs are K4 (or tetrahedron), K3,3, the Petersen graph, the cube, the Heawood graph, the Pappus graph, the Coxeter graph, the Tutte–Coxeter graph, the dodecahedron, the Desargues graph, Tutte 12-cage, the Biggs–Smith graph, and the Foster graph.

For a uniform polyhedron, the face of the dual polyhedron may be found from the original polyhedron's vertex figure using the Dorman Luke construction., p. 117; , p. 30. As an example, the illustration below shows the vertex figure (red) of the cuboctahedron being used to derive a face (blue) of the rhombic dodecahedron. Image:DormanLuke.

It is a rare zeolite mineral with a dodecahedron crystal form {110} and has a very large unit cell with a= 3.51 nanometers. The mineral information was described by Kamb and Oke (1960) which has Si/Al ratio of 3.0, a BaO range of 0.5-.4.1% and 18.5% of water content (Tscherinch and Wise, 1982).

Several black metal bands have risen to prominence from the Netherlands recently. Carach Angren, Cirith Gorgor, Israthoum, Dodecahedron, Ordo Draconis, Funeral Winds, Lugubre, Slechtvalk and Urfaust are some of the best-known. Israeli group Melechesh have also made the Netherlands their permanent base of operations. Viking/folk metal band Heidevolk are also gaining popularity.

3D model of a rhombidodecadodecahedron In geometry, the rhombidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U38. It has 54 faces (30 squares, 12 pentagons and 12 pentagrams), 120 edges and 60 vertices. It is given a Schläfli symbol t0,2{,5}, and by the Wythoff construction this polyhedron can also be named a cantellated great dodecahedron.

The dual of the crossed square cupola has 8 triangular and 4 kite faces: 240px Due to faces of the crossed square cupola passing close to its centre, this dual is very spiky in appearance. This also occurs for the dual uniform polyhedra known as the great pentakis dodecahedron (DU58) and medial inverted pentagonal hexecontahedron (DU60).

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.

It can be seen as a polyhedron compound of a great icosahedron and great stellated dodecahedron. It is one of five compounds constructed from a Platonic solid or Kepler-Poinsot solid, and its dual. It is a stellation of the great icosidodecahedron. It has icosahedral symmetry (Ih) and it has the same vertex arrangement as a great rhombic triacontahedron.

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.

Meglos is the second serial of the 18th season of the science fiction television series Doctor Who, which was first broadcast in four weekly parts on BBC1 from 27 September to 18 October 1980. In the serial, the Zolfa-Thuran plant Meglos steals a huge source of power on the planet Tigella known as the Dodecahedron.

Gravitation (also known as Gravity) is a mixed media work by the Dutch artist M. C. Escher completed in June 1952. It was first printed as a black-and-white lithograph and then coloured by hand in watercolour. It depicts a nonconvex regular polyhedron known as the small stellated dodecahedron. Each facet of the figure has a trapezoidal doorway.

Regular skew polyhedra can also be constructed in dimensions higher than 4 as embeddings into regular polytopes or honeycombs. For example, the regular icosahedron can be embedded into the vertices of the 6-demicube; this was named the regular skew icosahedron by H. S. M. Coxeter. The dodecahedron can be similarly embedded into the 10-demicube.

The convex regular icosahedron is usually referred to simply as the regular icosahedron, one of the five regular Platonic solids, and is represented by its Schläfli symbol {3, 5}, containing 20 triangular faces, with 5 faces meeting around each vertex. Its dual polyhedron is the regular dodecahedron {5, 3} having three regular pentagonal faces around each vertex.

For almost 2,000 years, the concept of a polyhedron as a convex solid had remained as developed by the ancient Greek mathematicians. During the Renaissance star forms were discovered. A marble tarsia in the floor of St. Mark's Basilica, Venice, depicts a stellated dodecahedron. Artists such as Wenzel Jamnitzer delighted in depicting novel star-like forms of increasing complexity.

In commercial reticulated foam, up to 98% of the faces are removed. The dodecahedron is sometimes given as the basic unit for these foams, but the most representative shape is a polyhedron with 13 faces. Cell size and cell size distribution are critical parameters for most applications. Porosity is typically 95%, but can be as high as 98%.

It is the only dodecahedral stellation with this property, apart from the dodecahedron itself. Its dual, the great icosahedron, is related in a similar fashion to the icosahedron. It is the only regular star polyhedron with a completely unique edge arrangement not shared by any other regular 3-polytope. Shaving the triangular pyramids off results in an icosahedron.

Every golden rhombic face has a face center, a vertex, and two edge centers of the original dodecahedron, with the edge centers forming the short diagonal. Each edge center is connected to two vertices and two face centers. Each face center is connected to five edge centers, and each vertex is connected to three edge centers.

This shape appears in a 1752 book by John Lodge Cowley, labeled as the dodecarhombus... As cited by . It is named after Stanko Bilinski, who rediscovered it in 1960.. Bilinski himself called it the rhombic dodecahedron of the second kind.. Bilinski's discovery corrected a 75-year-old omission in Evgraf Fedorov's classification of convex polyhedra with congruent rhombic faces.

A rhombic hexecontahedron can be constructed from a rhombic triacontahedron. A rhombic hexecontahedron can be constructed from a regular dodecahedron, by taking its vertices, its face centers and its edge centers and scaling them in or out from the body center to different extents. Thus, if the 20 vertices of a dodecahedron are pulled out to increase the circumradius by a factor of (ϕ+1)/2 ≈ 1.309, the 12 face centers are pushed in to decrease the inradius to (3-ϕ)/2 ≈ 0.691 of its original value, and the 30 edge centers are left unchanged, then a rhombic hexecontahedron is formed. (The circumradius is increased by 30.9% and the inradius is decreased by the same 30.9%.) Scaling the points by different amounts results in hexecontahedra with kite-shaped faces or other polyhedra.

The new campus was largely designed by A&D; Wejchert & Partners Architects and includes several notable structures, including the UCD Water Tower which was built in 1972 by John Paul Construction. The Tower won the 1979 Irish Concrete Society Award. It stands 60 metres high with a dodecahedron tank atop a pentagonal pillar. The Tower is part of the UCD Environmental Research Station.

The Schläfli symbol of a regular polyhedron is {p,q} if its faces are p-gons, and each vertex is surrounded by q faces (the vertex figure is a q-gon). For example, {5,3} is the regular dodecahedron. It has pentagonal (5 edges) faces, and 3 pentagons around each vertex. See the 5 convex Platonic solids, the 4 nonconvex Kepler-Poinsot polyhedra.

Salvador Dalí, influenced by the works of Matila Ghyka, explicitly used the golden ratio in his masterpiece, The Sacrament of the Last Supper. The dimensions of the canvas are a golden rectangle. A huge dodecahedron, in perspective so that edges appear in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.Hunt, Carla Herndon and Gilkey, Susan Nicodemus.

The dual figure of the octahedral compound, the compound of three cubes, is also shown in a later Escher woodcut, Waterfall, next to the same stellated rhombic dodecahedron.. The compound of three octahedra re-entered the mathematical literature more properly with the work of , who observed its existence and provided coordinates for its vertices. It was studied in more detail by and .

The building's exterior is a dodecahedron, while the interior is an octagon. It was built by the architects Guillermo Rossell de la Lama, Ramón Miquela Jáuregui and Joaquín Álvarez Ordonéz. The Polyforum mural of the "March of Humanity" by Siqueros acknowledges the potential of technological progress but is critical of the failure of the Mexican Revolution to achieve freedom and social justice.

Its name comes from a topological construction from the snub dodecahedron with the kis operator applied to the pentagonal faces. In this construction, all the vertices are computed to be the same distance from the center. The 80 of the triangles are equilateral, and 60 triangles from the pentagons are isosceles. It is a (2,1) geodesic polyhedron, made of all triangles.

3D model of a (uniform) pentagonal antiprism In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even- numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagons joined to each other by a ring of 10 triangles for a total of 12 faces. Hence, it is a non-regular dodecahedron.

The building was unfinished when Suarez died, and remained an unfinished skeleton for many years. The Polyforum Cultural Siqueiros was built in the grounds of the hotel, with murals by the painter David Alfaro Siqueiros. It was started early in the 1970s and completed before Siqueiros died in 1974. The building's exterior is a dodecahedron, while the interior is an octagon.

Three of the huts within the nominated boundary area are within about of each other. "The Lodge" hut, 2011 A stone hut known as "The Lodge", on lot 17, is the most substantial building in the area. It is 12-sided, a dodecahedron. The walls are made of "billy boulders", above which are horizontal logs between windows of various types using recycled materials.

Immersive Media's product platform and IP touch all areas of the production process, from capture, stitching, post production, distribution and play back. The 1st capture system was the Dodeca 2360, a camera named after the geodesic geometry of the Dodecahedron, on which the patent is based. The twelve-sided camera has eleven lenses which simultaneously record video. The twelfth side is the base of the camera.

The 25 great circles can be seen in 3 sets: 12, 9, and 4, each representing edges of a polyhedron projected onto a sphere. Nine great circles represent the edges of a disdyakis dodecahedron, the dual of a truncated cuboctahedron. Four more great circles represent the edges of a cuboctahedron, and the last twelve great circles connect edge-centers of the octahedron to centers of other triangles.

The painting portrays the friar and mathematician with a table filled with geometrical tools: slate, chalk, compass, a dodecahedron model. A rhombicuboctahedron, half-filled with water and characterized by a detailed triple reflection effect of the Ducal Palace of Urbino, is suspended from the ceiling. Pacioli is demonstrating a theorem by Euclid written in an open book. The closed book, with the inscription LI.RI.LUC.BUR.

As a spherical tiling It can also be seen as the compound of ten tetrahedra with full icosahedral symmetry (Ih). It is one of five regular compounds constructed from identical Platonic solids. It shares the same vertex arrangement as a dodecahedron. The compound of five tetrahedra represents two chiral halves of this compound (it can therefore be seen as a "compound of two compounds of five tetrahedra").

During the Hellenistic era, small, hollow bronze Roman dodecahedra were made and have been found in various Roman ruins in Europe. Their purpose is not certain. In 20th-century art, dodecahedra appear in the work of M. C. Escher, such as his lithographs Reptiles (1943) and Gravitation (1952). In Salvador Dalí's painting The Sacrament of the Last Supper (1955), the room is a hollow regular dodecahedron.

Every stellation of one polytope is dual, or reciprocal, to some facetting of the dual polytope. The regular star polyhedra can also be obtained by facetting the Platonic solids. listed the simpler facettings of the dodecahedron, and reciprocated them to discover a stellation of the icosahedron that was missing from the set of "59".. More have been discovered since, and the story is not yet ended.

For example, the tetrahedron is an alternated cube, h{4,3}. Diminishment is a more general term used in reference to Johnson solids for the removal of one or more vertices, edges, or faces of a polytope, without disturbing the other vertices. For example, the tridiminished icosahedron starts with a regular icosahedron with 3 vertices removed. Other partial truncations are symmetry-based; for example, the tetrahedrally diminished dodecahedron.

In mathematics, Seifert–Weber space (introduced by Herbert Seifert and Constantin Weber) is a closed hyperbolic 3-manifold. It is also known as Seifert–Weber dodecahedral space and hyperbolic dodecahedral space. It is one of the first discovered examples of closed hyperbolic 3-manifolds. It is constructed by gluing each face of a dodecahedron to its opposite in a way that produces a closed 3-manifold.

Canonical dual compound of cuboctahedron (light) and rhombic dodecahedron (dark). Pairs of edges meet on their common midsphere. Any convex polyhedron can be distorted into a canonical form, in which a unit midsphere (or intersphere) exists tangent to every edge, and such that the average position of the points of tangency is the center of the sphere. This form is unique up to congruences.

Dodecahedrane is a chemical compound, a hydrocarbon with formula , whose carbon atoms are arranged as the vertices (corners) of a regular dodecahedron. Each carbon is bound to three neighbouring carbon atoms and to a hydrogen atom. This compound is one of the three possible Platonic hydrocarbons, the other two being cubane and tetrahedrane. Dodecahedrane does not occur in nature and has no significant uses.

The original Hunt the Wumpus has the caves arranged as the vertices of a dodecahedron In early 1973, Gregory Yob was looking through some of the games published by the People's Computer Company (PCC), and grew annoyed that there were multiple games, including Hurkle and Mugwump, that had the player "hide and seek" in a 10 by 10 grid. Yob was inspired to make a game that used a non-grid pattern, where the player would move through points connected through some other type of topology. Yob came up with the name "Hunt the Wumpus" that afternoon, and decided from there that the player would traverse through rooms arranged in a non-grid pattern, with a monster called a Wumpus somewhere in them. Yob chose a dodecahedron because it was his favorite platonic solid, and because he had once made a kite shaped like one.

3D model of a great snub icosidodecahedron In geometry, the great snub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U57. It has 92 faces (80 triangles and 12 pentagrams), 150 edges, and 60 vertices. It can be represented by a Schläfli symbol sr{,3}, and Coxeter-Dynkin diagram . This polyhedron is the snub member of a family that includes the great icosahedron, the great stellated dodecahedron and the great icosidodecahedron.

The first half of the book described the sky and legends attached to it. The last half of the book contained star charts to be punched out and held in front of lamps, projecting stars in their proper relationships onto a wall or other smooth clear surface. Spitz dodecahedron planetarium projector (1953) A Spitz Junior home planetarium projector. About a million units were produced between 1954 and about 1972.

During the late 1930s, Armand Spitz, a part- time lecturer at The Fels Planetarium of the Franklin Institute in Philadelphia, had started developing his own projector for the planetarium. However, while designing an inexpensive planetarium projector, Spitz realized the difficulties involving the icosahedron shaped projector globe. He approached Albert Einstein for a solution. Following Einstein's suggestion, Spitz used a dodecahedron and managed to produce an inexpensive planetarium projector.

See for a detailed history of combinatorial group theory. A proto-form is found in the 1856 icosian calculus of William Rowan Hamilton, where he studied the icosahedral symmetry group via the edge graph of the dodecahedron. The foundations of combinatorial group theory were laid by Walther von Dyck, student of Felix Klein, in the early 1880s, who gave the first systematic study of groups by generators and relations.

The Megaminx is made in the shape of a dodecahedron, and has 12 faces and center pieces, 20 corner pieces, and 30 edge pieces. The face centers each have a single color, which identifies the color of that face in the solved state. The edge pieces have two colors, and the corner pieces have three. Each face contains a center piece, 5 corner pieces and 5 edge pieces.

For this placement of the segments, one vertex of the parallelohedron will itself be at the origin, and the rest will be at positions given by sums of certain subsets of these vectors. A parallelohedron with g vectors can in this way be parameterized by 3g coordinates, three for each vector, but only some of these combinations are valid (because of the requirement that certain triples of segments lie in parallel planes, or equivalently that certain triples of vectors are coplanar) and different combinations may lead to parallelohedra that differ only by a rotation, scaling transformation, or more generally by an affine transformation. When affine transformations are factored out, the number of free parameters that describe the shape of a parallelohedron is zero for a parallelepiped (all parallelepipeds are equivalent to each other under affine transformations), two for a hexagonal prism, three for a rhombic dodecahedron, four for an elongated dodecahedron, and five for a truncated octahedron.

Reptiles depicts a desk upon which is a 2D drawing of a tessellated pattern of reptiles and hexagons, Escher's 1939 Regular Division of the Plane. The reptiles at one edge of the drawing emerge into three dimensional reality, come to life and appear to crawl over a series of symbolic objects (a book on nature, a geometer's triangle, a three dimensional dodecahedron, a pewter bowl containing a box of matches and a box of cigarettes) to eventually re-enter the drawing at its opposite edge. Other objects on the desk are a potted cactus and yucca, a ceramic flask with a cork stopper next to a small glass of liquid, a book of JOB cigarette rolling papers, and an open handwritten note book of many pages. Although only the size of small lizards, the reptiles have protruding crocodile-like fangs, and the one atop the dodecahedron has a dragon-like puff of smoke billowing from its nostrils.

The five Platonic solids – the regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron – viewed as two-dimensional surfaces, have the property that any flag (a triple of a vertex, edge, and face that all meet each other) can be taken to any other flag by a symmetry of the surface. More generally, a map embedded in a surface with the same property, that any flag can be transformed to any other flag by a symmetry, is called a regular map. If a regular map is used to generate a clean dessin, and the resulting dessin is used to generate a triangulated Riemann surface, then the edges of the triangles lie along lines of symmetry of the surface, and the reflections across those lines generate a symmetry group called a triangle group, for which the triangles form the fundamental domains. For example, the figure shows the set of triangles generated in this way starting from a regular dodecahedron.

Great stellated dodecahedron enclosed by a skeletal icosahedron from Perspectiva corporum regularium The book focuses on the five Platonic solids, with the subtitles of its title page citing Plato's Timaeus and Euclid's Elements for their history. Each of these five shapes has a chapter, whose title page relates the connection of its polyhedron to the classical elements in medieval cosmology: fire for the tetrahedron, earth for the cube, air for the octahedron, and water for the icosahedron, with the dodecahedron representing the heavens, its 12 faces corresponding to the 12 symbols of the zodiac. Each chapter includes four engravings of polyhedra, each showing six variations of the shape including some of their stellations and truncations, for a total of 120 polyhedra. This great amount of variation, some of which obscures the original Platonic form of each polyhedron, demonstrates the theory of the time that all the variation seen in the physical world comes from the combination of these basic elements.

The dodecahedron is a regular polyhedron with Schläfli symbol {5,3}, having 3 pentagons around each vertex. In geometry, the Schläfli symbol is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more than three dimensions and discovered all their convex regular polytopes, including the six that occur in four dimensions.

Pyraminx Crystal, one face in the middle of a twist The puzzle consists of a dodecahedron sliced in such a way that each slice cuts through the centers of five different pentagonal faces. This cuts the puzzle into 20 corner pieces and 30 edge pieces, with 50 pieces in total. Each face consists of five corners and five edges. When a face is turned, these pieces and five additional edges move with it.

He points upward directing the viewer's attention to a dominating transparent torso with arms stretched outward spanning the width of the picture plane. The scene's setting is within a transparent dodecahedron or twelve-sided space as perceived in the pentagon- shaped windowpanes behind the table. In the background is a familiar landscape of Catalonia, which Dalí has included in his paintings numerous times, one example being his famous painting The Persistence of Memory.

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.

Topologically, the deltoidal hexecontahedron is identical to the nonconvex rhombic hexecontahedron. The deltoidal hexecontahedron can be derived from a dodecahedron (or icosahedron) by pushing the face centers, edge centers and vertices out to different radii from the body center. The radii are chosen so that the resulting shape has planar kite faces each such that vertices go to degree-3 corners, faces to degree-five corners, and edge centers to degree-four points.

The Megaminx, or Magic Dodecahedron, was invented by several people independently and produced by several different manufacturers with slightly different designs. Uwe Mèffert eventually bought the rights to some of the patents and continues to sell it in his puzzle shop under the Megaminx moniker.Jaap's puzzle page, Megaminx It is also known by the name Hungarian Supernova, invented by Dr. Cristoph Bandelow.twistypuzzles.com, Hungarian Supernova His version came out first, shortly followed by Meffert's Megaminx.

Gatineau Valley Historical Society By 1964, with a membership of 100 people, the club purchases an island for $12,000 near the site of the club's floating dock. The club consisted of two islands connected by a walkway, complete with cottage and two sleeping cabins. The Gatineau Boom Company donated lumber for a walkway to the island. The GRYC island clubhouse, which resembles a pinecone, was designed by James Strutt based on the rhombic dodecahedron.

Solem created a high-level programming language for controlling personal robots. In addition to initiating a laboratory program in artificial intelligence and robotics, Solem did "pioneering" calculations on the motility of microrobots (1994e). He showed unique mechanisms for self-assembly of motile microrobots based on Platonic solids, in particular the dodecahedron, which can assemble into a helix appropriate for propulsion at high-Reynolds number (2002). He described several microrobots for military applications (1996b).

In geometry, a plesiohedron is a special kind of space-filling polyhedron, defined as the Voronoi cell of a symmetric Delone set. Three-dimensional Euclidean space can be completely filled by copies of any one of these shapes, with no overlaps. The resulting honeycomb will have symmetries that take any copy of the plesiohedron to any other copy. The plesiohedra include such well- known shapes as the cube, hexagonal prism, rhombic dodecahedron, and truncated octahedron.

A 3-outerplanar graph, the graph of a rhombic dodecahedron. There are four vertices on the outside face, eight vertices on the second layer (light yellow), and two vertices on the third layer (darker yellow). Because of the symmetries of the graph, no other embedding has fewer layers. In graph theory, a k-outerplanar graph is a planar graph that has a planar embedding in which the vertices belong to at most k concentric layers.

There are distortions of the icosahedron that, while no longer regular, are nevertheless vertex-uniform. These are invariant under the same rotations as the tetrahedron, and are somewhat analogous to the snub cube and snub dodecahedron, including some forms which are chiral and some with Th-symmetry, i.e. have different planes of symmetry from the tetrahedron. The icosahedron is unique among the Platonic solids in possessing a dihedral angle not less than 120°.

The name truncated dodecadodecahedron is somewhat misleading: truncation of the dodecadodecahedron would produce rectangular faces rather than squares, and the pentagram faces of the dodecahedron would turn into truncated pentagrams rather than decagrams. However, it is the quasitruncation of the dodecadodecahedron, as defined by .. See especially the description as a quasitruncation on p. 411 and the photograph of a model of its skeleton in Fig. 114, Plate IV. For this reason, it is also known as the quasitruncated dodecadodecahedron.

This honeycomb's vertex figure is a tetrakis cube: 24 disphenoids meet at each vertex. The union of these 24 disphenoids forms a rhombic dodecahedron. Each edge of the tessellation is surrounded by either four or six disphenoids, according to whether it forms the base or one of the sides of its adjacent isosceles triangle faces respectively. When an edge forms the base of its adjacent isosceles triangles, and is surrounded by four disphenoids, they form an irregular octahedron.

Hall suggests that each bot may be in the shape of a dodecahedron with twelve arms extending outwards. Each arm would have four degrees of freedom. The foglets' bodies would be made of aluminum oxide rather than combustible diamond to avoid creating a fuel air explosive. Hall and his correspondents soon realized that utility fog could be manufactured en masse to occupy the entire atmosphere of a planet and replace any physical instrumentality necessary to human life.

Meglos abandons the Earthling, leaving a bemused man watching a cactus creature reassert himself in his laboratory. Meglos knows the Doctor has realigned the weapon. However, the creature is unable to stop the Doctor fleeing back to the TARDIS, taking the Earthling with him, and is also unable to persuade Grugger not to fire the weapon. From the TARDIS the Doctor and his friends witness the destruction of Zolfa-Thura, along with the Gaztaks, Meglos and the Dodecahedron.

Sound received awards in the Nob Yoshigahara Puzzle Design Competition in 2002 for the Block Box (first place and People's Choice Award), in 2003 for the Decorated Box (Honorable Mention), in 2004 for the Dodecahedron Box (Grand Prize and People's Choice Award), and in 2006 for the Maze Burr (Grand Prize and People's Choice Award). He received the Sam Loyd Award in 2009, which is a lifetime achievement award presented by the Association of Game and Puzzle Collectors.

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.

The "Paranoid Incarnation" was an overly cautious and extremely insane incarnation of The Nameless One. He was the incarnation that set up the traps within the Dodecahedron Journal. He was the incarnation that the Lady of Pain placed in her maze as punishment for killing people. He was also the incarnation who learned the language of Uyo and, after mastering it, killed off the linguist, Fin, to prevent people from being able to decode his journal writings.

By contrast, a highly nonspherical solid, the hexahedron (cube) represents "earth". These clumsy little solids cause dirt to crumble and break when picked up in stark difference to the smooth flow of water. Moreover, the cube's being the only regular solid that tessellates Euclidean space was believed to cause the solidity of the Earth. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarked, "...the god used [it] for arranging the constellations on the whole heaven".

A net for the regular dodecahedron Construction begins by choosing a size of the model, either the length of its edges or the height of the model. The size will dictate the material, the adhesive for edges, the construction time and the method of construction. The second decision involves colours. A single-colour cardboard model is easiest to construct -- and some models can be made by folding a pattern, called a net, from a single sheet of cardboard.

73 pound (33 kg) double crystal of gadolinite. Masses of coarsely crystallized fluorite up to weight were not rare, and some of these had very large faces of the cube and rhombic dodecahedron. Its color varied from dark green to puce and purple, and colorless transparent rough crystals having remarkably perfect cleavage were sometimes observed. Some of the fluorite was true chlorophane and exhibited a brilliant green light when strongly heated and viewed in the dark.

Monument and effigies, in Salisbury Cathedral, Wiltshire, of Sir Thomas Gorges (1536-1610) and his wife Helena Snakenborg (d.1635) He was buried in Salisbury Cathedral, Wiltshire, where survives (at the east end of the north choir aisle, on the north side of the Lady Chapel) his magnificent monument with recumbent effigies of himself and his wife erected in 1635 by his son Edward Gorges, 1st Baron Gorges,Per Latin inscription after the death of his widow. The sides of the elaborate canopy above the effigies, supported on four Solomonic columns, display sculpted framework polyhedra, including two cuboctahedra and an icosahedron and the canopy is topped by a celestial globe surmounted by a dodecahedron. These devices are possibly a reference to Leonardo da Vinci's drawings for Luca Pacioli (Divina Proportione, Paganini, Venice, 1509),Mathematical Gazetteer of the British Isles ultimately based on Plato's Timaeus in which each of the regular polyhedra (or Five Regular Solids) are assigned to the atomic structure of one of the Five Elements, with the dodecahedron representing the whole Celestial Sphere.

These subunits are conserved across many species, as the function of this complex is essential for the generation of ATP for all eukaryotes. The E3 binding protein directly interacts with the dihydrolipoamide transacetylase (E2) core, anchoring it to the complex. E3BP binds the I domain of E2 by its C-terminal I' domain. The composition of E2.E3BP was thought to be 60 E2 plus approximately 12 E3BP, however, equilibrium sedimentation and small angle x-ray scattering studies showed that the E3BP/E2 binding complex has a lower mass than the E2 subunit alone. Additionally, these studies showed that E3 binds to E2.E3BP outside the central dodecahedron of the PDH complex, and that this interaction creates a lower binding affinity for the E1 subunit. Together, these data support a substitution model, in which the smaller E3BP subunits replace the E2 subunits rather than adding to the 60-mer entire complex. The specific model illustrates 12 I domains of E2 being substituted by 12 I' domains of E3BP, thereby forming 6 dimer edges that are symmetrically located in the dodecahedron structure.

This was one of the first Armand Spitz planetariums. Concerned that the only planetariums then available were so expensive that few institutions could have them and few people would live near enough to visit, in 1947 Spitz completed design work on a very inexpensive planetarium model. The main problem, he discovered, was that creating a globe for stellar projection was very complex and expensive. Following a suggestion by Albert Einstein, Spitz used a dodecahedron as the "globe" equivalent for his star projector.

Only five planetariums existed in the United States before 1940. Concerned that the only planetariums then available were so expensive that few institutions could have them and few people would live near enough to visit, in 1947 Spitz completed design work on a very inexpensive planetarium model. The main problem, he discovered, was that creating a globe for stellar projection was very complex and expensive. Spitz used a dodecahedron as the "globe" equivalent for his star projector, a suggestion from Albert Einstein.

To see a plane, however, one has to look "past" the five pieces on top of it, all of which could/should have different colours than the plane being solved. If considering the pentagonal regions as faces, like in the great dodecahedron represented by Schläfli symbol {5,5/2}, then the requirement is for all faces to be monochrome (same color) and for opposite faces to share the same color. The puzzle does not turn smoothly, due to its unique design.Wray, C. G. (1981).

However, the Earthling fights back against his occupation, causing green cactus spikes to break out on his skin. When the Tigellans discover that the Dodecahedron is missing and sound the alarm, Meglos hides away, but the real Doctor arrives at the same time and is accused of theft. His bewilderment and charm are little defence as both Savants and Deons start to panic as the energy levels of the city start to fail. Lexa uses the situation to her own ends.

15th monohedral convex pentagonal type, discovered in 2015 In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon. A regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a regular pentagon, 108°, is not a divisor of 360°, the angle measure of a whole turn. However, regular pentagons can tile the hyperbolic plane and the sphere; the latter produces a tiling topologically equivalent to the dodecahedron.

Hackmanite dodecahedron from the Koksha Valley, Afghanistan Hackmanite is a variety of sodalite exhibiting tenebrescence. When hackmanite from Mont Saint- Hilaire (Quebec) or Ilímaussaq (Greenland) is freshly quarried, it is generally pale to deep violet but the color fades quickly to greyish or greenish white. Conversely, hackmanite from Afghanistan and the Myanmar Republic (Burma) starts off creamy white but develops a violet to pink-red color in sunlight. If left in a dark environment for some time, the violet will fade again.

The small cage again has the shape of a pentagonal dodecahedron (512), but the large one is a hexadecahedron (51264). Type II hydrates are formed by gases like O2 and N2. The unit cell of Type H consists of 34 water molecules, forming three types of cages – two small ones of different types, and one "huge". In this case, the unit cell consists of three small cages of type 512, two small ones of type 435663 and one huge of type 51268.

An alternative is to decrease the surface area to volume ratio of the pressurized volume, by using more anvils to converge upon a higher-order platonic solid, such as a dodecahedron. However, such a press would be complex and difficult to manufacture. Schematic of a BARS system The BARS apparatus is claimed to be the most compact, efficient, and economical of all the diamond-producing presses. In the center of a BARS device, there is a ceramic cylindrical "synthesis capsule" of about 2 cm3 in size.

The 120-cell consists of 120 dodecahedral cells. For visualization purposes, it is convenient that the dodecahedron has opposing parallel faces (a trait it shares with the cells of the tesseract and the 24-cell). One can stack dodecahedrons face to face in a straight line bent in the 4th direction into a great circle with a circumference of 10 cells. Starting from this initial ten cell construct there are two common visualizations one can use: a layered stereographic projection, and a structure of intertwining rings.

Roman dodecahedron Omnidirectional sound source Regular dodecahedral objects have found some practical applications, and have also played a role in the visual arts and in philosophy. Iamblichus states that Hippasus, a Pythagorean, perished in the sea, because he boasted that he first divulged "the sphere with the twelve pentagons."Florian Cajori, A History of Mathematics (1893) In Theaetetus, a dialogue of Plato, Plato was able to prove that there are just five uniform regular solids; these later became known as the platonic solids. Timaeus (c.

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.

Here, Pappus observed that a regular dodecahedron and a regular icosahedron could be inscribed in the same sphere such that their vertices all lay on the same 4 circles of latitude, with 3 of the icosahedron's 12 vertices on each circle, and 5 of the dodecahedron's 20 vertices on each circle. This observation has been generalized to higher- dimensional dual polytopes. # An addition by a later writer on another solution to the first problem of the book. Of Book IV the title and preface have been lost.

The Doctor and Romana break out of the loop by throwing it out of phase, and then land on Tigella in the middle of the hostile jungle. As the Doctor heads off to find Zastor, Romana stumbles across dangerous vegetation – deadly bell plants – and then the Gaztaks, waiting patiently for Meglos to return to their spaceship. She gives them the slip after a while and heads off to the city. Meglos has used his time as the Doctor to steal the Dodecahedron, shrinking it to minute size.

Let \phi be the golden ratio. The 12 points given by (0, \pm 1, \pm \phi) and cyclic permutations of these coordinates are the vertices of a regular icosahedron. Its dual regular dodecahedron, whose edges intersect those of the icosahedron at right angles, has as vertices the 8 points (\pm 1, \pm 1, \pm 1) together with the 12 points (0, \pm\phi, \pm 1/\phi) and cyclic permutations of these coordinates. All 32 points together are the vertices of a rhombic triacontahedron centered at the origin.

"The Six Platonic Solids", an image that humorously adds the Utah teapot to the five standard Platonic solids One famous ray-traced image, by James Arvo and David Kirk in 1987, shows six stone columns, five of which are surmounted by the Platonic solids (tetrahedron, cube, octahedron, dodecahedron, icosahedron). The sixth column supports a teapot. The image is titled "The Six Platonic Solids", with Arvo and Kirk calling the teapot "the newly discovered Teapotahedron". This image appeared on the covers of several books and computer graphic journals.

Among the generalized Petersen graphs are the n-prism G(n, 1), the Dürer graph G(6, 2), the Möbius-Kantor graph G(8, 3), the dodecahedron G(10, 2), the Desargues graph G(10, 3) and the Nauru graph G(12, 5). Four generalized Petersen graphs – the 3-prism, the 5-prism, the Dürer graph, and G(7, 2) – are among the seven graphs that are cubic, 3-vertex-connected, and well-covered (meaning that all of their maximal independent sets have equal size)..

The prototypical examples of exceptional objects arise in the classification of regular polytopes: in two dimensions, there is a series of regular n-gons for n ≥ 3\. In every dimension above 2, one can find analogues of the cube, tetrahedron and octahedron. In three dimensions, one finds two more regular polyhedra — the dodecahedron (12-hedron) and the icosahedron (20-hedron) — making five Platonic solids. In four dimensions, a total of six regular polytopes exist, including the 120-cell, the 600-cell and the 24-cell.

In chemistry, the dodecahedral molecular geometry describes the shape of compounds where eight atoms or groups of atoms or ligands are arranged around a central atom defining the vertices of a snub disphenoid (also known as a trigonal dodecahedron). This shape has D2d symmetry and is one of the three common shapes for octacoordinate transition metal complexes, along with the square antiprism and the bicapped trigonal prism.Wells A.F. (1984) Structural Inorganic Chemistry 5th edition Oxford Science Publications One example of the dodecahedral molecular geometry is the ion.

By ordering the solids selectively—octahedron, icosahedron, dodecahedron, tetrahedron, cube—Kepler found that the spheres could be placed at intervals corresponding to the relative sizes of each planet's path, assuming the planets circle the Sun. Kepler also found a formula relating the size of each planet's orb to the length of its orbital period: from inner to outer planets, the ratio of increase in orbital period is twice the difference in orb radius. However, Kepler later rejected this formula, because it was not precise enough.Caspar. Kepler, pp.

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.

Rebecca Kamen in front of her installation The Platonic Solids at the Chemical Heritage Foundation, 2011 The Platonic Solids was inspired by Plato's conception of the five classical elements: earth, air, fire, water, and ether. In Plato's work Timaeus (ca. 350 BCE), the five forms of matter are related to elemental solids and shapes (the cube, the octahedron, the tetrahedron, the icosahedron, and the dodecahedron). In Kamen's work these regular polyhedra, created from fiberglass rods and sheets of mylar, are held against the larger plane of the wall, demonstrating "tension and compression".

Following a demonstration at an astronomical conference at the Harvard-Smithsonian Center for Astrophysics, Spitz received considerable publicity, and began marketing his Model A planetarium for $500. These were sold to the various American military academies, small museums, schools, and even to King Farouk of Egypt. Within a few years, Spitz introduced the model A-1, which incorporated the Sun, Moon, and five naked eye planets, still using the dodecahedron shape for the star projector. Later a model A-2 came out, projecting more stars (the model A only gave stars brighter than magnitude 4.3).

This shows that the Penrose tiling has a scaling self- similarity, and so can be thought of as a fractal. Penrose originally discovered the P1 tiling in this way, by decomposing a pentagon into six smaller pentagons (one half of a net of a dodecahedron) and five half- diamonds; he then observed that when he repeated this process the gaps between pentagons could all be filled by stars, diamonds, boats and other pentagons. By iterating this process indefinitely he obtained one of the two P1 tilings with pentagonal symmetry.

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 same applies to star polyhedra, although here we must be careful to make the net for only the visible outer surface. If this net is drawn on cardboard, or similar foldable material (for example, sheet metal), the net may be cut out, folded along the uncut edges, joined along the appropriate cut edges, and so forming the polyhedron for which the net was designed. For a given polyhedron there may be many fold-out nets. For example, there are 11 for the cube, and over 900000 for the dodecahedron.

Some interesting fold-out nets of the cube, octahedron, dodecahedron and icosahedron are available here. Numerous children's toys, generally aimed at the teen or pre-teen age bracket, allow experimentation with regular polygons and polyhedra. For example, klikko provides sets of plastic triangles, squares, pentagons and hexagons that can be joined edge-to-edge in a large number of different ways. A child playing with such a toy could re-discover the Platonic solids (or the Archimedean solids), especially if given a little guidance from a knowledgeable adult.

With shapes different from the Chinese cross the level of difficulty reached levels of up to 100 moves for the first piece to be removed, a scale humans would struggle to grasp. The peak of this development is a puzzle in which the addition of a few pieces doubles the number of moves. Prior to the 2003 publication of the RD Design Project by Owen, Charnley and Strickland, puzzles without right angles could not be efficiently analyzed by computers. Stewart Coffin has been creating puzzles based upon the rhombic dodecahedron since the 1960s.

Low, Brink, and Robbins spacewalk to the asteroid and set the charges. While they are successful in altering the orbit of Attila, they find the inside of the asteroid appears hollow, and proceed to explore. When they enter a central chamber, they are trapped as the asteroid transforms into a dodecahedron pod and rapidly accelerates away into deep space. When the three recover and can exit the pod, they find themselves on an alien planet, on a central island surrounded by five smaller, spire-shaped islands; in the game's novelization, they name the planet Cocytus.

The cube is the cell of the only regular tiling of three-dimensional Euclidean space. It is also unique among the Platonic solids in having faces with an even number of sides and, consequently, it is the only member of that group that is a zonohedron (every face has point symmetry). The cube can be cut into six identical square pyramids. If these square pyramids are then attached to the faces of a second cube, a rhombic dodecahedron is obtained (with pairs of coplanar triangles combined into rhombic faces).

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).

There is a prominent zigzag pattern of chains of Na polyhedra extending in the c direction on the outer layer of the slab. The Na1 octahedron shares an edge with the Na2 augmented octahedron which shares a face with the Na3 triangular dodecahedron. This forms a linear trimer that extends in the [011] direction. This trimer is then links by edge-sharing between the Na3 and a1 polyhedra to another trimer extending in the [0-11] direction. This motif continues to form a [Na3Φ14] zigzag chain extending in the c direction.

Armenian Heritage Park The Armenian Heritage Park is dedicated to the victims of the Armenian Genocide and acknowledges the history of Boston as a port of entry for immigrants worldwide, and celebrates those who have migrated to Massachusetts shores and contributed to American life and culture. The Park consists of two key features surrounded by seating, brick paving and landscaping. The Abstract Sculpture, a split dodecahedron, is mounted on a Reflecting Pool, represents the immigrant experience. The Labyrinth, a circular winding path paved in granite and set in lawn, celebrates life's journey.

Icosahedral symmetry fundamental domains soccer ball, a common example of a spherical truncated icosahedron, has full icosahedral symmetry. A regular icosahedron has 60 rotational (or orientation-preserving) symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation. A regular dodecahedron has the same set of symmetries, since it is the dual of the icosahedron. The set of orientation-preserving symmetries forms a group referred to as A5 (the alternating group on 5 letters), and the full symmetry group (including reflections) is the product A5 × Z2.

The dodecahedral conjecture in geometry is intimately related to sphere packing. László Fejes Tóth, a 20th-century Hungarian geometer, considered the Voronoi decomposition of any given packing of unit spheres. He conjectured in 1943 that the minimal volume of any cell in the resulting Voronoi decomposition was at least as large as the volume of a regular dodecahedron circumscribed to a unit sphere.. Thomas Callister Hales and Sean McLaughlin proved the conjecture in 1998,. following the same strategy that led Hales to his proof of the Kepler conjecture.

There is doubt regarding the mathematical interpretation of these objects, as many have non-platonic forms, and perhaps only one has been found to be a true icosahedron, as opposed to a reinterpretation of the icosahedron dual, the dodecahedron.The Scottish Solids Hoax, It is also possible that the Etruscans preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery near Padua (in Northern Italy) in the late 19th century of a dodecahedron made of soapstone, and dating back more than 2,500 years (Lindemann, 1987).

The music video was co-directed by Takumi Shiga alongside the band's long-time stylist Hisashi "Momo" Kitazawa, and was unveiled on YouTube on September 28, 2011, the same day as the album's release. The video features a white dodecahedron in a dark room, where Sakanaction are playing inside. Each band member is wearing a mask in the shape of a large eyeball, making their faces invisible. Over the top of the shape, bright patterned lights are cast, sometimes showing brief glimpses of the band member's faces, primarily of Yamaguchi.

That is, it is a unit distance graph. The simplest non-orientable surface on which the Petersen graph can be embedded without crossings is the projective plane. This is the embedding given by the hemi-dodecahedron construction of the Petersen graph. The projective plane embedding can also be formed from the standard pentagonal drawing of the Petersen graph by placing a cross-cap within the five-point star at the center of the drawing, and routing the star edges through this cross-cap; the resulting drawing has six pentagonal faces.

Since every apex graph is linkless embeddable, this shows that there are graphs that are linkless embeddable but not YΔY-reducible and therefore that there are additional forbidden minors for the YΔY-reducible graphs. Robertson's apex graph is shown in the figure. It can be obtained by connecting an apex vertex to each of the degree-three vertices of a rhombic dodecahedron, or by merging two diametrally opposed vertices of a four-dimensional hypercube graph. Because the rhombic dodecahedron's graph is planar, Robertson's graph is an apex graph.

The solids were ordered with the innermost being the octahedron, followed by the icosahedron, dodecahedron, tetrahedron, and finally the cube, thereby dictating the structure of the solar system and the distance relationships between the planets by the Platonic solids. In the end, Kepler's original idea had to be abandoned, but out of his research came his three laws of orbital dynamics, the first of which was that the orbits of planets are ellipses rather than circles, changing the course of physics and astronomy. He also discovered the Kepler solids.

In 2003, lack of structure on the largest scales (above 60 degrees) in the cosmic microwave background as observed for one year by the WMAP spacecraft led to the suggestion, by Jean-Pierre Luminet of the Observatoire de Paris and colleagues, that the shape of the universe is a Poincaré sphere."Is the universe a dodecahedron?", article at PhysicsWorld. In 2008, astronomers found the best orientation on the sky for the model and confirmed some of the predictions of the model, using three years of observations by the WMAP spacecraft.

There are three ways to do this gluing consistently. Opposite faces are misaligned by 1/10 of a turn, so to match them they must be rotated by 1/10, 3/10 or 5/10 turn; a rotation of 3/10 gives the Seifert–Weber space. Rotation of 1/10 gives the Poincaré homology sphere, and rotation by 5/10 gives 3-dimensional real projective space. With the 3/10-turn gluing pattern, the edges of the original dodecahedron are glued to each other in groups of five.

In China, oncolytic adenovirus is an approved cancer treatment. Specific modifications on fibre proteins are used to target Adenovirus to certain cell types; a major effort is made to limit hepatotoxicity and prevent multiple organ failure. Adenovirus dodecahedron can qualify as a potent delivery platform for foreign antigens to human myeloid dendritic cells (MDC), and that it is efficiently presented by MDC to M1-specific CD8+ T lymphocytes. Adenovirus has been used for delivery of CRISPR/Cas9 gene editing systems, but high immune reactivity to viral infection has posed challenges in use for patients.

By ordering the solids correctly—octahedron, icosahedron, dodecahedron, tetrahedron, and cube—Kepler found that the spheres correspond to the relative sizes of each planet's path around the Sun, generally varying from astronomical observations by less than 10%. Kepler also found a formula relating the size of each planet's orbit to the length of its orbital period: from inner to outer planets, the ratio of increase in orbital period is twice the difference in orb radius. However, Kepler later rejected this formula because it was not precise enough.Caspar. Kepler, pp.

One face of an uncut octahedral diamond, showing trigons (of positive and negative relief) formed by natural chemical etching Diamonds occur most often as euhedral or rounded octahedra and twinned octahedra known as macles. As diamond's crystal structure has a cubic arrangement of the atoms, they have many facets that belong to a cube, octahedron, rhombicosidodecahedron, tetrakis hexahedron or disdyakis dodecahedron. The crystals can have rounded off and unexpressive edges and can be elongated. Diamonds (especially those with rounded crystal faces) are commonly found coated in nyf, an opaque gum- like skin.

In 2003, lack of structure on the largest scales (above 60 degrees) in the cosmic microwave background as observed for one year by the WMAP spacecraft led to the suggestion, by Jean- Pierre Luminet of the Observatoire de Paris and colleagues, that the shape of the universe is a Poincaré sphere."Is the universe a dodecahedron?", article at PhysicsWorld. In 2008, astronomers found the best orientation on the sky for the model and confirmed some of the predictions of the model, using three years of observations by the WMAP spacecraft.

The elongated dodecahedron, a zonohedron. Its eight red parallelogram faces correspond to ordinary points of a five-line arrangement; an equivalent form of the Sylvester–Gallai theorem states that every zonohedron has at least one parallelogram face. Arrangements of lines have a combinatorial structure closely connected to zonohedra, polyhedra formed as the Minkowski sum of a finite set of line segments, called generators. In this connection, each pair of opposite faces of a zonohedron corresponds to a crossing point of an arrangement of lines in the projective plane, with one line for each generator.

In 3-dimensional hyperbolic space, the dihedral angle of a polyhedron depends on its size. The regular hyperbolic honeycombs thus include two with four or five dodecahedra meeting at each edge; their dihedral angles thus are π/2 and 2π/5, both of which are less than that of a Euclidean dodecahedron. Apart from this effect, the hyperbolic honeycombs obey the same topological constraints as Euclidean honeycombs and polychora. The 4 compact and 11 paracompact regular hyperbolic honeycombs and many compact and paracompact uniform hyperbolic honeycombs have been enumerated.

For the sake of consistency, we consider the vertices of the regular polyhedra as speaker positions, which makes the twelve-vertex icosahedron the next in the list.Unfortunately, in the literature the icosahedral layout is commonly called a dodecahedron and vice versa, without justification as to why we should now consider faces rather than vertices. If suitable rigging options are available, it is capable of second-order full- sphere reproduction. A good and slightly more practical alternative is a horizontal hexagon complemented by two twisted triangles on floor and ceiling.

The Barth Sextic may be visualized in three dimensions as featuring 50 finite and 15 infinite ordinary double points (nodes). Referring to the figure, the 50 finite ordinary double points are arrayed as the vertices of 20 roughly tetrahedral shapes oriented such that the bases of these four-sided "outward pointing" shapes form the triangular faces of a regular icosidodecahedron. To these 30 icosidodecahedral vertices are added the summit vertices of the 20 tetrahedral shapes. These 20 points themselves are the vertices of a concentric regular dodecahedron circumscribed about the inner icosidodecahedron.

His one published game, Hunt the Wumpus (1975), written while he was attending University of Massachusetts Dartmouth, is one of the earliest adventure games. While living in Palo Alto, California, Yob came across logic games on a mainframe computer named Hurkle, Snark, and Mugwump. Each of these games was based on a 10 × 10 grid, and Yob recognized that a puzzle game on a computer could have a far more complex structure. He created the world for Wumpus in the shape of a dodecahedron, in part because as a child he made a kite with that shape.

A Ho-Mg-Zn icosahedral quasicrystal formed as a pentagonal dodecahedron, the dual of the icosahedron In 1984, Israeli chemist Dan Shechtman found an aluminum-manganese alloy having five-fold symmetry, in breach of crystallographic convention at the time which said that crystalline structures could only have two-, three-, four-, or six-fold symmetry. Due to fear of the scientific community's reaction, it took him two years to publish the results for which he was awarded the Nobel Prize in Chemistry in 2011. Since this time, hundreds of quasicrystals have been reported and confirmed. They exist in many metallic alloys (and some polymers).

Leonardo's illustration of a dodecahedron from Pacioli's Divina proportione (1509) Divina proportione (Divine proportion), a three-volume work by Luca Pacioli, was published in 1509. Pacioli, a Franciscan friar, was known mostly as a mathematician, but he was also trained and keenly interested in art. Divina proportione explored the mathematics of the golden ratio. Though it is often said that Pacioli advocated the golden ratio's application to yield pleasing, harmonious proportions, Livio points out that the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions.

For , every -critical graph (that is, every odd cycle) can be generated as a -constructible graph such that all of the graphs formed in its construction are also -critical. For , this is not true: a graph found by as a counterexample to Hajós's conjecture that -chromatic graphs contain a subdivision of , also serves as a counterexample to this problem. Subsequently, -critical but not -constructible graphs solely through -critical graphs were found for all . For , one such example is the graph obtained from the dodecahedron graph by adding a new edge between each pair of antipodal vertices.

The real Doctor has by now been able to prove that he did not steal the artefact and that there is a doppelgänger at work. Lexa realises her mistake but does not live long to regret it when she is shot dead while protecting Romana from a wounded Gaztak who was left behind. The Doctor, Romana, Caris and Deedrix head with K9 for the TARDIS, determined to follow the Gaztak ship. Grugger’s ship touches down on Zolfa-Thura and Meglos wastes no time in restoring the Dodecahedron to full size and placing it at a spot equidistant between the Screens.

It can also be seen as a nonuniform truncated icosahedron with pyramids augmented to the pentagonal and hexagonal faces with heights adjusted until the dihedral angles are zero, and the two pyramid type side edges are equal length. This construction is expressed in the Conway polyhedron notation jtI with join operator j. Without the equal edge constraint, the wide rhombi are kites if limited only by the icosahedral symmetry. joined truncated icosahedron The sixty broad rhombic faces in the rhombic enneacontahedron are identical to those in the rhombic dodecahedron, with diagonals in a ratio of 1 to the square root of 2.

A copy of Perspectiva corporum regularium in the Metropolitan Museum of Art, open to one of the pages depicting variations of the dodecahedron (Perspective of the Regular Solids) is a book of perspective drawings of polyhedra by German Renaissance goldsmith Wenzel Jamnitzer, with engravings by Jost Amman, published in 1568. Despite its Latin title, Perspectiva corporum regularium is written mainly in the German language. It was "the most lavish of the perspective books published in Germany in the late sixteenth century" and was included in several royal art collections. It may have been the first work to depict chiral icosahedral symmetry.

Other evidence suggests that he may have only been familiar with the tetrahedron, cube, and dodecahedron and that the discovery of the octahedron and icosahedron belong to Theaetetus, a contemporary of Plato. In any case, Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that no other convex regular polyhedra exist. The Platonic solids are prominent in the philosophy of Plato, their namesake. Plato wrote about them in the dialogue Timaeus 360 B.C. in which he associated each of the four classical elements (earth, air, water, and fire) with a regular solid.

A positively curved universe is described by elliptic geometry, and can be thought of as a three-dimensional hypersphere, or some other spherical 3-manifold (such as the Poincaré dodecahedral space), all of which are quotients of the 3-sphere. Poincaré dodecahedral space is a positively curved space, colloquially described as "soccerball-shaped", as it is the quotient of the 3-sphere by the binary icosahedral group, which is very close to icosahedral symmetry, the symmetry of a soccer ball. This was proposed by Jean-Pierre Luminet and colleagues in 2003"Is the universe a dodecahedron?", article at PhysicsWeb.

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.

Many of the exceptional objects in mathematics and physics have been found to be connected to each other. Developments such as the Monstrous moonshine conjectures show how, for example, the Monster group is connected to string theory. The theory of modular forms shows how the algebra E8 is connected to the Monster group. (In fact, well before the proof of the Monstrous moonshine conjecture, the elliptic j-function was discovered to encode the representations of E8.) Other interesting connections include how the Leech lattice is connected via the Golay code to the adjacency matrix of the dodecahedron (another exceptional object).

A Hamiltonian cycle around a network of six vertices In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a Hamiltonian path that is a cycle. Determining whether such paths and cycles exist in graphs is the Hamiltonian path problem, which is NP-complete. Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the icosian game, now also known as Hamilton's puzzle, which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron.

The icosian game is a mathematical game invented in 1857 by William Rowan Hamilton. The game's object is finding a Hamiltonian cycle along the edges of a dodecahedron such that every vertex is visited a single time, and the ending point is the same as the starting point. The puzzle was distributed commercially as a pegboard with holes at the nodes of the dodecahedral graph and was subsequently marketed in Europe in many forms. The motivation for Hamilton was the problem of symmetries of an icosahedron, for which he invented icosian calculus—an algebraic tool to compute the symmetries.

The use of the Penrose stairs is paralleled by Escher's Ascending and Descending (1960), where instead of the flow of water, two lines of monks endlessly march uphill or downhill around the four flights of stairs. The two support towers continue above the aqueduct and are topped by two compound polyhedra, revealing Escher's interest in mathematics as an artist. The one on the left is a compound of three cubes. The one on the right is a stellation of a rhombic dodecahedron (or a compound of three non-regular octahedra) and is known as Escher's solid.

The A Series As noted, Spitz wanted to create a projector more affordable than the German Zeiss "all optical" projectors. Thus, all of his projectors used large "star balls" that relied on the pinhole lens principle, where star images became smaller (more realistic) as the starlight source (in center of the star ball) was more distant from the star-ball surface. Larger holes drilled into the star ball resulted in larger dots on the dome; thus practically all such projectors used lenses for the larger holes (brighter stars) to condense the dot. The earlier mentioned A series used a dodecahedron star "ball" for easier manufacture.

Most planetaria ignore Uranus as being at best marginally visible to the naked eye. A great boost to the popularity of the planetarium worldwide was provided by the Space Race of the 1950s and 60s when fears that the United States might miss out on the opportunities of the new frontier in space stimulated a massive program to install over 1,200 planetaria in U.S. high schools. Early Spitz star projector Armand Spitz recognized that there was a viable market for small inexpensive planetaria. His first model, the Spitz A, was designed to project stars from a dodecahedron, thus reducing machining expenses in creating a globe.

360 B.C.), as a personage of Plato's dialogue, associates the other four platonic solids with the four classical elements, adding that there is a fifth solid pattern which, though commonly associated with the regular dodecahedron, is never directly mentioned as such; "this God used in the delineation of the universe."Plato, Timaeus, Jowett translation [line 1317–8]; the Greek word translated as delineation is diazographein, painting in semblance of life. Aristotle also postulated that the heavens were made of a fifth element, which he called aithêr (aether in Latin, ether in American English). Regular dodecahedra have been used as dice and probably also as divinatory devices.

Cubic graphs arise naturally in topology in several ways. For example, if one considers a graph to be a 1-dimensional CW complex, cubic graphs are generic in that most 1-cell attaching maps are disjoint from the 0-skeleton of the graph. Cubic graphs are also formed as the graphs of simple polyhedra in three dimensions, polyhedra such as the regular dodecahedron with the property that three faces meet at every vertex. Representation of a planar embedding as a graph-encoded map An arbitrary graph embedding on a two-dimensional surface may be represented as a cubic graph structure known as a graph-encoded map.

Shiga had only four days to create the projections before the video shoot, so contracted the work to an artist he previous collaborated with called Masaya Yoshida, as well as an art creation company. The projections of the members' faces were taken from video chats with the band members before the video recording. The dodecahedron was an idea of Kitazawa's, as he considered it one of the most beautiful shapes, and felt that a beautiful shape would work well as a metaphor for the human brain. The shape is made of scrim attached to a wire frame, in order for both the members and the projections to be seen.

The generalized Petersen graphs also include the n-prism G(n,1) the Dürer graph G(6,2), the Möbius-Kantor graph G(8,3), the dodecahedron G(10,2), the Desargues graph G(10,3) and the Nauru graph G(12,5). The Petersen family consists of the seven graphs that can be formed from the Petersen graph by zero or more applications of Δ-Y or Y-Δ transforms. The complete graph K6 is also in the Petersen family. These graphs form the forbidden minors for linklessly embeddable graphs, graphs that can be embedded into three-dimensional space in such a way that no two cycles in the graph are linked.

It includes new material on knotted polyhedra and on rings of regular octahedra and regular dodecahedra; as the ring of dodecahedra forms the outline of a golden rhombus, it can be extended to make skeletal pentagon-faced versions of the convex polyhedra formed from the golden rhombus, including the Bilinski dodecahedron, rhombic icosahedron, and rhombic triacontahedron. The second edition also includes the Császár polyhedron and Szilassi polyhedron, toroidal polyhedra with non-regular faces but with pairwise adjacent vertices and faces respectively, and constructions by Alaeglu and Giese of polyhedra with irregular but congruent faces and with the same numbers of edges at every vertex.

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.

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.

There is only one triangle, however, and the triangle has definite properties. In this way, truth is sought within mathematics and philosophy in a congruous way. Euclid's Elements could be thought of as a document whose objective is to construct a dodecahedron and an icosahedron (Propositions 16 and 17 book XIII). Appollonius' On Conics Book I could be thought of as a document whose objective is to construct a pair of hyperbolas from two bisecting lines (Proposition 50 of book I). Propositions have historically been used in logic and mathematics to work towards solving a problem, and these fields both reflect that in their foundations through Euclid and Aristotle.

Partition of the graph of a rhombic dodecahedron into two linear forests, showing that its linear arboricity is two In graph theory, a branch of mathematics, the linear arboricity of an undirected graph is the smallest number of linear forests its edges can be partitioned into. Here, a linear forest is an acyclic graph with maximum degree two; that is, it is a disjoint union of path graphs. Linear arboricity is a variant of arboricity, the minimum number of forests into which the edges can be partitioned. The linear arboricity of any graph of maximum degree \Delta is known to be at least \lceil\Delta/2\rceil and is conjectured to be at most \lceil(\Delta+1)/2\rceil.

Zastor and Deedrix are arrested in a Deon coup, with other Savants expelled to the hostile surface of the planet, while the Doctor himself is prepared for sacrifice to Ti. The doors of the city are sealed with Meglos trapped inside, with a hostage Savant named Caris for company. She soon gets the upper hand when the Earthling tries another bout of resistance. In a subsequent mix-up, Romana overpowers Caris, letting Meglos escape and reunite with the Gaztaks, who have staged an attack on the city to rescue him. With the miniaturized Dodecahedron in his possession, the pirates blast off back to Zolfa-Thura – though three Gaztaks, half the crew, have been lost.

The number of sides of each face is twice the number of lines that cross in the arrangement. For instance, the elongated dodecahedron shown is a zonohedron with five generators, two pairs of opposite hexagon faces, and four pairs of opposite parallelogram faces. In the corresponding five-line arrangement, two triples of lines cross (corresponding to the two pairs of opposite hexagons) and the remaining four pairs of lines cross at ordinary points (corresponding to the four pairs of opposite parallelograms). An equivalent statement of the Sylvester–Gallai theorem, in terms of zonohedra, is that every zonohedron has at least one parallelogram face (counting rectangles, rhombuses, and squares as special cases of parallelograms).

In the isometric system, the most common types of twins are the Spinel Law (twin plane, parallel to an octahedron), [111] where the twin axis is perpendicular to an octahedral face, and the Iron Cross [001] which is the interpenetration of two pyritohedrons a subtype of dodecahedron. In the hexagonal system, calcite shows the contact twin laws {0001} and {0112}. Quartz shows the Brazil Law {1120}, and Dauphiné Law [0001] which are penetration twins caused by transformation and Japan Law {1122} which is often caused by accidents during growth. In the tetragonal system, cyclical contact twins are the most commonly observed type of twin, such as in rutile titanium dioxide and cassiterite tin oxide.

The symmetries of the 3-D surface of the 600-cell are somewhat difficult to visualize due to both the large number of tetrahedral cells, and the fact that the tetrahedron has no opposing faces or vertices. One can start by realizing the 600-cell is the dual of the 120-cell. One may also notice that the 600-cell also contains the vertices of a dodecahedron, which with some effort can be seen in most of the below perspective projections. A three-dimensional model of the 600-cell, in the collection of the Institut Henri Poincaré, was photographed in 1934–1935 by Man Ray, and formed part of two of his later "Shakesperean Equation" paintings.. See in particular mathematical object mo-6.2, p.

In a finite, connected, simple, planar graph, any face (except possibly the outer one) is bounded by at least three edges and every edge touches at most two faces; using Euler's formula, one can then show that these graphs are sparse in the sense that if v ≥ 3: :e\leq 3v-6. A Schlegel diagram of a regular dodecahedron, forming a planar graph from a convex polyhedron. Euler's formula is also valid for convex polyhedra. This is no coincidence: every convex polyhedron can be turned into a connected, simple, planar graph by using the Schlegel diagram of the polyhedron, a perspective projection of the polyhedron onto a plane with the center of perspective chosen near the center of one of the polyhedron's faces.

The 26-fullerene graph has D_{3h} prismatic symmetry, the same group of symmetries as the triangular prism. This symmetry group has 12 elements; it has six symmetries that arbitrarily permute the three hexagonal faces of the graph and preserve the orientation of its planar embedding, and another six orientation-reversing symmetries. The number of fullerenes with a given even number of vertices grows quickly in the number of vertices; 26 is the largest number of vertices for which the fullerene structure is unique. The only two smaller fullerenes are the graph of the regular dodecahedron (a fullerene with 20 vertices) and the graph of the truncated hexagonal trapezohedron (a 24-vertex fullerene), which are the two types of cells in the Weaire–Phelan structure.

A space-filling tessellation, the trapezo-rhombic dodecahedral honeycomb, can be made by translated copies of this cell. Each "layer" is a hexagonal tiling, or a rhombille tiling, and alternate layers are connected by shifting their centers and rotating each polyhedron so the rhombic faces match up. :320px:260px In the special case that the long sides of the trapezoids equals twice the length of the short sides, the solid now represents the 3D Voronoi cell of a sphere in a Hexagonal Close Packing (HCP), next to Face-Centered- Cubic an optimal way to stack spheres in a lattice. It is therefore similar to the rhombic dodecahedron, which can be represented by turning the lower half of the picture at right over an angle of 60 degrees.

Another method based on the diffusion model is the hex-path algorithm, developed by R. Andrew Russel for underground chemical odor localization with a buried probe controlled by a robotic manipulator. The probe moves at a certain depth along the edges of a closely packed hexagonal grid. At each state junction , there are two paths (left and right) for choosing, and the robot will take the path that leads to higher concentration of the odor based on the previous two junction states odor concentration information , . In the 3D version of the hex-path algorithm, the dodecahedron algorithm, the probe moves in a path that corresponds to a closely packed dodecahedra, so that at each state point there are three possible path choices.

In three dimensions, it is not possible for a geometrically chiral polytope to have finitely many finite faces. For instance, the snub cube is vertex-transitive, but its flags have more than two orbits, and it is neither edge-transitive nor face-transitive, so it is not symmetric enough to meet the formal definition of chirality. The quasiregular polyhedra and their duals, such as the cuboctahedron and the rhombic dodecahedron, provide another interesting type of near-miss: they have two orbits of flags, but are mirror-symmetric, and not every adjacent pair of flags belongs to different orbits. However, despite the nonexistence of finite chiral three-dimensional polyhedra, there exist infinite three-dimensional chiral skew polyhedra of types {4,6}, {6,4}, and {6,6}.

Hunt the Wumpus is a text-based adventure game developed by Gregory Yob in 1973. In the game, the player moves through a series of connected caves, arranged in a dodecahedron, as they hunt a monster named the Wumpus. The turn- based game has the player trying to avoid fatal bottomless pits and "super bats" that will move them around the cave system; the goal is to fire one of their "crooked arrows" through the caves to kill the Wumpus. Yob created the game in early 1973 due to his annoyance at the multiple hide-and-seek games set in caves in a grid pattern, and it and multiple variations were sold via mail order by Yob and the People's Computer Company.

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".

For instance, the complete graph is a toroidal graph: it is not planar but can be embedded in a torus, with each face of the embedding being a triangle. This embedding has the Heawood graph as its dual graph.. The same concept works equally well for non- orientable surfaces. For instance, can be embedded in the projective plane with ten triangular faces as the hemi-icosahedron, whose dual is the Petersen graph embedded as the hemi-dodecahedron.. Even planar graphs may have nonplanar embeddings, with duals derived from those embeddings that differ from their planar duals. For instance, the four Petrie polygons of a cube (hexagons formed by removing two opposite vertices of the cube) form the hexagonal faces of an embedding of the cube in a torus.

The features of the cube and its dual octahedron correspond one-for-one with dimensions reversed. There are many distinct but interrelated dualities in which geometric or topological objects correspond to other objects of the same type, but with a reversal of the dimensions of the features of the objects. A classical example of this is the duality of the platonic solids, in which the cube and the octahedron form a dual pair, the dodecahedron and the icosahedron form a dual pair, and the tetrahedron is self-dual. The dual polyhedron of any of these polyhedra may be formed as the convex hull of the center points of each face of the primal polyhedron, so the vertices of the dual correspond one-for-one with the faces of the primal.

The Klein quartic is related to various other surfaces. Geometrically, it is the smallest Hurwitz surface (lowest genus); the next is the Macbeath surface (genus 7), and the following is the First Hurwitz triplet (3 surfaces of genus 14). More generally, it is the most symmetric surface of a given genus (being a Hurwitz surface); in this class, the Bolza surface is the most symmetric genus 2 surface, while Bring's surface is a highly symmetric genus 4 surface – see isometries of Riemann surfaces for further discussion. Algebraically, the (affine) Klein quartic is the modular curve X(7) and the projective Klein quartic is its compactification, just as the dodecahedron (with a cusp in the center of each face) is the modular curve X(5); this explains the relevance for number theory.

During the earlier years of its operation, the planetarium originally used simple folding chairs situated around a projection module, but in the 1980s the chairs were replaced by rows of curved benches arranged in semicircles on either side. In 2012, the planetarium was upgraded with the installation of digital hardware and software, greatly increasing capabilities, and allowing for the projection of specialized 360° video. However, the analog hardware originally used in the 1960s, including a control panel and a dodecahedron with optical pinholes used for star projection, are on permanent display on the main floor of the museum. The planetarium was further renovated between 2018–2019, including the replacement of the benches with modern seating in a more traditional theater arrangement, and rotating the hemispherical projection screen forward, both expanding audience capacity and improving comfortability of the domed theater.

An accident apparently directed René-Just Haüy's attention to what became a new field in natural history, crystallography. Haüy was examining a broken specimen of calcareous spar in the collection of Jacques de France de Croisset. (According to some accounts, Haüy dropped the specimen and caused it to break.) He became intrigued by the perfectly smooth plane of the fracture. Pearwood model of rock crystal rhomboid, made by René- Just Haüy, Teylers Museum Integrant molecules form a pentagonal dodecahedron of pyrite, Traité de minéralogie (1801) Studying the fragments inspired Haüy to make further experiments in crystal cutting. Breaking down crystals to the smallest pieces possible, Haüy concluded that each type of crystal has a fundamental primitive, nucleus or “integrant molecule” of a particular shape, that could not be broken further without destroying both the physical and chemical nature of the crystal.

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,.

The Prion star system contains two habitable planets which have supported civilisations: Zolfa-Thura, a desert world devoid seemingly of life structures bar five giant screens; and Tigella, a jungle world inhabited by the humanoid, white haired Tigellans. The structure of Tigellan society is based on two castes: the scientific Savants, led by the earnest Deedrix; and the religiously fanatical Deons, led by Lexa. The latter worship the Dodecahedron, a mysterious twelve-sided crystal which they see as a gift from the god Ti. The Savants, however, have utilised its power as an energy source for their entire civilisation. The planet’s leader, Zastor, mediates between the two factions, whose tensions have grown greater as the energy source has begun to fluctuate. When Zastor’s old friend the Doctor gets in touch, the weary leader invites him back to Tigella to investigate and help.

Aristotle added a fifth element, aithēr (aether in Latin, "ether" in English) and postulated that the heavens were made of this element, but he had no interest in matching it with Plato's fifth solid.Wildberg (1988): Wildberg discusses the correspondence of the Platonic solids with elements in Timaeus but notes that this correspondence appears to have been forgotten in Epinomis, which he calls "a long step towards Aristotle's theory", and he points out that Aristotle's ether is above the other four elements rather than on an equal footing with them, making the correspondence less apposite. Euclid completely mathematically described the Platonic solids in the Elements, the last book (Book XIII) of which is devoted to their properties. Propositions 13–17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order.

The game is turn-based; each cave is given a number by the game, and each turn begins with the player being told which cave they are in and which caves are connected to it by tunnels. The player then elects to either move to one of those connected caves or shoot one of their five "crooked arrows", named for their ability to change direction while in flight. Each cave is connected to three others, and the system as a whole is equivalent to a dodecahedron. Numbering of the caves in Hunt the Wumpus The caves are in complete darkness, so the player cannot see into adjacent caves; instead, upon moving to a new empty cave, the game describes if they can smell a Wumpus, hear a bat, or feel a draft from a pit in one of the connected caves.

A rhombicuboctahedron drawn by Leonardo da Vinci, 1509, four centuries before Escher Escher's interest in geometry is well known, but he was also an avid amateur astronomer, and in the early 1940s he became a member of the Dutch Association for Meteorology and Astronomy. He owned a 6 cm refracting telescope, and recorded several observations of binary stars. The use of polyhedra to model heavenly bodies can be traced back to Plato, who in the Timaeus identified the regular dodecahedron with the shape of the heavens and its 12 faces with the constellations of the zodiac.. Later, Johannes Kepler theorized that the distribution of distances of the planets from the sun could be explained by the shapes of the five Platonic solids, nested within each other. Escher kept a model of this system of nested polyhedra, and regularly depicted polyhedra in his artworks relating to astronomy and other worlds.

A map graph (top), the cocktail party graph K2,2,2,2, defined by corner adjacency of eight regions in the plane (lower left), or as the half-square of a planar bipartite graph (lower right, the graph of the rhombic dodecahedron) In graph theory, a branch of mathematics, a map graph is an undirected graph formed as the intersection graph of finitely many simply connected and internally disjoint regions of the Euclidean plane. The map graphs include the planar graphs, but are more general. Any number of regions can meet at a common corner (as in the Four Corners of the United States, where four states meet), and when they do the map graph will contain a clique connecting the corresponding vertices, unlike planar graphs in which the largest cliques have only four vertices.. Another example of a map graph is the king's graph, a map graph of the squares of the chessboard connecting pairs of squares between which the chess king can move.

The diagonal of the unit cube is . This projection of the Bilinski dodecahedron is a rhombus with diagonal ratio . The square root of 3 can be found as the leg length of an equilateral triangle that encompasses a circle with a diameter of 1. If an equilateral triangle with sides of length 1 is cut into two equal halves, by bisecting an internal angle across to make a right angle with one side, the right angle triangle's hypotenuse is length one and the sides are of length and . From this the trigonometric function tangent of 60° equals , and the sine of 60° and the cosine of 30° both equal . The square root of 3 also appears in algebraic expressions for various other trigonometric constants, includingJulian D. A. Wiseman Sin and Cos in Surds the sines of 3°, 12°, 15°, 21°, 24°, 33°, 39°, 48°, 51°, 57°, 66°, 69°, 75°, 78°, 84°, and 87°.

Geometric group theory grew out of combinatorial group theory that largely studied properties of discrete groups via analyzing group presentations, that describe groups as quotients of free groups; this field was first systematically studied by Walther von Dyck, student of Felix Klein, in the early 1880s, while an early form is found in the 1856 icosian calculus of William Rowan Hamilton, where he studied the icosahedral symmetry group via the edge graph of the dodecahedron. Currently combinatorial group theory as an area is largely subsumed by geometric group theory. Moreover, the term "geometric group theory" came to often include studying discrete groups using probabilistic, measure-theoretic, arithmetic, analytic and other approaches that lie outside of the traditional combinatorial group theory arsenal. In the first half of the 20th century, pioneering work of Max Dehn, Jakob Nielsen, Kurt Reidemeister and Otto Schreier, J. H. C. Whitehead, Egbert van Kampen, amongst others, introduced some topological and geometric ideas into the study of discrete groups.

This work has proven central to the modern study of classical field theories such as electromagnetism, and to the development of quantum mechanics. In pure mathematics, he is best known as the inventor of quaternions. William Rowan Hamilton's scientific career included the study of geometrical optics, classical mechanics, adaptation of dynamic methods in optical systems, applying quaternion and vector methods to problems in mechanics and in geometry, development of theories of conjugate algebraic couple functions (in which complex numbers are constructed as ordered pairs of real numbers), solvability of polynomial equations and general quintic polynomial solvable by radicals, the analysis on Fluctuating Functions (and the ideas from Fourier analysis), linear operators on quaternions and proving a result for linear operators on the space of quaternions (which is a special case of the general theorem which today is known as the Cayley–Hamilton theorem). Hamilton also invented "icosian calculus", which he used to investigate closed edge paths on a dodecahedron that visit each vertex exactly once.

A version was released for the HP-41C calculator in 1981, and a commercial port for the TI-99/4A home computer was published by Texas Instruments in 1980, with graphics and a torus shape of caves displayed in a grid pattern instead of a dodecahedron. Hunt the Wumpus has been cited as an early example of a survival horror game; the book Vampires and Zombies claims that it was an early example of the genre, while the paper "Restless dreams in Silent Hill" states that "from a historical perspective the genre's roots lie in Hunt the Wumpus". Other sources, however, such as the book The World of Scary Video Games, claim that the game lacks elements needed for a "horror" game, as the player hunts rather than is hunted by the Wumpus, and nothing in the game is explicitly intended to frighten the player, making it more of an early adventure or puzzle game. Kevin Cogger of 1Up.

Ancient writers refer to other works of Apollonius that are no longer extant: # Περὶ τοῦ πυρίου, On the Burning-Glass, a treatise probably exploring the focal properties of the parabola # Περὶ τοῦ κοχλίου, On the Cylindrical Helix (mentioned by Proclus) # A comparison of the dodecahedron and the icosahedron inscribed in the same sphere # Ἡ καθόλου πραγματεία, a work on the general principles of mathematics that perhaps included Apollonius's criticisms and suggestions for the improvement of Euclid's Elements # Ὠκυτόκιον ("Quick Bringing-to-birth"), in which, according to Eutocius, Apollonius demonstrated how to find closer limits for the value of than those of Archimedes, who calculated as the upper limit and as the lower limit # an arithmetical work (see Pappus) on a system both for expressing large numbers in language more everyday than that of Archimedes' The Sand Reckoner and for multiplying these large numbers # a great extension of the theory of irrationals expounded in Euclid, Book x., from binomial to multinomial and from ordered to unordered irrationals (see extracts from Pappus' comm. on Eucl. x., preserved in Arabic and published by Woepke, 1856).

The triangulation of the sphere with (2,3,5) triangle group, generated by using the regular dodecahedron to construct a clean dessin The triangulation of the hyperbolic plane with (2,3,7) triangle group generated as the universal cover of the Klein quartic A vertex in a dessin has a graph-theoretic degree, the number of incident edges, that equals its degree as a critical point of the Belyi function. In the example above, all white points have degree two; dessins with the property that each white point has two edges are known as clean, and their corresponding Belyi functions are called pure. When this happens, one can describe the dessin by a simpler embedded graph, one that has only the black points as its vertices and that has an edge for each white point with endpoints at the white point's two black neighbors. For instance, the dessin shown in the figure could be drawn more simply in this way as a pair of black points with an edge between them and a self-loop on one of the points.

Broken lances lying along perspective lines in Paolo Uccello's The Battle of San Romano, 1438 Small stellated dodecahedron, from De divina proportione by Luca Pacioli, woodcut by Leonardo da Vinci. Venice, 1509 Albrecht Dürer's 1514 engraving Melencolia, with a truncated triangular trapezohedron and a magic square Rencontre dans la porte tournante by Man Ray, 1922, with helix Four- dimensional geometry in Painting 2006-7 by Tony Robbin Quintrino by Bathsheba Grossman, 2007, a sculpture with dodecahedral symmetry Heart by Hamid Naderi Yeganeh, 2014, using a family of trigonometric equations This is a list of artists who actively explored mathematics in their artworks. Art forms practised by these artists include painting, sculpture, architecture, textiles and origami. Some artists such as Piero della Francesca and Luca Pacioli went so far as to write books on mathematics in art. Della Francesca wrote books on solid geometry and the emerging field of perspective, including De Prospectiva Pingendi (On Perspective for Painting), Trattato d’Abaco (Abacus Treatise), and De corporibus regularibus (Regular Solids),Piero della Francesca, De Prospectiva Pingendi, ed.

The planar graphs and the apex graphs are linklessly embeddable, as are the graphs obtained by Y-Δ transforms from these graphs. The YΔY reducible graphs are the graphs that can be reduced to a single vertex by Y-Δ transforms, removal of isolated vertices and degree-one vertices, and compression of degree-two vertices; they are also minor-closed, and include all planar graphs. However, there exist linkless graphs that are not YΔY reducible, such as the apex graph formed by connecting an apex vertex to every degree-three vertex of a rhombic dodecahedron.. There also exist linkless graphs that cannot be transformed into an apex graph by Y-Δ transforms, removal of isolated vertices and degree-one vertices, and compression of degree-two vertices: for instance, the ten-vertex crown graph has a linkless embedding, but cannot be transformed into an apex graph in this way. Related to the concept of linkless embedding is the concept of knotless embedding, an embedding of a graph in such a way that none of its simple cycles form a nontrivial knot.

This shape was called a Siamese dodecahedron in the paper by Hans Freudenthal and B. L. van der Waerden (1947) which first described the set of eight convex deltahedra.. The dodecadeltahedron name was given to the same shape by , referring to the fact that it is a 12-sided deltahedron. There are other simplicial dodecahedra, such as the hexagonal bipyramid, but this is the only one that can be realized with equilateral faces. Bernal was interested in the shapes of holes left in irregular close-packed arrangements of spheres, so he used a restrictive definition of deltahedra, in which a deltahedron is a convex polyhedron with triangular faces that can be formed by the centers of a collection of congruent spheres, whose tangencies represent polyhedron edges, and such that there is no room to pack another sphere inside the cage created by this system of spheres. This restrictive definition disallows the triangular bipyramid (as forming two tetrahedral holes rather than a single hole), pentagonal bipyramid (because the spheres for its apexes interpenetrate, so it cannot occur in sphere packings), and icosahedron (because it has interior room for another sphere).

Its boundary is composed of 600 tetrahedral cells with 20 meeting at each vertex. Together they form 1200 triangular faces, 720 edges, and 120 vertices. The edges form 72 flat regular decagons. Each vertex of the 600-cell is a vertex of six such decagons. The mutual distances of the vertices, measured in degrees of arc on the circumscribed hypersphere, only have the values 36° = , 60° = , 72° = , 90° = , 108° = , 120° = , 144° = , and 180° = . Departing from an arbitrary vertex V one has at 36° and 144° the 12 vertices of an icosahedron, at 60° and 120° the 20 vertices of a dodecahedron, at 72° and 108° again the 12 vertices of an icosahedron, at 90° the 30 vertices of an icosidodecahedron, and finally at 180° the antipodal vertex of V.S.L. van Oss (1899); F. Buekenhout and M. Parker (1998) These can be seen in the H3 Coxeter plane projections with overlapping vertices colored. Just like the icosidodecahedron can be partitioned into 6 central decagons (60 edge = 6×10), the 600-cell can be partitioned into 72 decagons (720 edges = 72 × 10). :640px Its vertex figure is an icosahedron, and its dual polytope is the 120-cell, with which it can form a compound.