Dividing the icosahedron further into triangles (or, more formally, applying a hexagonal lattice to the icosahedron and then replacing each hexagon with six triangles) and positioning proteins in the corners of those triangles provided a more general and accurate picture of what these kinds of viruses looked like.
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They projected this archipelago onto an icosahedron consisting of 20 equilateral triangles that could be rearranged to visualize geospatial information like air and sea routes.
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James began the "Portal Icosahedron" series in 2011, and the sculptures have been showcased all over the world in cities including LA, London, and Seattle.
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Among its critics is Mathias Woo, executive director of the experimental theater company Zuni Icosahedron and a former member of the cultural district's arts and cultural advisory group.
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After it caught on fire in 1976, and its acrylic bubble was burned away from the steel icosahedron, it was transformed into an environmental museum, which is now called Montréal Biosphère.
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It turns out that [shoves glasses up on nose] it's an idea from Plato, who thought that when water was totally pure, it was an icosahedron, which is a 20-sided polyhedron.
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I have to give soccer balls at least a B. Because the soccer ball shape—all those little polygons put together—when you make a soccer ball it's actually something called a truncated icosahedron.
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The movie takes its name from Plato's idea that in its purest form, water takes the shape of an icosahedron, a 20-sided polyhedron, evoking the idea that beauty, and humanity, has many faces.
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Sound engineers can take that image, feed it into a corresponding compact spherical loudspeaker array (dubbed an icosahedron), and recreate the sound exactly, right down to how the sound waves reflect off the walls of the performance space.
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It's as if the crystalline infinite evoked by "Portal Icosahedron" and previous objects is not art's ability to vacillate between chaos and order or to explore some yet-unearthed psychological insight — but merely our limitless greed for shiny objects.
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While the MotherBox charges devices wirelessly, the truncated icosahedron-shaped device itself needs to be plugged in (although the company is offering a smaller, portable battery pack version that doesn't need constant power, at the expense of a far more limited range).
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Singapore's Supertree Grove vertical gardens The CCTV Headquarters in Beijing's central business district A 20-faced icosahedron A truss arch, commonly used in bridge-building A suspension bridge built with a combination of Mola kits Arata Isozaki's Art Tower Mito located in central Japan Singapore's Supertree Grove vertical gardens The CCTV Headquarters in Beijing's central business district A 20-faced icosahedron A truss arch, commonly used in bridge-building A suspension bridge built with a combination of Mola kits Arata Isozaki's Art Tower Mito located in central Japan Despite advances in digital technology, the structural engineering syllabus has remained largely unchanged for close to a century.
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Other uniform duals whose exterior surfaces are stellations of the icosahedron are the medial triambic icosahedron and the great triambic icosahedron.
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It can be seen as an icosahedron with triangular pyramids augmented to each face; that is, it is the Kleetope of the icosahedron. This interpretation is expressed in the name, triakis. :160px If the icosahedron is augmented by tetrahedral without removing the center icosahedron, one gets the net of an icosahedral pyramid.
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3D model of a truncated great icosahedron In geometry, the truncated great icosahedron (or great truncated icosahedron) is a nonconvex uniform polyhedron, indexed as U55. It has 32 faces (12 pentagrams and 20 hexagons), 90 edges, and 60 vertices. It is given a Schläfli symbol t{3,} or t0,1{3,} as a truncated great icosahedron.
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Jessen's icosahedron Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, is a non-convex polyhedron with the same number of vertices, edges and faces as the regular icosahedron. Its faces meet only in right angles, even though they cannot all be made parallel to the coordinate planes. It is named for Børge Jessen who investigated it in 1967.
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If the faces of the pentagonal antiprism are all regular, it is a semiregular polyhedron. It can also be considered as a parabidiminished icosahedron, a shape formed by removing two pentagonal pyramids from a regular icosahedron leaving two nonadjacent pentagonal faces; a related shape, the metabidiminished icosahedron (one of the Johnson solids), is likewise form from the icosahedron by removing two pyramids, but its pentagonal faces are adjacent to each other. The two pentagonal faces of either shape can be augmented with pyramids to form the icosahedron.
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In geometry, the tridiminished icosahedron is one of the Johnson solids (J63). The name refers to one way of constructing it, by removing three pentagonal pyramids from a regular icosahedron, which replaces three sets of five triangular faces from the icosahedron with three mutually adjacent pentagonal faces.
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In geometry, the gyroelongated pentagonal pyramid is one of the Johnson solids (J11). As its name suggests, it is formed by taking a pentagonal pyramid and "gyroelongating" it, which in this case involves joining a pentagonal antiprism to its base. It can also be seen as a diminished icosahedron, an icosahedron with the top (a pentagonal pyramid, J2) chopped off by a plane. Other Johnson solids can be formed by cutting off multiple pentagonal pyramids from an icosahedron: the pentagonal antiprism and metabidiminished icosahedron (two pyramids removed), and the tridiminished icosahedron (three pyramids removed).
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In geometry, the augmented tridiminished icosahedron is one of the Johnson solids (J64). It can be obtained by joining a tetrahedron to another Johnson solid, the tridiminished icosahedron.
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The external surface of the small triambic icosahedron (removing the parts of each hexagonal face that are surrounded by other faces, but interpreting the resulting disconnected plane figures as still being faces) coincides with one of the stellations of the icosahedron. (1st Edn University of Toronto (1938)) If instead, after removing the surrounded parts of each face, each resulting triple of coplanar triangles is considered to be three separate faces, then the result is one form of the triakis icosahedron, formed by adding a triangular pyramid to each face of an icosahedron. The dual polyhedron of the small triambic icosahedron is the small ditrigonal icosidodecahedron. As this is a uniform polyhedron, the small triambic icosahedron is a uniform dual.
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Three golden rectangles in an icosahedron The convex hull of two opposite edges of a regular icosahedron forms a golden rectangle. The twelve vertices of the icosahedron can be decomposed in this way into three mutually-perpendicular golden rectangles, whose boundaries are linked in the pattern of the Borromean rings..
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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.
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Jessen's icosahedron In Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, the 12 isosceles faces are arranged differently so that the figure is non-convex and has right dihedral angles. It is scissors congruent to a cube, meaning that it can be sliced into smaller polyhedral pieces that can be rearranged to form a solid cube.
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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.
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Geodesic grids may use the dual polyhedron of the geodesic polyhedron, which is the Goldberg polyhedron. Goldberg polyhedra are made up of hexagons and (if based on the icosahedron) 12 pentagons. One implementation that uses an icosahedron as the base polyhedron, hexagonal cells, and the Snyder equal-area projection is known as the Icosahedron Snyder Equal Area (ISEA) grid.
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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.
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3D model of a tridyakis icosahedron In geometry, the tridyakis icosahedron is the dual polyhedron of the nonconvex uniform polyhedron, icositruncated dodecadodecahedron. It has 44 vertices, 180 edges, and 120 scalene triangular faces.
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240px The triakis icosahedron has numerous stellations, including this one.
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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.
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The closo-carboranes are chemical compounds with shape very close to icosahedron. Icosahedral twinning also occurs in crystals, especially nanoparticles. Many borides and allotropes of boron contain boron B12 icosahedron as a basic structure unit.
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100px The Icosahedron colored as a snub tetrahedron has chiral symmetry.
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This arguably makes the icosahedron the "roundest" of the platonic solids.
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In geometry, the metabidiminished icosahedron is one of the Johnson solids (J62). The name refers to one way of constructing it, by removing two pentagonal pyramids from a regular icosahedron, replacing two sets of five triangular faces of the icosahedron with two adjacent pentagonal faces. If two pentagonal pyramids are removed to form nonadjacent pentagonal faces, the result is instead the pentagonal antiprism.
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Jessen's icosahedron is vertex-transitive (or isogonal), meaning that it has symmetries taking any vertex to any other vertex. Its dihedral angles are all right angles. One can use it as the basis for the construction of a large family of polyhedra with right dihedral angles, formed by gluing copies of Jessen's icosahedron together on their equilateral-triangle faces. net for Jessen's icosahedron, suitable for making a (shaky) physical model Although it is not a flexible polyhedron, Jessen's icosahedron is also not infinitesimally rigid; that is, it is a "shaky polyhedron".
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Mathematical model of the tensegrity icosahedron Different shapes of tensegrity icosahedra, depending on the ratio between the lengths of the tendons and the struts. The polyhedron which corresponds directly to the geometry of the tensegrity icosahedron is called the Jessen's icosahedron. Its spherical dynamics were of special interest to Buckminster Fuller, who referred to its expansion-contraction transformations around a stable equilibrium as jitterbug motion. The following is a mathematical model for figures related to the tensegrity icosahedron, explaining why it is a stable construction, albeit with infinitesimal mobility.
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The rhombic icosahedron is a zonohedron made up of 20 congruent rhombs. It can be derived from the rhombic triacontahedron by removing 10 middle faces. Even though all the faces are congruent, the rhombic icosahedron is not face-transitive.
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The cuboctahedral structure maintains the cubic-closed pack and symmetry of fcc. This can be thought of as redefining the unit cell into a more complicated cell. Each edge of the cuboctahedron represents a peripheral Au–Au bond. The cuboctahedron has 24 edges while the icosahedron has 30 edges; the transition from cuboctahedron to icosahedron is favored since the increase in bonds contributes to the overall stability of the icosahedron structure.
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Zuni Icosahedron is a Hong Kong-based international experimental theatre company. Founded in 1982, Zuni is one of the nine major professional performing arts groups subsidised by the HKSAR government and has produced more than 190 original productions of alternative theatre and multimedia performances. As well as theatre works, Zuni is also active in video, sound experimentation and installation art, as well as working in the areas of arts education, arts criticism, cultural policy research and international cultural exchange.Zuni Icosahedron official website The name Zuni Icosahedron expresses two essential elements of its theatre works: colour (zuni) and form (icosahedron).
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R. Buckminster Fuller and Japanese cartographer Shoji Sadao designed a world map in the form of an unfolded icosahedron, called the Fuller projection, whose maximum distortion is only 2%. The American electronic music duo ODESZA use a regular icosahedron as their logo.
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Model of an icosahedron made with metallic spheres and magnetic connectors The 12 edges of a regular octahedron can be subdivided in the golden ratio so that the resulting vertices define a regular icosahedron. This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly subdividing each edge into the golden mean along the direction of its vector. The five octahedra defining any given icosahedron form a regular polyhedral compound, while the two icosahedra that can be defined in this way from any given octahedron form a uniform polyhedron compound. Regular icosahedron and its circumscribed sphere.
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Convex regular icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes meaning "twenty" and meaning "seat". The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non-similar shapes of icosahedra, some of them being more symmetrical than others.
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It can be seen as the "lid" of an icosahedron; the rest of the icosahedron forms a gyroelongated pentagonal pyramid, J11 More generally an order-2 vertex- uniform pentagonal pyramid can be defined with a regular pentagonal base and 5 isosceles triangle sides of any height.
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The full symmetry group of the icosahedron (including reflections) is known as the full icosahedral group, and is isomorphic to the product of the rotational symmetry group and the group C2 of size two, which is generated by the reflection through the center of the icosahedron.
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Inside a Magic 8-Ball, various answers to yes–no questions are inscribed on a regular icosahedron.
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The structure is face-centered cubic, with space group Fmc (No. 226), Pearson symbol cF1936 and lattice constant a = 2.3440(6) nm. There are 13 boron sites B1–B13 and one yttrium site. The B1 sites form one B12 icosahedron and the B2–B9 sites make up another icosahedron.
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A regular icosahedron has 59 stellations 59 (fifty-nine) is the natural number following 58 and preceding 60.
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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.
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3D model of a great icosahedron In geometry, the great icosahedron is one of four Kepler-Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol {3,} and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence. The great icosahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the (n-1)-D simplex faces of the core nD polytope (equilateral triangles for the great icosahedron, and line segments for the pentagram) until the figure regains regular faces. The grand 600-cell can be seen as its four-dimensional analogue using the same process.
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The tridiminished icosahedron is the vertex figure of the snub 24-cell, a uniform 4-polytope (4-dimensional polytope).
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In popular craft culture, large sparkleballs can be made using a icosahedron pattern and plastic, styrofoam or paper cups.
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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.
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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.
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This unit can be observed in β-tetragonal boron and is a modification of the B20 unit of α-AlB12 (or B19 unit in early reports). The B20 unit is a twinned icosahedron made from B13 to B22 sites with two vacant sites and one B atom (B23) bridging both sides of the unit. The twinned icosahedron is shown in figure 18a. B23 was treated as an isolated atom in the early reports; it is bonded to each twinned icosahedra through B18 and to another icosahedron through B5 site.
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It may be numbered from "0" to "9" twice (in which form it usually serves as a ten-sided die, or d10), but most modern versions are labeled from "1" to "20". An icosahedron is the three-dimensional game board for Icosagame, formerly known as the Ico Crystal Game. An icosahedron is used in the board game Scattergories to choose a letter of the alphabet. Six letters are omitted (Q, U, V, X, Y, and Z). In the Nintendo 64 game Kirby 64: The Crystal Shards, the boss Miracle Matter is a regular icosahedron.
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The best known is the (convex, non-stellated) regular icosahedron—one of the Platonic solids—whose faces are 20 equilateral triangles.
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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.
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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.
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For the polar reciprocals of the regular and uniform polytopes, the dual facets will be polar reciprocals of the original's vertex figure. For example, in four dimensions, the vertex figure of the 600-cell is the icosahedron; the dual of the 600-cell is the 120-cell, whose facets are dodecahedra, which are the dual of the icosahedron.
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The tridyakis icosahedron is the dual polyhedron of the icositruncated dodecadodecahedron. It has 44 vertices, 180 edges, and 120 scalene triangular faces.
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Piero della Francesca's image of a truncated icosahedron from his book De quinque corporibus regularibus The truncated icosahedron was known to Archimedes, who classified the 13 Archimedean solids in a lost work. All we know of his work on these shapes comes from Pappus of Alexandria, who merely lists the numbers of faces for each: 12 pentagons and 20 hexagons, in the case of the truncated icosahedron. The first known image and complete description of a truncated icosahedron is from a rediscovery by Piero della Francesca, in his 15th-century book De quinque corporibus regularibus, which included five of the Archimedean solids (the five truncations of the regular polyhedra). The same shape was depicted by Leonardo da Vinci, in his illustrations for Luca Pacioli's plagiarism of della Francesca's book in 1509.
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The stellation diagram for the icosahedron with the central triangle marked for the original icosahedron The Fifty-Nine Icosahedra is a book written and illustrated by H. S. M. Coxeter, P. Du Val, H. T. Flather and J. F. Petrie. It enumerates certain stellations of the regular convex or Platonic icosahedron, according to a set of rules put forward by J. C. P. Miller. First published by the University of Toronto in 1938, a Second Edition reprint by Springer-Verlag followed in 1982. Tarquin's 1999 Third Edition included new reference material and photographs by K. and D. Crennell.
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An unusual linkage is depicted in figure 8b, where two B12-I5 icosahedra connect via two B atoms of each icosahedron forming an imperfect square. The boron framework of YB41Si1.2 can be described as a layered structure where two boron networks (figures 9a,b) stack along the z-axis. One boron network consists of 3 icosahedra I1, I2 and I3 and is located in the z = 0 plane; another network consists of the icosahedron I5 and the B12Si3 polyhedron and lies at z = 0.5. The icosahedron I4 bridges these networks, and thus its height along the z-axis is 0.25.
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Matila Ghyka, The Geometry of Art and Life (1977), p.68 The 600-cell is regarded as the 4-dimensional analog of the icosahedron, since it has five tetrahedra meeting at every edge, just as the icosahedron has five triangles meeting at every vertex. It is also called a tetraplex (abbreviated from "tetrahedral complex") and polytetrahedron, being bounded by tetrahedral cells.
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A hemi-icosahedron is an abstract regular polyhedron, containing half the faces of a regular icosahedron. It can be realized as a projective polyhedron (a tessellation of the real projective plane by 10 triangles), 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.
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Icosahedron This polyhedron can be constructed from an icosahedron with the 12 vertices truncated (cut off) such that one third of each edge is cut off at each of both ends. This creates 12 new pentagon faces, and leaves the original 20 triangle faces as regular hexagons. Thus the length of the edges is one third of that of the original edges.
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In geometry, an octadecahedron (or octakaidecahedron) is a polyhedron with 18 faces. No octadecahedron is regular; hence, the name does not commonly refer to one specific polyhedron. In chemistry, "the octadecahedron" commonly refers to a specific structure with C2v symmetry, the edge-contracted icosahedron, formed from a regular icosahedron with one edge contracted. It is the shape of the closo-boranate ion [B11H11]2−.
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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.
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3D model of a regular icosahedron In geometry, a regular icosahedron ( or ) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces. It has five equilateral triangular faces meeting at each vertex. It is represented by its Schläfli symbol {3,5}, or sometimes by its vertex figure as 3.3.3.3.
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Buckminsterfullerene C60 has 60 carbon atoms in each molecule, arranged in a truncated icosahedron. The first fullerene to be discovered was buckminsterfullerene C60, an allotrope of carbon with 60 atoms in each molecule, arranged in a truncated icosahedron. This ball is known as a buckyball, and looks like a soccer ball. The atomic number of neodymium is 60, and cobalt-60 (60Co) is a radioactive isotope of cobalt.
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600-cell: polytope {3,3,5} Twenty irregular tetrahedra pack with a common vertex in such a way that the twelve outer vertices form a regular icosahedron. Indeed, the icosahedron edge length l is slightly longer than the circumsphere radius r (l ≈ 1.05r). There is a solution with regular tetrahedra if the space is not Euclidean, but spherical. It is the polytope {3,3,5}, using the Schläfli notation, also known as the 600-cell.
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These 10, along with the great circles from projections of two other polyhedra, form the 31 great circles of the spherical icosahedron used in construction of geodesic domes.
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Donohue (1982) commented that the number of atoms in the unit cell did not appear to be icosahedrally related (the icosahedron being a motif common to boron structures).
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For example, there are exactly 2 nodes in C that are mapped to the blue node in H. However, C is not a bipartite double cover of H or any other graph; it is not a bipartite graph. If we replace one triangle by a square in H the resulting graph has four distinct double covers. Two of them are bipartite but only one of them is the Kronecker cover. :Image:Covering-graph-4.svg As another example, the graph of the icosahedron is a double cover of the complete graph K6; to obtain a covering map from the icosahedron to K6, map each pair of opposite vertices of the icosahedron to a single vertex of K6.
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Earth was associated with the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron. There was intuitive justification for these associations: the heat of fire feels sharp and stabbing (like little tetrahedra). Air is made of the octahedron; its minuscule components are so smooth that one can barely feel it. Water, the icosahedron, flows out of one's hand when picked up, as if it is made of tiny little balls.
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Continuous transformation pausing at the vertex position of Jessen's icosahedron Jessen's icosahedron is one of a continuous series of icosahedra with 8 regular faces and 12 isosceles faces, described by H. S. M. Coxeter in 1948. The shapes in this family range from cuboctahedron to regular octahedron (as limit cases), which can be inscribed in a regular octahedron. The twisting, expansive-contractive transformations between members of this family were named Jitterbug transformations by Buckminster Fuller.
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Volume rendering of Geodesic grid applied in atmosphere simulation using Global Cloud Resolving Model (GCRM). The combination of grid illustration and volume rendering of vorticity (yellow tubes) . Note that for the purpose of clear illustration in the image, the grid is coarser than the actual one used to generate the vorticity.The icosahedron A highly divided geodesic polyhedron based on the icosahedron A highly divided Goldberg polyhedron: the dual of the above image.
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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°.
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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.
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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.
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East Wing West Wing () is a Hong Kong Drama Series by Zuni Icosahedron, Edward Lam and Mathias Woo production. This drama is a funny and ironic criticism of Hong Kong politics.
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These icosahedra arrange in a thirteen-icosahedron unit (B12)12B12 which is called supericosahedron. The icosahedron formed by the B1 site atoms is located at the center of the supericosahedron. The supericosahedron is one of the basic units of the boron framework of YB66. There are two types of supericosahedra: one occupies the cubic face centers and another, which is rotated by 90°, is located at the center of the cell and at the cell edges.
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For example, fullerene is a cluster of 60 carbon atoms arranged as the vertices of a truncated icosahedron, and decaborane is a cluster of 10 boron atoms forming an incomplete icosahedron, surrounded by 14 hydrogen atoms. The term is most commonly used for ensembles consisting of several atoms of the same element, or of a few different elements, bonded in a three-dimensional arrangement. Transition metals and main group elements form especially robust clusters.Inorganic Chemistry Huheey, JE, 3rd ed.
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Full Icosahedral symmetry has 15 mirror planes (seen as cyan great circles on this sphere) meeting at order , , angles, dividing a sphere into 120 triangle fundamental domains. There are 6 5-fold axes (blue), 10 3-fold axes (red), and 15 2-fold axes (magenta). The vertices of the regular icosahedron exist at the 5-fold rotation axis points. The rotational symmetry group of the regular icosahedron is isomorphic to the alternating group on five letters.
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3D model of a great disdyakis triacontahedron The great disdyakis triacontahedron (or trisdyakis icosahedron) is a nonconvex isohedral polyhedron. It is the dual of the great truncated icosidodecahedron. Its faces are triangles.
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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.
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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.
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The birectified order-5 120-cell honeycomb constructed by all rectified 600-cells, with octahedron and icosahedron cells, and triangle faces with a 5-5 duoprism vertex figure and has extended symmetry 5,3,3,5.
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An icosahedron can also be called a gyroelongated pentagonal bipyramid. It can be decomposed into a gyroelongated pentagonal pyramid and a pentagonal pyramid or into a pentagonal antiprism and two equal pentagonal pyramids.
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David Tuch was among the first to describe a solution to this problem. The idea is best understood by conceptually placing a kind of geodesic dome around each image voxel. This icosahedron provides a mathematical basis for passing a large number of evenly spaced gradient trajectories through the voxel--each coinciding with one of the apices of the icosahedron. Basically, we are now going to look into the voxel from a large number of different directions (typically 40 or more).
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3D model of a truncated icosahedron In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It has 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices and 90 edges. It is the Goldberg polyhedron GPV(1,1) or {5+,3}1,1, containing pentagonal and hexagonal faces. This geometry is associated with footballs (soccer balls) typically patterned with white hexagons and black pentagons.
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This yields a coarse-resolution equal area grid called the resolution 1 grid. It consists of 20 hexagons on the surface of the sphere and 12 pentagons centered on the 12 vertices of the icosahedron.
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However, the icosahedron is not bipartite, so it is not the bipartite double cover of K6. Instead, it can be obtained as the orientable double cover of an embedding of K6 on the projective plane.
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Hemicube is derived from a cube by equating opposite vertices, edges, and faces. It has 4 vertices, 6 edges, and 3 faces. Grünbaum also discovered the 11-cell, a four-dimensional self-dual object whose facets are not icosahedra, but are "hemi-icosahedra" -- that is, they are the shape one gets if one considers opposite faces of the icosahedra to be actually the same face . The hemi-icosahedron has only 10 triangular faces, and 6 vertices, unlike the icosahedron, which has 20 and 12.
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The Impossiball is made in the shape of an icosahedron that has been rounded out to a sphere, and has 20 pieces, all of them corners. The puzzle has twelve circles located at the vertices of the icosahedron, and on the six-color version opposite circles are the same color. Because of the rounded shape of the puzzle, the pieces move up and down as they are rotated. It is also possible to remove one piece, turning the puzzle into a spherical version of the 15 puzzle.
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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.
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The descriptions of the gods require that certain attributes be represented by adjacent vertices on the implied icosahedron. Faith, in particular, is adjacent to Goodness and Mercy (because Evren is the Goddess of Faith, Mercy, and Goodness), as well as Chaos and Impatience (Akhran), Charity (Promenthas), and Greed (Benario). Since Faith is adjacent to at least six other attributes, while a vertex of an icosahedron is only adjacent to five other vertices, the attributes ascribed to the gods are inconsistent with the implied global structure.
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A black-and-white patterned spherical truncated icosahedron design, brought to prominence by the Adidas Telstar, has become an icon of the sport. Many different designs of balls exist, varying both in appearance and physical characteristics.
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In 1985 their work was repeated by Harold Kroto, James R. Heath, Sean O'Brien, Robert Curl, and Richard Smalley, who proposed the truncated icosahedron structure for the prominent C60 molecule, and proposed the name "buckminsterfullerene" for it.
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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.
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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.
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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.
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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).
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Three linked golden rectangles in a regular icosahedron A realization of the Borromean rings by three mutually perpendicular golden rectangles can be found within a regular icosahedron by connecting three opposite pairs of its edges. Every three unknotted polygons in Euclidean space may be combined, after a suitable scaling transformation, to form the Borromean rings. If all three polygons are planar, then scaling is not needed. More generally, Matthew Cook has conjectured that any three unknotted simple closed curves in space, not all circles, can be combined without scaling to form the Borromean rings.
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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.
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Wheeler was an Invited Speaker of the ICM in 1924 at Toronto.Wheeler, Albert Harry (1924) "Certain forms of the icosahedron and a method for deriving and designating higher polyhedra." In Proceedings of the International Mathematical Congress, Toronto, vol.
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While "Zuni" stands for a colour between green and blue, "Icosahedron" is a twenty-sided object as well as a highly infectious virus.Lilley, R. (1990). Staging Hong Kong: Gender and Performance in Transition. Honolulu: University of Hawai’i Press.
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An icosahedron. Geometric combinatorics is related to convex and discrete geometry, in particular polyhedral combinatorics. It asks, for example, how many faces of each dimension a convex polytope can have. Metric properties of polytopes play an important role as well, e.g.
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The 31 great circles can be seen in 3 sets: 15, 10, and 6, each representing edges of a polyhedron projected onto a sphere. Fifteen great circles represent the edges of a disdyakis triacontahedron, the dual of a truncated icosidodecahedron. Six more great circles represent the edges of an icosidodecahedron, and the last ten great circles come from the edges of the uniform star dodecadodecahedron, making pentagrams with vertices at the edge centers of the icosahedron. There are 62 points of intersection, positioned at the 12 vertices, and center of the 30 edges, and 20 faces of a regular icosahedron.
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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.
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It was shown by that the Schönhardt polyhedron can be generalized to other polyhedra, combinatorially equivalent to antiprisms, that cannot be triangulated. These polyhedra are formed by connecting regular k-gons in two parallel planes, twisted with respect to each other, in such a way that k of the 2k edges that connect the two k-gons have concave dihedrals. Another polyhedron that cannot be triangulated is Jessen's icosahedron, combinatorially equivalent to a regular icosahedron. In a different direction, constructed a polyhedron that shares with the Schönhardt polyhedron the property that it has no internal diagonals.
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One can also divide the edges of an octahedron in the ratio of the golden mean to define the vertices of an icosahedron. This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly partitioning each edge into the golden mean along the direction of its vector. There are five octahedra that define any given icosahedron in this fashion, and together they define a regular compound. Octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by Buckminster Fuller.
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The locations of the vertices of a regular icosahedron can be described using spherical coordinates, for instance as latitude and longitude. If two vertices are taken to be at the north and south poles (latitude ±90°), then the other ten vertices are at latitude ±arctan() ≈ ±26.57°. These ten vertices are at evenly spaced longitudes (36° apart), alternating between north and south latitudes. This scheme takes advantage of the fact that the regular icosahedron is a pentagonal gyroelongated bipyramid, with D5d dihedral symmetry—that is, it is formed of two congruent pentagonal pyramids joined by a pentagonal antiprism.
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In geometry, a decagonal bipyramid is one of the infinite set of bipyramids, dual to the infinite prisms. If a decagonal bipyramid is to be face- transitive, all faces must be isosceles triangles. It is an icosahedron, but not the regular one.
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The truncated 600-cell consists of 600 truncated tetrahedra and 120 icosahedra. The truncated tetrahedral cells are joined to each other via their hexagonal faces, and to the icosahedral cells via their triangular faces. Each icosahedron is surrounded by 20 truncated tetrahedra.
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It has 10 triangular faces, 15 edges, and 6 vertices. It is also related to the nonconvex uniform polyhedron, the tetrahemihexahedron, which could be topologically identical to the hemi- icosahedron if each of the 3 square faces were divided into two triangles.
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It is related to the order-5 tesseractic honeycomb, {4,3,3,5}, and order-5 120-cell honeycomb, {5,3,3,5}. It is topologically similar to the finite 5-orthoplex, {3,3,3,4}, and 5-simplex, {3,3,3,3}. It is analogous to the 600-cell, {3,3,5}, and icosahedron, {3,5}.
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Hafner, I. and Zitko, T. Introduction to golden rhombic polyhedra. Faculty of Electrical Engineering, University of Ljubljana, Slovenia. This figure is another spacefiller, and can also occur in non-periodic spacefillings along with the rhombic triacontahedron, the rhombic icosahedron and rhombic hexahedra.
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Extending work of Max Brückner, Wheeler actually constructed previously unknown polyhedra. In particular, he produced new stellations of the icosahedron. This achievement impressed Coxeter, who noted Wheeler's achievement in the text. Wheeler continued teaching high school mathematics in Worcester until his retirement.
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The CeO12 core defines an icosahedron. Ce4+ is a strong one-electron oxidizing agent. In terms of its redox potential (E° ~ 1.61 V vs. N.H.E.) it is even stronger oxidizing agent than Cl2 (E° ~ 1.36 V). Few shelf-stable reagents are stronger oxidants.
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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.
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The well-studied tobacco mosaic virus is an example of a helical virus.Collier p. 37 ; Icosahedral: Most animal viruses are icosahedral or near-spherical with chiral icosahedral symmetry. A regular icosahedron is the optimum way of forming a closed shell from identical sub- units.
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Du Val devised a symbolic notation for identifying sets of congruent cells, based on the observation that they lie in "shells" around the original icosahedron. Based on this he tested all possible combinations against Miller's rules, confirming the result of Coxeter's more analytical approach.
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Wells, D. The Penguin Dictionary of Curious and Interesting Geometry, London: Penguin, (1991). p. 161. However, the resulting polyhedron does not have right-angled dihedrals. The vertices of Jessen's icosahedron are perturbed from these positions in order to give all the dihedrals right angles.
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Since the tensegrity icosahedron represents an extremal point of the above relation, it has infinitesimal mobility: a small change in the length s of the tendon (e.g. by stretching the tendons) results in a much larger change of the distance 2d of the struts.
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Cartesian coordinates for the vertices of a truncated icosahedron centered at the origin are all even permutations of: :(0, ±1, ±3φ) :(±1, ±(2 + φ), ±2φ) :(±φ, ±2, ±(2φ + 1)) where φ = is the golden mean. The circumradius is ≈ 4.956 and the edges have length 2.
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The triakis icosahedron, a polyhedron in which every edge has endpoints with total degree at least 13 In graph theory and polyhedral combinatorics, areas of mathematics, Kotzig's theorem is the statement that every polyhedral graph has an edge whose two endpoints have total degree at most 13. An extreme case is the triakis icosahedron, where no edge has smaller total degree. The result is named after Anton Kotzig, who published it in 1955 in the dual form that every convex polyhedron has two adjacent faces with a total of at most 13 sides. It was named and popularized in the west in the 1970s by Branko Grünbaum.
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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.
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This configuration became common throughout Continental Europe in the 1960s, and was publicised worldwide by the Adidas Telstar, the official ball of the 1970 World Cup. This design in often referenced when describing the truncated icosahedron Archimedean solid, carbon buckyballs, or the root structure of geodesic domes.
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In contrast, in Gram-positive bacteria (e.g. Bacillus stearothermophilus) and eukaryotes the central PDC core contains 60 E2 molecules arranged into an icosahedron. Eukaryotes also contain 12 copies of an additional core protein, E3 binding protein (E3BP). The exact location of E3BP is not completely clear.
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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.
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Soccer ball (right) has 20 hexagons and icosahedral symmetry as in a truncated icosahedron. Mechanism of clathrin-mediated endocytosis. The clathrin triskelion is composed of three clathrin heavy chains interacting at their C-termini, each ~190 kDa heavy chain has a ~25 kDa light chain tightly bound to it.
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'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.
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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 .
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A variation of the icosahedron was used as the basis of the honeycomb wheels (made from a polycast material) used by the Pontiac Motor Division between 1971 and 1976 on its Trans Am and Grand Prix. This shape was also the configuration of the lenses used for focusing the explosive shock waves of the detonators in both the gadget and Fat Man atomic bombs. The truncated icosahedron can also be described as a model of the Buckminsterfullerene (fullerene) (C60), or "buckyball", molecule – an allotrope of elemental carbon, discovered in 1985. The diameter of the football and the fullerene molecule are 22 cm and about 0.71 nm, respectively, hence the size ratio is ≈31,000,000:1.
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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.
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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.
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Buckminsterfullerene is a type of fullerene with the formula C60. It has a cage-like fused-ring structure (truncated icosahedron) that resembles a soccer ball, made of twenty hexagons and twelve pentagons. Each carbon atom has three bonds. It is a black solid that dissolves in hydrocarbon solvents to produce a violet solution.
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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.
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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.
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It is an icosahedrally symmetric virus with a unique triangulation number (T) of 31. At the 12 fivefold symmetrical positions of the icosahedron protrude 'turrets' that extend 13 nanometers (nm) above the capsid surface. The turrets have an average diameter of 24 nm. The center of each turret contains a ~3-nm channel.
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In 1972-3, Brother Patrick, a monk of Prinknash, contributed the design and manufacture of a number of items at Clifton Cathedral, Bristol. These included the votive candelabra hanging in the Lady Chapel, made from twenty stainless steel equilateral triangles (a regular icosahedron), and the decorative screens to the Blessed Sacrament Chapel.
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His publications covers a wide range of topics in graph theory and combinatorics: convex polyhedra, quasigroups, special decompositions into Hamiltonian paths, Latin squares, decompositions of complete graphs, perfect systems of difference sets, additive sequences of permutations, tournaments and combinatorial games theory. The triakis icosahedron, a polyhedron in which every edge has endpoints with total degree at least 13 One of his results, known as Kotzig's theorem, is the statement that every polyhedral graph has an edge whose two endpoints have total degree at most 13. An extreme case is the triakis icosahedron, where no edge has smaller total degree. Kotzig published the result in Slovakia in 1955, and it was named and popularized in the west by Branko Grünbaum in the mid-1970s.
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Unit cell of B4C. The green sphere and icosahedra consist of boron atoms, and black spheres are carbon atoms. Fragment of the B4C crystal structure. Boron carbide has a complex crystal structure typical of icosahedron-based borides. There, B12 icosahedra form a rhombohedral lattice unit (space group: Rm (No. 166), lattice constants: a = 0.56 nm and c = 1.212 nm) surrounding a C-B-C chain that resides at the center of the unit cell, and both carbon atoms bridge the neighboring three icosahedra. This structure is layered: the B12 icosahedra and bridging carbons form a network plane that spreads parallel to the c-plane and stacks along the c-axis. The lattice has two basic structure units – the B12 icosahedron and the B6 octahedron.
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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.
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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}.
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It is one of three semiregular 4-polytopes made of two or more cells which are Platonic solids, discovered by Thorold Gosset in his 1900 paper. He called it a octicosahedric for being made of octahedron and icosahedron cells. E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC600.
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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.
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The truncated icosahedron or Buckminsterfullerene graph has a traditional connectivity of 3, and an algebraic connectivity of 0.243. The algebraic connectivity of a graph G can be positive or negative, even if G is a connected graph. Furthermore, the value of the algebraic connectivity is bounded above by the traditional (vertex) connectivity of the graph.J.
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The great icosahedron is one of the four regular star Kepler-Poinsot polyhedra. Its Schläfli symbol is {3, }. Like the convex form, it also has 20 equilateral triangle faces, but its vertex figure is a pentagram rather than a pentagon, leading to geometrically intersecting faces. The intersections of the triangles do not represent new edges.
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Of these, many have a single face in each of the 20 face planes and so are also icosahedra. The great icosahedron is among them. Other stellations have more than one face in each plane or form compounds of simpler polyhedra. These are not strictly icosahedra, although they are often referred to as such.
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Fig. 20. ScB17C0.25 crystal structure viewed along the a-axis. The icosahedron layers alternatively stack along the c-axis in the order I1–I2–I1–I2–I1. Very small amount of carbon is sufficient to stabilize "ScB17C0.25". This compound has a broad composition range, namely ScB16.5+xC0.2+y with x ≤ 2.2 and y ≤ 0.44.
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The original Tuttminx in its solved state. A Tuttminx ( or ) is a Rubik's Cube-like twisty puzzle, in the shape of a truncated icosahedron. It was invented by Lee Tutt in 2005.Twisty Puzzles: Tuttminx, 2005 It has a total of 150 movable pieces to rearrange, compared to 20 movable pieces of the Rubik’s Cube.
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The icosahedral shell encapsulating enzymes and labile intermediates are built of different types of proteins with BMC domains. In 1904, Ernst Haeckel described a number of species of Radiolaria, including Circogonia icosahedra, whose skeleton is shaped like a regular icosahedron. A copy of Haeckel's illustration for this radiolarian appears in the article on regular polyhedra.
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1587 was adapted into a play by Zuni Icosahedron director Mathias Woo, which premièred in Hong Kong in 1999. The second production was in 2006, after Woo and Towards the Republic screenwriter Zhang Jianwei (張建偉) re-wrote the script by adding a considerable amount of Kun opera and other elements. There was a third run in 2008.
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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.
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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.
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Mathematical Sciences Building A 113,000 sq ft building built in 1970 at a cost of $7.2 million to house Mathematics and Computer Science departments, as well as university computer systems. In 1976 a glass sculpture of an Icosahedron made by Dominick Labino was installed. In 2018 a Digital forensics lab began construction on the third floor of the building.
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The crest's form was inspired by the truncated icosahedron pattern of a football. In 1993, the mythical flying horse Pegasus was added to the crest. The supporters opposed this change and the crest was restored to its original form in 1996. Groningen's official colours are green and white, derived from the arms of the city of Groningen.
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Stellation is the process of extending the faces or edges of a polyhedron until they meet to form a new polyhedron. It is done symmetrically so that the resulting figure retains the overall symmetry of the parent figure. In their book The Fifty-Nine Icosahedra, Coxeter et al. enumerated 58 such stellations of the regular icosahedron.
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Relationship between the ionic radius of trivalent rare-earth ion and chemical composition of icosahedron-based rare-earth borides. YB25 and YB50 decompose without melting that hinders their growth as single crystals by the floating zone method. However, addition of a small amount of Si solves this problem and results in single crystals with the stoichiometry of YB41Si1.2.
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3D model of a snub square antiprism In geometry, the snub square antiprism is one of the Johnson solids (J85). It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids, although it is a relative of the icosahedron that has fourfold symmetry instead of threefold.
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Various arrangements of capsomeres are: 1) Icosahedral, 2) Helical, and 3) Complex. 1) Icosahedral- An icosahedron is a polyhedron with 12 vertices and 20 faces. Two types of capsomeres constitute the icosahedral capsid: pentagonal (pentons) at the vertices and hexagonal (hexons) at the faces. There are always twelve pentons, but the number of hexons varies among virus groups.
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The 12-color Dogic The Dogic () is an icosahedron-shaped puzzle like the Rubik's Cube. The 5 triangles meeting at its tips may be rotated, or 5 entire faces (including the triangles) around the tip may be rotated. It has a total of 80 movable pieces to rearrange, compared to the 20 pieces in the Rubik's Cube.
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Construction of the Au13 icosahedron. Bulk gold exhibits a face-centered cubic (fcc) structure. As gold particle size decreases the fcc structure of gold transforms into a centered-icosahedral structure illustrated by . It can be shown that the fcc structure can be extended by a half unit cell in order to make it look like a cuboctahedral structure.
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Although Albrecht Dürer omitted this shape from the other Archimedean solids listed in his 1525 book on polyhedra, Underweysung der Messung, a description of it was found in his posthumous papers, published in 1538. Johannes Kepler later rediscovered the complete list of the 13 Archimedean solids, including the truncated icosahedron, and included them in his 1609 book, Harmonices Mundi.
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Ptolemaic Egypt die Icosahedral dice with twenty sides have been used since ancient times.Cromwell, Peter R. "Polyhedra" (1997) Page 327. In several roleplaying games, such as Dungeons & Dragons, the twenty-sided die (d20 for short) is commonly used in determining success or failure of an action. This die is in the form of a regular icosahedron.
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Their names given here were given by John Conway, extending Cayley's names for the Kepler-Poinsot polyhedra: along with stellated and great, he adds a grand modifier. Conway offered these operational definitions: #stellation – replaces edges by longer edges in same lines. (Example: a pentagon stellates into a pentagram) #greatening – replaces the faces by large ones in same planes. (Example: an icosahedron greatens into a great icosahedron) #aggrandizement – replaces the cells by large ones in same 3-spaces. (Example: a 600-cell aggrandizes into a grand 600-cell) John Conway names the 10 forms from 3 regular celled 4-polytopes: pT=polytetrahedron {3,3,5} (a tetrahedral 600-cell), pI=polyicoshedron {3,5,} (an icosahedral 120-cell), and pD=polydodecahedron {5,3,3} (a dodecahedral 120-cell), with prefix modifiers: g, a, and s for great, (ag)grand, and stellated.
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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.
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"A World Map on a Regular Icosahedron by Gnomonic Projection." Geographical Review 33 (4): 605. (A jab at Mercator and proposal for a replacement.) As early as 1943, Stewart notes this phenomenon and compares the quest for the perfect projection to "squaring the circle or making pi come out even"Stewart, John Q. (1943). "The Use and Abuse of Map Projections".
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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.
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Crystal structure of YB25. Black and green spheres indicate Y and B atoms, respectively. The structure of yttrium borides with B/Y ratio of 25 and above consists of a network of B12 icosahedra. The boron framework of YB25 is one of the simplest among icosahedron-based borides – it consists of only one kind of icosahedra and one bridging boron site.
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Collier pp. 40, 42 Hexons are in essence flat and pentons, which form the 12 vertices, are curved. The same protein may act as the subunit of both the pentamers and hexamers or they may be composed of different proteins. ; Prolate: This is an icosahedron elongated along the fivefold axis and is a common arrangement of the heads of bacteriophages.
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Another unusual feature of YB41Si1.2 is the 100% occupancy of the Y site. In most icosahedron-based metal borides, metal sites have rather low site occupancy, for example, about 50% for YB66 and 60–70% for REAlB14. When the Y site is replaced by rare-earth elements, REB41Si1.2 can have an antiferromagnetic-like ordering because of this high site occupancy.
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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.
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Many polyhedra are also coloured such that no same-coloured faces touch each other along an edge or at a vertex. :For example, a 20-face icosahedron can use twenty colours, one colour, ten colours, or five colours, respectively. An alternative way for polyhedral compound models is to use a different colour for each polyhedron component. Net templates are then made.
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Retrieved: 2013-10-14. Most modern Association footballs are stitched from 32 panels of waterproofed leather or plastic: 12 regular pentagons and 20 regular hexagons. The 32-panel configuration is the spherical polyhedron corresponding to the truncated icosahedron; it is spherical because the faces bulge from the pressure of the air inside. The first 32-panel ball was marketed by Select in the 1950s in Denmark.
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VP1, VP2, and VP3 form the major part of the protein capsid. The much smaller VP4 protein has a more extended structure, and lies at the interface between the capsid and the RNA genome. There are 60 copies of each of these proteins assembled as an icosahedron. Antibodies are a major defense against infection with the epitopes lying on the exterior regions of VP1-VP3.
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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.
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One of the possible responses of the Magic 8-Ball. The Magic 8-Ball is a hollow plastic sphere resembling a black-and-white 8-ball. Its standard size is larger than an ordinary pool ball, but it has been made in various sizes. Inside the ball, a cylindrical reservoir contains a white plastic icosahedron die floating in approximately 100mL of alcohol dyed dark blue.
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The structure shown, is built of 36 triskelia, one of which is shown in blue. Another common assembly is a truncated icosahedron. To enclose a vesicle, exactly 12 pentagons must be present in the lattice. In a cell, clathrin triskelion in the cytoplasm binds to an adaptor protein that has bound membrane, linking one of its three feet to the membrane at a time.
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The 32-panel configuration is the spherical polyhedron corresponding to the truncated icosahedron; it is spherical because the faces bulge from the pressure of the air inside. The first 32-panel ball was marketed by Select in the 1950s in Denmark. This configuration became common throughout Continental Europe in the 1960s, and was publicised worldwide by the Adidas Telstar, the official ball of the 1970 World Cup.
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Anthony Wong Yiu-ming (; born 16 June 1962) is a Hong Kong singer, songwriter, actor, record producer and political activist. He rose to prominence as the vocalist for the Cantopop duo Tat Ming Pair during the 1980s before embarking on a solo career. He also performed and collaborated with the theatre group Zuni Icosahedron. Wong is the director for music production company People Mountain People Sea.
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In the early 20th century, Ernst Haeckel described a number of species of Radiolaria, some of whose skeletons are shaped like various regular polyhedra (Haeckel, 1904). Examples include Circoporus octahedrus, Circogonia icosahedra, Lithocubus geometricus and Circorrhegma dodecahedra; the shapes of these creatures are indicated by their names. The outer protein shells of many viruses form regular polyhedra. For example, HIV is enclosed in a regular icosahedron.
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Although Miller did not contribute to the book directly, he was a close colleague of Coxeter and Petrie. His contribution is immortalised in his set of rules for defining which stellation forms should be considered "properly significant and distinct":Coxeter, du Val, et al (Third Edition 1999) Pages 15-16. :(i) The faces must lie in twenty planes, viz., the bounding planes of the regular icosahedron.
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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.
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Fig. 1. (a) B6 octahedron, (b) B12 cuboctahedron and (c) B12 icosahedron. In metal borides, the bonding of boron varies depending on the atomic ratio B/M. Diborides have B/M = 2, as in the well-known superconductor MgB2; they crystallize in a hexagonal AlB2-type layered structure. Hexaborides have B/M = 6 and form a three-dimensional boron framework based on a boron octahedron (Fig. 1a).
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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.
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In geometry, the order-7 truncated triangular tiling, sometimes called the hyperbolic soccerball,HOW TO BUILD YOUR OWN HYPERBOLIC SOCCER BALL MODEL is a semiregular tiling of the hyperbolic plane. There are two hexagons and one heptagon on each vertex, forming a pattern similar to a conventional soccer ball (truncated icosahedron) with heptagons in place of pentagons. It has Schläfli symbol of t{3,7}.
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The tetravalency (4-connectedness) of carbon excludes an icosahedron because 5 edges meet at each vertex. True pentavalent carbon is unlikely; methanium, nominally , usually exists as . The hypothetical icosahedral lacks hydrogen so it is not a hydrocarbon; it is also an ion. Both icosahedral and octahedral structures have been observed in boron compounds such as the dodecaborate ion and some of the carbon-containing carboranes.
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The unit cell is orthorhombic and its most salient feature is four boron-containing icosahedra. Each icosahedron contains 12 boron atoms. Eight more boron atoms connect the icosahedra to the other elements in the unit cell. The occupancy of metal sites in the lattice is lower than one, and thus, while the material is usually identified with the formula AlMgB14, its chemical composition is closer to Al0.75Mg0.75B14.
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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.
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The largest strictly-convex deltahedron is the regular icosahedron This is a truncated tetrahedron with hexagons subdivided into triangles. This figure is not a strictly-convex deltahedron since coplanar faces are not allowed within the definition. In geometry, a deltahedron (plural deltahedra) is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek upper case delta (Δ), which has the shape of an equilateral triangle.
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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.
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The number turns up frequently in geometry, particularly in figures with pentagonal symmetry. The length of a regular pentagon's diagonal is times its side. The vertices of a regular icosahedron are those of three mutually orthogonal golden rectangles. There is no known general algorithm to arrange a given number of nodes evenly on a sphere, for any of several definitions of even distribution (see, for example, Thomson problem).
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Icosahedral capsid of an adenovirus Virus capsid T-numbers The icosahedral structure is extremely common among viruses. The icosahedron consists of 20 triangular faces delimited by 12 fivefold vertexes and consists of 60 asymmetric units. Thus, an icosahedral virus is made of 60N protein subunits. The number and arrangement of capsomeres in an icosahedral capsid can be classified using the "quasi-equivalence principle" proposed by Donald Caspar and Aaron Klug.
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Perspective of anatomy from De Prospectiva Pingendi. An icosahedron in perspective from De Prospectiva pingendi De Prospectiva pingendi (On the Perspective of painting) is the earliest and only pre-1500 Renaissance treatise solely devoted to the subject of perspective. It was written by the Italian master Piero della Francesca in the mid-1470s to 1480s, and possibly by about 1474. Despite its Latin title, the opus is written in Italian.
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350 = 2 × 52 × 7, primitive semiperfect number, divisible by the number of primes below it, nontotient, a truncated icosahedron of frequency 6 has 350 hexagonal faces and 12 pentagonal faces. 350.org is an international environmental organization. 350 is the number of cubic inches displaced in the most common form of the Small Block Chevrolet V8. The number of seats in the Congress of Deputies (Spain) is 350.
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The minimum number of identical capsomeres required for each triangular face is 3, which gives 60 for the icosahedron. Many viruses, such as rotavirus, have more than 60 capsomers and appear spherical but they retain this symmetry. To achieve this, the capsomeres at the apices are surrounded by five other capsomeres and are called pentons. Capsomeres on the triangular faces are surrounded by six others and are called hexons.
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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.
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The most familiar spherical polyhedron is the soccer ball, thought of as a spherical truncated icosahedron. The next most popular spherical polyhedron is the beach ball, thought of as a hosohedron. Some "improper" polyhedra, such as the hosohedra and their duals the dihedra, exist as spherical polyhedra but have no flat-faced analogue. In the examples below, {2, 6} is a hosohedron and {6, 2} is the dual dihedron.
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Stellation diagram with numbered face sets Cell diagram with Du Val notation for cells Before Coxeter, only Brückner and Wheeler had recorded any significant sets of stellations, although a few such as the great icosahedron had been known for longer. Since publication of The 59, Wenninger published instructions on making models of some; the numbering scheme used in his book has become widely referenced, although he only recorded a few stellations.
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Today's footballs are much more complex than past footballs. Most modern footballs consist of twelve regular pentagonal and twenty regular hexagonal panels positioned in a truncated icosahedron spherical geometry. Some premium-grade 32-panel balls use non- regular polygons to give a closer approximation to sphericality. The inside of the football is made up of a latex or butyl rubber bladder which enables the football to be pressurised.
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Buckminsterfullerene is a truncated icosahedron with 60 vertices and 32 faces (20 hexagons and 12 pentagons where no pentagons share a vertex) with a carbon atom at the vertices of each polygon and a bond along each polygon edge. The van der Waals diameter of a molecule is about 1.01 nanometers (nm). The nucleus to nucleus diameter of a molecule is about 0.71 nm. The molecule has two bond lengths.
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Icosahedron-20 sided polyhedron. All six genera in the family Phycodnaviridae have similar virion structure and morphology. They are large virions that can range between 100–220 nm in diameter. They have a double-stranded DNA genome, and a protein core surrounded by a lipid bilayer and an icosahedral capsid. The capsid has 2, 3 and 5 fold axis of symmetry with 20 equilateral triangle faces composing of protein subunits.
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Increasing the number of atoms that comprise the carbon skeleton leads to a geometry that increasingly approximates a sphere, and the space enclosed in the carbon "cage" increases. This trend continues with buckyballs or spherical fullerene (C60). Although not a Platonic hydrocarbon, buckminsterfullerene has the shape of a truncated icosahedron, an Archimedean solid. The concept can also be extended to regular Euclidean tilings, with the hexagonal tiling producing graphane.
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The Mukhopadhyay module can form any equilateral polyhedron. Each unit has a middle crease that forms an edge, and triangular wings that form adjacent stellated faces. For example, a cuboctahedral assembly has 24 units, since the cuboctahedron has 24 edges. Additionally, bipyramids are possible, by folding the central crease on each module outwards or convexly instead of inwards or concavely as for the icosahedron and other stellated polyhedra.
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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.
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3D model of a small retrosnub icosicosidodecahedron In geometry, the small retrosnub icosicosidodecahedron (also known as a retrosnub disicosidodecahedron, small inverted retrosnub icosicosidodecahedron, or retroholosnub icosahedron) is a nonconvex uniform polyhedron, indexed as U72. It has 112 faces (100 triangles and 12 pentagrams), 180 edges, and 60 vertices. It is given a Schläfli symbol ß{,5}. The 40 non-snub triangular faces form 20 coplanar pairs, forming star hexagons that are not quite regular.
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The prolate structure of a typical head on a bacteriophage An elongated icosahedron is a common shape for the heads of bacteriophages. Such a structure is composed of a cylinder with a cap at either end. The cylinder is composed of 10 elongated triangular faces. The Q number (or Tmid), which can be any positive integer, specifies the number of triangles, composed of asymmetric subunits, that make up the 10 triangles of the cylinder.
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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.
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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.
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The structure's covering later burned, but the structure itself still stands and, under the name Biosphère, currently houses an interpretive museum about the Saint Lawrence River. In the 1970s, Zomeworks licensed plans for structures based on other geometric solids, such as the Johnson solids, Archimedean solids, and Catalan solids.Geodesic domes are most often based on Platonic solids, particularly the icosahedron. These structures may have some faces that are not triangular, being squares or other polygons.
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Woo joined Zuni Icosahedron since 1988. Over the years he has created more than 70 theatre works. He is particularly outstanding in his creation of the Multimedia Music Theatre Series, which is a conglomeration of dexterous uses of theatre space, texts, video images and cutting-edge multimedia technologies. His endeavours in this music theatre series are groundbreaking, and his attempt in such theatre productions is the first of its kind in Hong Kong.
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The familiar 32-panel football design is sometimes referenced to describe the truncated icosahedron Archimedean solid, carbon buckyballs or the root structure of geodesic domes. There are a number of different types of football balls depending on the match and turf including: training footballs, match footballs, professional match footballs, beach footballs, street footballs, indoor footballs, turf balls, futsal footballs and mini/skills footballs.Soccer Balls , Soccer , 14 October 2013. Retrieved: 2013-10-14.
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The B80 boron cluster occupies the large space between four supericosahedra as described in the REB66 section. On the other hand, the 2-dimensional supericosahedron networks in the Sc4.5–xB57–y+zC3.5–z crystal structure stack in-phase along the z-axis. Instead of the B80 cluster, a pair of the I2 icosahedra fills the open space staying within the supericosahedron network, as shown in figure 28 where the icosahedron I2 is colored in yellow.
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Fisher, Irving (1943). "A World Map on a Regular Icosahedron by Gnomonic Projection." Geographical Review 33 (4): 605. Because of its very common usage, the Mercator projection has been supposed to have influenced people's view of the world, and because it shows countries near the Equator as too small when compared to those of Europe and North America, it has been supposed to cause people to consider those countries as less important.
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The same triangle forms half of a golden rectangle. It may also be found within a regular icosahedron of side length c: the shortest line segment from any vertex V to the plane of its five neighbors has length a, and the endpoints of this line segment together with any of the neighbors of V form the vertices of a right triangle with sides a, b, and c.nLab: pentagon decagon hexagon identity.
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The basic design of the Dogic is an icosahedron cut into 60 triangular pieces around its 12 tips and 20 face centers. All 80 pieces can move relative to each other. There are also a good number of internal moving pieces inside the puzzle, which are necessary to keep it in one piece as its surface pieces are rearranged. The 10-color Dogic, with the two possible layers of rotation slightly twisted.
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National Council for Science and the Environment. eds. S. Draggan and C. Cleveland Viral structures are built of repeated identical protein subunits known as capsomeres, and the icosahedron is the easiest shape to assemble using these subunits. A regular polyhedron is used because it can be built from a single basic unit protein used over and over again; this saves space in the viral genome. Various bacterial organelles with an icosahedral shape were also found.
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Euclid's Elements gave what amount to ruler-and-compass constructions for the five Platonic solids.See, for example, Euclid's Elements. However, the merely practical question of how one might draw a straight line in space, even with a ruler, might lead one to question what exactly it means to "construct" a regular polyhedron. (One could ask the same question about the polygons, of course.) Net for icosahedron The English word "construct" has the connotation of systematically building the thing constructed.
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Empedocles roots became the four classical elements of Greek philosophy. Plato (427–347 BC) took over the four elements of Empedocles. In the Timaeus, his major cosmological dialogue, the Platonic solid associated with water is the icosahedron which is formed from twenty equilateral triangles. This makes water the element with the greatest number of sides, which Plato regarded as appropriate because water flows out of one's hand when picked up, as if it is made of tiny little balls.
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Mathias WOO Yan Wai Mathias Woo Yan Wai (Chinese: 胡恩威; born 1968) joined the arts collective Zuni Icosahedron in 1988 and is now the Executive Director cum Co-Artistic Director of the group. Woo is renowned for his creative career in multimedia theatre as a scriptwriter, director, designer, producer as well as curator with a portfolio of more than 70 original theatre works, which are best known for their unique rendering of space and technology.
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We use "n-tuple" tessellations to add more evenly spaced apices to the original icosahedron (20 faces)--an idea that also had its precedents in paleomagnetism research several decades earlier. We just want to know which direction lines turn up the maximum anisotropic diffusion measures. If there is a single tract, there will be just two maxima pointing in opposite directions. If two tracts cross in the voxel, there will be two pairs of maxima, and so on.
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Therefore, the circles 1 and 2 intersect – a contradiction.See also Lemma 3.1 in A highly symmetrical realization of the kissing number 12 in three dimensions is by aligning the centers of outer spheres with vertices of a regular icosahedron. This leaves slightly more than 0.1 of the radius between two nearby spheres. In three dimensions, the kissing number is 12, but the correct value was much more difficult to establish than in dimensions one and two.
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Fig. 17. The boron-rich corner of the Sc-B-C phase diagram. Scandium has the smallest atomic and ionic (3+) radii (1.62 and 0.885 Å, respectively) among the rare-earth elements. It forms several icosahedron-based borides which are not found for other rare-earth elements; however, most of them are ternary Sc-B-C compounds. There are many boron-rich phases in the boron-rich corner of Sc-B-C phase diagram, as shown in figure 17.
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New York: W. A. Benjamin, 1963. Print. However, the boron arrangements can be classified as fragments of either the icosahedron or the octahedron because the bond angles are actually between 105° and 90°. The comparison of the diffraction data from X-ray diffraction and electron diffraction gave suspected bond lengths and angles: B1—B2 = 1.84 Å, B1—B3= 1.71 Å, B2—B1—B4= 98 ̊, B—H = 1.19 Å, B1—Hμ = 1.33 Å, B2—Hμ =1.43 Å.
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Throughout his career, Fuller had experimented with incorporating tensile components in his work, such as in the framing of his dymaxion houses. Snelson's 1948 innovation spurred Fuller to immediately commission a mast from Snelson. In 1949, Fuller developed a tensegrity- icosahedron based on the technology, and he and his students quickly developed further structures and applied the technology to building domes. After a hiatus, Snelson also went on to produce a plethora of sculptures based on tensegrity concepts.
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Subsequently, Bauersfeld developed the Model 2 with 8,956 stars, and full latitude capability. Over a dozen were installed before World War II again suspended planetarium work. These inter-war planetariums were constructed in Berlin and Düsseldorf in Germany, as well as Rome, Paris, Chicago, Los Angeles and New York. The Zeiss I planetarium in Jena is also considered the first geodesic dome derived from the icosahedron, 26 years before Buckminster Fuller reinvented and popularized this design.
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Cartesian coordinates for the vertices of a truncated great icosahedron centered at the origin are all the even permutations of : (±1, 0, ±3/τ) : (±2, ±1/τ, ±1/τ3) : (±(1+1/τ2), ±1, ±2/τ) where τ = (1+√5)/2 is the golden ratio (sometimes written φ). Using 1/τ2 = 1 − 1/τ one verifies that all vertices are on a sphere, centered at the origin, with the radius squared equal to 10−9/τ. The edges have length 2.
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An ideal regular octahedron in the Poincaré ball model of hyperbolic space (sphere at infinity not shown). All dihedral angles of this shape are right angles. Animation of an ideal icosahedron in the Klein model of hyperbolic space In three-dimensional hyperbolic geometry, an ideal polyhedron is a convex polyhedron all of whose vertices are ideal points, points "at infinity" rather than interior to three-dimensional hyperbolic space. It can be defined as the convex hull of a finite set of ideal points.
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Snyder equal-area projection is used in the ISEA (Icosahedral Snyder Equal Area) discrete global grids. The first projection studies was conducted by John P. Snyder in the 1990s. Snyder, J. P. (1992), “An Equal-Area Map Projection for Polyhedral Globes”, Cartographica, 29(1), 10-21. urn:doi:10.3138/27H7-8K88-4882-1752. It is a modified Lambert azimuthal equal-area projection, most adequate to the polyhedral globe, a truncated icosahedron with 32 same-area faces (20 hexagons and 12 pentagons).
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In 1855, the U.S. inventor Charles Goodyear – who had patented vulcanised rubber – exhibited a spherical football, with an exterior of vulcanised rubber panels, at the Paris Exhibition Universelle. The ball was to prove popular in early forms of football in the U.S.soccerballworld.com, (no date) "Charles Goodyear's Soccer Ball" Downloaded 30/11/06. The iconic ball with a regular pattern of hexagons and pentagons (see truncated icosahedron) did not become popular until the 1960s, and was first used in the World Cup in 1970.
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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).
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Construction from the vertices of a truncated octahedron, showing internal rectangles. The coordinates of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form (2, 1, 0). These coordinates represent the truncated octahedron with alternated vertices deleted. This construction is called a snub tetrahedron in its regular icosahedron form, generated by the same operations carried out starting with the vector (ϕ, 1, 0), where ϕ is the golden ratio.
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There are more than 4 sites in total among, say, B5–B8 sites, but many of them are equivalent by symmetry and thus do not have an individual label. The B10 polyhedron has not been observed previously and it is shown in figure 23. The icosahedron I2 has a boron-carbon mixed-occupancy site B,C6 whose occupancy is B/C=0.58/0.42. Remaining 3 boron-carbon mixed-occupancy sites are bridge sites; C and Si sites are also bridge sites.
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Because very small changes in its edge lengths can cause much bigger changes in its angles, physical models of the polyhedron appear to be flexible. As with the simpler Schönhardt polyhedron, the interior of Jessen's icosahedron cannot be triangulated into tetrahedra without adding new vertices. However, because it has Dehn invariant equal to zero, it is scissors-congruent to a cube, meaning that it can be sliced into smaller polyhedral pieces that can be rearranged to form a solid cube.
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The [B12H12]2− anion's B12 core is a regular icosahedron. The [B12H12]2− as a whole also has icosahedral molecular symmetry, and it belongs to the molecular point group Ih. Its icosahedral shape is consistent with the classification of this cage as "closo" in polyhedral skeletal electron pair theory. Crystals of Cs2B12H12 feature Cs+ ions in contact with twelve hydrides provided by four B12H122−. The B-B bond distances are 178 pm, and the B-H distances are 112 pm.
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IcoSoku is a puzzle invented in 2009 by Andrea Mainini and sold by Recent Toys International, which is based in the Netherlands. It won several awards in 2010. The puzzle frame is a blue plastic icosahedron, and the pieces are 20 white equilateral-triangular snap-in tiles with black dots and 12 yellow pins for the corners. The pins are printed with the numbers from 1 to 12, and the triangular tiles have up to three dots in each of their corners.
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The simplest shape made of these pyramids, often called "Toshie's Jewel" (shown on the right), is named after origami enthusiast Toshie Takahama. It is a three-unit hexahedron built around the notional scaffold of a flat equilateral triangle (two "faces", three edges); the protruding tab/pocket flaps are simply reconnected on the underside, resulting in two triangular pyramids joined at the base, a triangular bipyramid. The most popular intermediate model is the triakis icosahedron, shown below. It requires 30 units to build.
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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.
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Brückner's photo of the final stellation of the icosahedron, a stellated polyhedron first studied by Brückner Photo of polyhedra models by Brückner. Johannes Max Brückner (5 August 1860 – 1 November 1934) was a German geometer, known for his collection of polyhedral models. Brückner was born on August 5, 1860 in Hartau, in the Kingdom of Saxony, a town that is now part of Zittau, Germany. He completed a Ph.D. at Leipzig University in 1886, supervised by Felix Klein and Wilhelm Scheibner, with a dissertation concerning conformal maps.
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The first part, Compendio divina proportione (Compendium on the Divine Proportion), studies the golden ratio from a mathematical perspective (following the relevant work of Euclid) and explores its applications to various arts, in seventy-one chapters. Pacioli points out that golden rectangles can be inscribed by an icosahedron, and in the fifth chapter, gives five reasons why the golden ratio should be referred to as the "Divine Proportion": #Its value represents divine simplicity. #Its definition invokes three lengths, symbolizing the Holy Trinity. #Its irrationality represents God's incomprehensibility.
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Boron framework of YAlB14 is one of the simplest among icosahedron-based borides – it consists of only one kind of icosahedra and one bridging boron site. The bridging boron site is tetrahedrally coordinated by four boron atoms. Those atoms are another boron atom in the counter bridge site and three equatorial boron atoms of one of three B12 icosahedra. Aluminium atoms are separated by 0.2911 nm and are arranged in lines parallel to the x-axis, whereas yttrium atoms are separated by 0.3405 nm.
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Loop Subdivision of an icosahedron (top) after one and after two refinement steps In computer graphics, Loop subdivision surface is an approximating subdivision scheme developed by Charles Loop in 1987 for triangular meshes. Loop subdivision surfaces are defined recursively, dividing each triangle into four smaller ones. The method is based on a quartic box spline, which generate C2 continuous limit surfaces everywhere except at extraordinary vertices where they are C1 continuous. Geologists have also applied Loop Subdivision Surfaces to erosion on mountain faces, specifically in the Appalachians.
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Clusters ranged in molecular masses, but Kroto and Smalley found predominance in a C60 cluster that could be enhanced further by allowing the plasma to react longer. They also discovered that the C60 molecule formed a cage-like structure, a regular truncated icosahedron. For this discovery Curl, Kroto, and Smalley were awarded the 1996 Nobel Prize in Chemistry. The experimental evidence, a strong peak at 720 atomic mass units, indicated that a carbon molecule with 60 carbon atoms was forming, but provided no structural information.
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The most common examples are the curves X(N), X0(N), and X1(N) associated with the subgroups Γ(N), Γ0(N), and Γ1(N). The modular curve X(5) has genus 0: it is the Riemann sphere with 12 cusps located at the vertices of a regular icosahedron. The covering X(5) → X(1) is realized by the action of the icosahedral group on the Riemann sphere. This group is a simple group of order 60 isomorphic to A5 and PSL(2, 5).
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"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.
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The Euler characteristic of a sphere, triangulated like an icosahedron, is . Since every compact oriented 2-manifold can be triangulated by small geodesic triangles, it follows that : \int_M K dA = 2\pi\,\chi(M) where denotes the Euler characteristic of the surface. In fact if there are faces, edges and vertices, then and the left hand side equals . This is the celebrated Gauss–Bonnet theorem: it shows that the integral of the Gaussian curvature is a topological invariant of the manifold, namely the Euler characteristic.
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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.
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The name Icos comes from icosahedron, a 20-sided polyhedron, which is the shape of many viruses, and was chosen because the founders originally thought retroviruses might be involved in inflammation. The founders raised $33 million in July 1990 from many investors, including Bill Gates – who at the time was the largest shareholder, with 10% of the equity. The company initially had temporary offices in downtown Seattle, but moved to Bothell in September 1990. Icos went public on June 6, 1991, raising $36 million.
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Horizontal position may be expressed directly in geographic coordinates (latitude and longitude) for global models or in a map projection planar coordinates for regional models. The German weather service is using for its global ICON model (icosahedral non-hydrostatic global circulation model) a grid based on a regular icosahedron. Basic cells in this grid are triangles instead of the four corner cells in a traditional latitude-longitude grid. The advantage is that, different from a latitude-longitude cells are everywhere on the globe the same size.
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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.
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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".
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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.
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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.
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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.
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The support has a substantial effect on the electronic structure of gold clusters. Metal hydroxide supports such as Be(OH)2, Mg(OH)2, and La(OH)3, with gold clusters of < 1.5 nm in diameter constitute highly active catalysts for CO oxidation at 200 K (-73 °C). By means of techniques such as HR-TEM and EXAFS, it has been proven that the activity of these catalysts is due exclusively to clusters with 13 atoms arranged in an icosahedron structure. Furthermore, the metal loading should exceed 10 wt% for the catalysts to be active.
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He used these diagrams to construct an 11-fold cover of the Riemann sphere by itself, with monodromy group PSL(2,11), following earlier constructions of a 7-fold cover with monodromy PSL(2,7) connected to the Klein quartic in . These were all related to his investigations of the geometry of the quintic equation and the group A5 ≅ PSL(2,5), collected in his famous 1884/88 Lectures on the Icosahedron. The three surfaces constructed in this way from these three groups were much later shown to be closely related through the phenomenon of trinity.
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She ended her contract with What's Music in 2001 and signed on with EMI Taiwan. The first album with her new record label was titled I Just Want to Tell You, and it was produced by Yosinori Kameda, a famous Japanese music producer. In 2001, she decided to broaden her professional career by acting in her first TV drama with F4, Taiwan's boy-band equivalent to 'N Sync. She further broadened her acting career in 2003 by performing on stage with Zuni Icosahedron, an independent cultural collective concentrating on alternative theater and multi-media performances.
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Similar geometries occur for PSL(2,n) and more general groups for other modular curves. More exotically, there are special connections between the groups PSL(2,5) (order 60), PSL(2,7) (order 168) and PSL(2,11) (order 660), which also admit geometric interpretations – PSL(2,5) is the symmetries of the icosahedron (genus 0), PSL(2,7) of the Klein quartic (genus 3), and PSL(2,11) the buckyball surface (genus 70). These groups form a "trinity" in the sense of Vladimir Arnold, which gives a framework for the various relationships; see trinities for details.
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The B6 octahedron is smaller than the B12 icosahedron; therefore, rare-earth elements can reside in the space created by the replacement. The stacking sequences of B4C, REB15.5CN, REB22C2N and REB28.5C4 are shown in figures 12a, b, c and d, respectively. High-resolution transmission electron microscopy (HRTEM) lattice images of the latter three compounds, added to Fig. 12, do confirm the stacking sequence of each compound. The symbols 3T, 12R and 15R in brackets indicate the number of layers necessary to complete the stacking sequence, and T and R refer to trigonal and rhombohedral.
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ScB19+xSiy has a tetragonal crystal structure with space group P41212 (No. 92) or P43212 and lattice constants of a, b = 1.03081(2) and c = 1.42589(3) nm; it is isotypic to the α-AlB12 structure type. There are 28 atomic sites in the unit cell, which are assigned to 3 scandium atoms, 24 boron atoms and one silicon atom. Atomic coordinates, site occupancies and isotropic displacement factors are listed in table VI. The boron framework of ScB19+xSiy is based on one B12 icosahedron and one B22 unit.
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This stabilization technique allowed the synthesis of some other boron-rich rare- earth borides. Albert and Hillebrecht reviewed binary and selected ternary boron compounds containing main-group elements, namely, borides of the alkali and alkaline-earth metals, aluminum borides and compounds of boron and the nonmetals C, Si, Ge, N, P, As, O, S and Se. They, however, excluded the described here icosahedron-based rare-earth borides. Note that rare-earth elements have d- and f-electrons that complicates chemical and physical properties of their borides. Werheit et al.
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German stamp featuring Jamnitzer's work A later work on perspective, Artes Excelençias de la Perspectiba (1688) by P. Gómez de Alcuña, was heavily influenced by Jamnitzer. A 2008 German postage stamp, issued to commemorate the 500th anniversary of Jamnitzer's birth, included a reproduction of one of the pages of the book, depicting two polyhedral cones tilted towards each other. The full sheet of ten stamps also includes another figure from the book, a skeletal icosahedron. A French edition of Perspectiva corporum regularium, edited by Albert Flocon, was published by Brieux in 1964.
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A similar but distinct phenomenon is that of exceptional objects (and exceptional isomorphisms), which occurs when there are a "small" number of exceptions to a general pattern (such as a finite set of exceptions to an otherwise infinite rule). By contrast, in cases of pathology, often most or almost all instances of a phenomenon are pathological (e.g., almost all real numbers are irrational). Subjectively, exceptional objects (such as the icosahedron or sporadic simple groups) are generally considered "beautiful", unexpected examples of a theory, while pathological phenomena are often considered "ugly", as the name implies.
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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.
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The typical framework building blocks are polyhedral units, with 4-, 5-, 6- or 7-coordinate metal centres. These units share edges and/or vertices, or, less commonly, faces (such as in the ion , which has face-shared octahedra with Mo atoms at the vertices of an icosahedron). The most common unit for polymolybdates is the octahedral {MoO6} unit, often distorted by the Mo atom being off-centre to give one shorter Mo–O bond. Some polymolybdates contain pentagonal bipyramidal units; these are the key building blocks in the molybdenum blues.
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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.
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A stellated icosahedron made from custom papers Modular origami consists of putting a number of identical pieces together to form a complete model. Normally the individual pieces are simple but the final assembly may be tricky. Many of the modular origami models are decorative folding balls like kusudama, the technique differs though in that kusudama allows the pieces to be put together using thread or glue. Chinese paper folding includes a style called Golden Venture Folding where large numbers of pieces are put together to make elaborate models.
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This generalizes the construction of a line graph, in which every edge of the multigraph is replaced by a vertex. Fuzzy linear interval graphs are constructed in the same way as fuzzy circular interval graphs, but on a line rather than on a circle. Chudnovsky and Seymour classify arbitrary connected claw-free graphs into one of the following: # Six specific subclasses of claw-free graphs. Three of these are line graphs, proper circular arc graphs, and the induced subgraphs of an icosahedron; the other three involve additional definitions.
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During this time, the AVN model was extended to the end of the forecast period, eliminating the need of the MRF and thereby replacing it. In late 2002, the AVN model was renamed the Global Forecast System (GFS). The German Weather Service has been running their global hydrostatic model, the GME, using a hexagonal icosahedral grid since 2002. The GFS is slated to eventually be supplanted by the Flow- following, finite-volume Icosahedral Model (FIM), which like the GME is gridded on a truncated icosahedron, in the mid-2010s.
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For instance, the Grötzsch graph, the Mycielskian of a five-vertex cycle- graph, is factor-critical.. Every 2-vertex-connected claw-free graph with an odd number of vertices is factor-critical. For instance, the 11-vertex graph formed by removing a vertex from the regular icosahedron (the graph of the gyroelongated pentagonal pyramid) is both 2-connected and claw-free, so it is factor-critical. This result follows directly from the more fundamental theorem that every connected claw-free graph with an even number of vertices has a perfect matching..
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Chemical graph theory concerns the graph-theoretic structure of molecules and other clusters of atoms. Both the Errera graph itself and its dual graph are relevant in this context. Atoms of metals such as gold can form clusters in which a central atom is surrounded by twelve more atoms, in the pattern of an icosahedron. Another, larger, type of cluster can be formed by coalescing two of these icosahedral clusters, so that the central atom of each cluster becomes one of the boundary atoms for the other cluster.
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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.
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The stellation of a polyhedron extends the faces of a polyhedron into infinite planes and generates a new polyhedron that is bounded by these planes as faces and the intersections of these planes as edges. The Fifty Nine Icosahedra enumerates the stellations of the regular icosahedron, according to a set of rules put forward by J. C. P. Miller, including the complete stellation. The Du Val symbol of the complete stellation is H, because it includes all cells in the stellation diagram up to and including the outermost "h" layer.
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Modular origami A stellated icosahedron made from custom papers Modular origami or unit origami is a paperfolding technique which uses two or more sheets of paper to create a larger and more complex structure than would be possible using single-piece origami techniques. Each individual sheet of paper is folded into a module, or unit, and then modules are assembled into an integrated flat shape or three-dimensional structure by inserting flaps into pockets created by the folding process. These insertions create tension or friction that holds the model together.
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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).
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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.
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Ancient philosophy posited a set of classical elements to explain observed patterns in nature. These elements originally referred to earth, water, air and fire rather than the chemical elements of modern science. The term 'elements' (stoicheia) was first used by the Greek philosopher Plato in about 360 BCE in his dialogue Timaeus, which includes a discussion of the composition of inorganic and organic bodies and is a speculative treatise on chemistry. Plato believed the elements introduced a century earlier by Empedocles were composed of small polyhedral forms: tetrahedron (fire), octahedron (air), icosahedron (water), and cube (earth).
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Young artists, e.g. arts graduates, in their twenties are renting studios in Fo Tan while large art groups and cultural bodies like Zuni Icosahedron, Artist Commune, Videotage, 1a Space and On & On Theatre Workshop Company are based in the Cattle Depot Artist Village. At Fo Tan, artists are generally more scattered, individualistic and reluctant to be labelled, while artists in the Village tend to be more organised, forming an aggregate body. They described themselves as simply a group of studios but the sense of community is already strong enough for them to hold an open studio each year.
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The order-5 cubic honeycomb has a related alternated honeycomb, ↔ , with icosahedron and tetrahedron cells. The honeycomb is also one of four regular compact honeycombs in 3D hyperbolic space: There are fifteen uniform honeycombs in the [5,3,4] Coxeter group family, including the order-5 cubic honeycomb as the regular form: The order-5 cubic honeycomb is in a sequence of regular polychora and honeycombs with icosahedral vertex figures. It is also in a sequence of regular polychora and honeycombs with cubic cells. The first polytope in the sequence is the tesseract, and the second is the Euclidean cubic honeycomb.
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Wong released his first solo album Faith, Hope & Love (信望愛) in January 1992. This album began his long-time musical partnership with musician Jason Choi (蔡德才), along with lyrics by Yiu- fai Chow (周耀輝), Kam Kwok-leung (甘國亮), Jimmy Ngai (魏紹恩), and Zuni Icosahedron member Pia Ho (何秀萍). Wong desired to develop an eclectic and electronic sound for his first solo record, incorporating influences from acoustic, traditional Chinese and Indian music. The album acted as a cathartic release of Wong's feelings towards religious symbolism and his previous beliefs.
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The CaMV particle is an icosahedron with a diameter of 52 nm built from 420 capsid protein (CP) subunits arranged with a triangulation T = 7, which surrounds a solvent-filled central cavity. CaMV contains a circular double-stranded DNA molecule of about 8.0 kilobases, interrupted by nicks that result from the actions of RNAse H during reverse transcription. These nicks come from the Met-tRNA, and two RNA primers used in reverse transcription. After entering the host cell, these single stranded "nicks" in the viral DNA are repaired, forming a supercoiled molecule that binds to histones.
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An orthogonal polyhedron is one all of whose faces meet at right angles, and all of whose edges are parallel to axes of a Cartesian coordinate system. (Jessen's icosahedron provides an example of a polyhedron meeting one but not both of these two conditions.) Aside from the rectangular boxes, orthogonal polyhedra are nonconvex. They are the 3D analogs of 2D orthogonal polygons, also known as rectilinear polygons. Orthogonal polyhedra are used in computational geometry, where their constrained structure has enabled advances on problems unsolved for arbitrary polyhedra, for example, unfolding the surface of a polyhedron to a polygonal net..
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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.
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Nitrogen atoms strengthen the bonding in the c-plane by bridging three icosahedra, like C atoms in the C-B-C chain. Figure 13 depicts the c-plane network revealing the alternate bridging of the boron icosahedra by N and C atoms. Decreasing the number of the B6 octahedra diminishes the role of nitrogen because the C-B-C chains start bridging the icosahedra. On the other hand, in MgB9N the B6 octahedron layer and the B12 icosahedron layer stack alternatively and there is no C-B-C chains; thus only N atoms bridge the B12 icosahedra.
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The opposite edge of the supertetrahedron is parallel to the b-axis and the icosahedra on this edge form a chain along the b-axis. As shown in figure 19, there are wide tunnels surrounded by the icosahedron arrangement along the a- and b-axes. The tunnels are filled by the B22 units which strongly bond to the surrounding icosahedra; the connection of the B22 units is helical and it runs along the c-axis as shown in figure 19b. Scandium atoms occupy the voids in the boron network as shown in figure 19c, and the Si atoms bridge the B22 units.
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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.
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Astrological dice are a specialized set of three 12-sided dice for divination; the first die represents planets, the Sun, the Moon, and the nodes of the Moon, the second die represents the 12 zodiac signs, and the third represents the 12 houses. A specialized icosahedron die provides the answers of the Magic 8-Ball, conventionally used to provide answers to yes-or-no questions. Dice can be used to generate random numbers for use in passwords and cryptography applications. The Electronic Frontier Foundation describes a method by which dice can be used to generate passphrases.
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Lipscomb and his graduate students further determined the molecular structure of boranes (compounds of boron and hydrogen) using X-ray crystallography in the 1950s and developed theories to explain their bonds. Later he applied the same methods to related problems, including the structure of carboranes (compounds of carbon, boron, and hydrogen). Longuet-Higgins and Roberts discussed the electronic structure of an icosahedron of boron atoms and of the borides MB6. The mechanism of the three-center two-electron bond was also discussed in a later paper by Longuet-Higgins, and an essentially equivalent mechanism was proposed by Eberhardt, Crawford, and Lipscomb.
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A geodesic grid is a global Earth reference that uses triangular tiles based on the subdivision of a polyhedron (usually the icosahedron, and usually a Class I subdivision) to subdivide the surface of the Earth. Such a grid does not have a straightforward relationship to latitude and longitude, but conforms to many of the main criteria for a statistically valid discrete global grid. Primarily, the cells' area and shape are generally similar, especially near the poles where many other spatial grids have singularities or heavy distortion. The popular Quaternary Triangular Mesh (QTM) falls into this category.
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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.
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The Architecture is Art Festival (AIAF) is the biannual art festival presented by Zuni Icosahedron since 2009. With architecture as the central topic, a series of multimedia performances, traditional Chinese opera, exhibitions, seminars, conferences and student workshops are held at the Hong Kong Cultural Centre and other venues. As Rossella Ferrari writes in her review in TDR, "(AIAF)...is a celebration of architecture and theatre or, rather, of architecture in theatre and architecture as theatre—the first of its kind in Asia and possibly worldwide."Ferrari, Rossella (2011) 'The Stage as a Drawing Board: Zuni Icosahedron’s Architecture Is Art Festival.
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Buckminsterfullerene is the smallest fullerene molecule containing pentagonal and hexagonal rings in which no two pentagons share an edge (which can be destabilizing, as in pentalene). It is also most common in terms of natural occurrence, as it can often be found in soot. The empirical formula of buckminsterfullerene is and its structure is a truncated icosahedron, which resembles an association football ball of the type made of twenty hexagons and twelve pentagons, with a carbon atom at the vertices of each polygon and a bond along each polygon edge. The van der Waals diameter of a buckminsterfullerene molecule is about 1.1 nanometers (nm).
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The Megavirus particle exhibits a protein capsid diameter of 440 nanometres (as seen by electron microscopy on thin sections of epoxy resin inclusions), enclosed into a solid mesh of bacterial-like capsular material 75 nm to 100 nm thick. The capsid appears hexagonal, but its icosahedral symmetry is imperfect, due to the presence of the “stargate”, at a single specific vertex of the icosahedron. The stargate is a five-pronged star structure forming the portal through which the internal core of the particle is delivered to the host's cytoplasm. This core is enclosed within two lipid membranes in the particle, also containing a large and diverse complement of viral proteins (e.g.
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More generally, every planar graph of minimum degree at least three either has an edge of total degree at most 12, or at least 60 edges that (like the edges in the triakis icosahedron) connect vertices of degrees 3 and 10. If all triangular faces of a polyhedron are vertex-disjoint, there exists an edge with smaller total degree, at most eight. Generalizations of the theorem are also known for graph embeddings onto surfaces with higher genus. The theorem cannot be generalized to all planar graphs, as the complete bipartite graphs K_{1,n-1} and K_{2,n-2} have edges with unbounded total degree.
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The Herschel graph is named after British astronomer Alexander Stewart Herschel, who wrote an early paper concerning William Rowan Hamilton's icosian game: the Herschel graph describes the smallest convex polyhedron for which this game has no solution. However, Herschel's paper described solutions for the Icosian game only on the graphs of the regular tetrahedron and regular icosahedron; it did not describe the Herschel graph.. The name "the Herschel graph" makes an early appearance in a graph theory textbook by John Adrian Bondy and U. S. R. Murty, published in 1976. However, the graph itself was described earlier, for instance by H. S. M. Coxeter.
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He exhaustively cataloged the central and surface angles of these spherical triangles and their related chord factors. Fuller was continually on the lookout for ways to connect the dots, often purely speculatively. As an example of "dot connecting" he sought to relate the 120 basic disequilibrium LCD triangles of the spherical icosahedron to the plane net of his A module.(915.11Fig. 913.01, Table 905.65) The Jitterbug Transformation provided a unifying dynamic in this work, with much significance attached to the doubling and quadrupling of edges that occurred, when a cuboctahedron is collapsed through icosahedral, octahedral and tetrahedral stages, then inside- outed and re-expanded in a complementary fashion.
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Conway calls it a kisdeltille,John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table) constructed as a kis operation applied to a triangular tiling (deltille). In Japan the pattern is called asanoha for hemp leaf, although the name also applies to other triakis shapes like the triakis icosahedron and triakis octahedron. It is the dual tessellation of the truncated hexagonal tiling which has one triangle and two dodecagons at each vertex. :320px It is one of eight edge tessellations, tessellations generated by reflections across each edge of a prototile..
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Adidas Telstar-style ball, with the familiar black and white spherical truncated icosahedron pattern, introduced in 1970 A football, soccer ball, football ball, or association football ball is the ball used in the sport of association football. The name of the ball varies according to whether the sport is called "football", "soccer", or "association football". The ball's spherical shape, as well as its size, weight, and material composition, are specified by Law 2 of the Laws of the Game maintained by the International Football Association Board. Additional, more stringent standards are specified by FIFA and subordinate governing bodies for the balls used in the competitions they sanction.
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For example, three boron atoms make up a triangle where they share two electrons to complete the so-called three-center bonding. Boron polyhedra, such as B6 octahedron, B12 cuboctahedron and B12 icosahedron, lack two valence electrons per polyhedron to complete the polyhedron-based framework structure. Metal atoms need to donate two electrons per boron polyhedron to form boron-rich metal borides. Thus, boron compounds are often regarded as electron-deficient solids. Icosahedral B12 compounds include α-rhombohedral boron (B13C2), β-rhombohedral boron (MeBx, 23≤x), α-tetragonal boron (B48B2C2), β-tetragonal boron (β-AlB12), AlB10 or AlC4B24, YB25, YB50, YB66, NaB15 or MgAlB14, γ-AlB12, BeB3 and SiB6. Fig. 2.
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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.
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Most species of leafhoppers produce hollow spherical brochosomes, 0.2–0.7 micrometres in diameter, with a honeycombed outer wall. They often consist of 20 hexagonal and 12 pentagonal cells, making the outline of each brochosome approximating a truncated icosahedron – the geometry of a soccer ball and a C60 buckminsterfullerene molecule. The chemical composition of brochosomes includes several kinds of proteinsRakitov R., Moysa A. A., Kopylov A. T., Moshkovskii S. A., Peters R. S., Meusemann K., Misof B., Dietrich C. H., Johnson K. P., Podsiadlowski L., Walden K. K. O. (2018) Brochosomins and other novel proteins from brochosomes of leafhoppers (Insecta, Hemiptera, Cicadellidae). Insect Biochemistry and Molecular Biology, 94, 10-17.
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425–6 An alternative approach is to consider the pelvis part of an integrated mechanical system based on the tensegrity icosahedron as an infinite element. Such a system is able to withstand omnidirectional forces—ranging from weight-bearing to childbearing—and, as a low energy requiring system, is favoured by natural selection. Levin (2003), A Different Approach to the Mechanics of the Human Pelvis: Tensegrity (See conclusions.) The pelvic inclination angle is the single most important element of the human body posture and is adjusted at the hips. It is also one of the rare things that can be measured at the assessment of the posture.
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Common building blocks for the shell of alpha- carboxysomes are called CsoS1A/B/C (BMC-H), CsoS4A/B (BMC-P), and CsoS1D (BMC-T). CsoS4A/B were the first BMC-P proteins to be experimentally demonstrated as minor components of the BMC shell (only 12 pentamers are required to cap the vertices of an icosahedron). CsoS1D is first BMC-T which has been structurally characterized; it is also the first example of dimerization of two BMC building blocks in a face-to-face fashion to create a tiny cage. The CsoS1D cage has gated pore at both end, which is proposed to facilitate large metabolites crossing the shell.
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Zack then activates his drones near Europa to defeat the escorts and destroy the Icebreaker, resulting in the alien ships powering down and falling to Earth. An icosahedron arises from Europa and identifies itself as the Emissary, a machine that was created by a galactic community of civilizations called the Sodality. The Emissary had orchestrated the entire situation as a test to see if humanity could exist peacefully with their group, and declares that Earth has passed. Zack accepts the membership in the Sodality on behalf of Earth, and the third wave of alien ships arrive to aid the survivors and to restore the planet.
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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.
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The balls used in association football and team handball are perhaps the best-known example of a spherical polyhedron analog to the truncated icosahedron, found in everyday life. The ball comprises the same pattern of regular pentagons and regular hexagons, but it is more spherical due to the pressure of the air inside and the elasticity of the ball. This ball type was introduced to the World Cup in 1970 (starting in 2006, this iconic design has been superseded by alternative patterns). Geodesic domes are typically based on triangular facetings of this geometry with example structures found across the world, popularized by Buckminster Fuller.
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If two bilunabirotundae are aligned this way on opposite sides of the icosidodecahedron, then two vertices of the bilunabirotundae meet in the very center of the icosidodecahedron. The other two clusters of faces of the bilunabirotunda, the lunes (each lune featuring two triangles adjacent to opposite sides of one square), can be aligned with a congruent patch of faces on the rhombicosidodecahedron. If two bilunabirotundae are aligned this way on opposite sides of the rhombicosidodecahedron, then a cube can be put between the bilunabirotundae at the very center of the rhombicosidodecahedron. Each of the two pairs of adjacent pentagons (each pair of pentagons sharing an edge) can be aligned with the pentagonal faces of a metabidiminished icosahedron as well.
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For example, Volume 25 featured an illustrated diagram of the 2018 lunar calendar, and Volume 18 included a twenty-sided icosahedron globe. Thematically, Smith Journal "moves where it will and that really is anywhere; columns from aging rock-stars, photo essays, features on primitive skills practitioners and investigations into international (bee) hive heists." For instance, Volume 25 profiled photographer Daniel George, whose portfolio consists of household items used for target practice; an article discussing the cultural re-evaluation of 1980s hair metal music; and an interview with former NASA scientist Robert Lang who now designs origami. Previous issues included interviews with American musician Henry Rollins, and professional safe-cracker Jeff Sitar, an historical article on Soviet rocket architect Galina Balashova, and Brazil's illegal hot air balloon subculture.
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The yellow smiley returned the following year, but was now enclosed in a circle surrounded by the words "LEEDS UNITED AFC". In the 1978–79 season, a new badge appeared that was similar to that of the previous season, except now the words "LEEDS UNITED AFC" enclosed a stylised peacock (a reference to the club's nickname, "The Peacocks") rather than the yellow smiley. In 1984, another badge was introduced which lasted until 1998, making it the longest lived of the modern era. The distinctive rose and ball badge used the traditional blue, gold and white colours, and incorporated the White Rose of York, the club's name, and a football (a truncated icosahedron similar to the Adidas Telstar, but in Leeds colours) in the core section.
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Kepler Star in Oslo Airport, Gardermoen, 2000 Vebjørn's next public arts project was the Kepler Star, a permanent 45 meter high art installation by the Oslo Airport. Created to honor Doctors Without Borders for winning the 1999 Nobel Peace Prize, the star itself is based on a design from Johannes Kepler, further combined with an icosahedron - a polyhedron with 20 faces and one of the five platonic bodies, and consists of a skeleton made of steel with crinkled glass. The star sits on three thirty meter high concrete pillars; inspired by the Nunataken in Queen Maud Land Vebjørn saw during his expedition to Antarctica in 1996. Since then, the Kepler Star has seen many uses including being lit pink for breast cancer awareness month in October 2014.
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An icosahedron constructed with Geomag rods and spheres In May 1998, Claudio Vicentelli, experienced in technical applications of permanent magnets and the creator of the first (1st) product deposited for the magnetic circuit, and the brand name "GEOMAG" (which has developed the characters in the logo with chained circles). The patent defines a particular circuit obtained by the coupling of bars, of which each consists of two magnets at the ends and are connected with a metal pin and metallic spheres. This structure guarantees high performance using the minimum amount of magnetic material, which is very expensive. In August of the same year Vincelli grants the license to market and manufacture the products worldwide to the Sardinian company, Plastwood S.r.l.
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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.
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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.
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After teaching at a grammar school in Zwickau, he moved to the gymnasium in Bautzen. Brückner is known for making many geometric models, particularly of stellated and uniform polyhedra, which he documented in his book Vielecke und Vielflache: Theorie und Geschichte (Polygons and polyhedra: Theory and History, Leipzig: B. G. Teubner, 1900).Vielecke und Vielflache on Internet Archive, accessed 2015-12-01.. The shapes first studied in this book include the final stellation of the icosahedron and the compound of three octahedra, made famous by M. C. Escher's print Stars.. Coxeter's analysis of Stars is on pp. 61–62. Joseph Malkevitch lists the publication of this book, which documented all that was known on polyhedra at the time, as one of 25 milestones in the history of polyhedra.
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If the twinned icosahedra were independent without twinning then B23 would be a bridge site linking three icosahedra. However, because of twinning, B23 shifts closer to the twinned icosahedra than another icosahedron; thus B23 is currently treated as a member of the twinned icosahedra. In ScB19+xSiy, the two B24 sites which correspond to the vacant sites in the B20 unit are partially occupied; thus, the unit should be referred to as a B22 cluster which is occupied by about 20.6 boron atoms. Scandium atoms occupy 3 of 5 Al sites of α-AlB12, that is Sc1, Sc2 and Sc3 correspond to Al4, Al1 and Al2 sites of α-AlB12, respectively. The Al3 and Al5 sites are empty for ScB19+xSiy, and the Si site links two B22 units.
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Atomic coordinates, site occupancies and isotropic displacement factors are listed in table IX. More than 500 atoms are available in the unit cell. In the crystal structure, there are six structurally independent icosahedra I1–I6, which are constructed from B1–B12, B13–B24, B25–B32, B33–B40, B41–B44 and B45–B56 sites, respectively; B57–B62 sites form a B8 polyhedron. The Sc4.5–xB57–y+zC3.5–z crystal structure is layered, as shown in figure 26. This structure has been described in terms of two kinds of boron icosahedron layers, L1 and L2. L1 consists of the icosahedra I3, I4 and I5 and the C65 "dimer", and L2 consists of the icosahedra I2 and I6. I1 is sandwiched by L1 and L2 and the B8 polyhedron is sandwiched by L2.
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'Mary, the woman of Faith', bronze, by Terry Jones, Clifton Cathedral The votive candelabra hanging in the Lady Chapel [H on Plan] is made from twenty stainless steel equilateral triangles (a regular icosahedron) and was designed and made by Brother Patrick, of Prinknash Abbey in Gloucestershire.History Tour, Clifton Cathedral site It features in a pop music video (see Appearances in Media section). Also in the Lady Chapel is a Lampedusa Cross [J on Plan]. This is made from wood from migrant boats destroyed in the Mediterranean and recovered from Lampedusa, Italy between 2012 and 2016. It is similar to the British Museum’s Lampedusa Cross and is intended to reflect Pope Francis’s 2017 ‘Share the Journey’ exhortation for the Church to care for, and show solidarity to, all migrants and asylum seekers.
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Modified Airocean projection The Terra 1-to-1 mod implements several map projections by default; including the transverse Mercator, sinusoidal, Equal Earth, equirectangular, and Dymaxion or Airocean projections, and a modified version of the latter, made specifically for the project, named Modified Airocean or Airocean-edit. PippenFTS released a YouTube video titled We are now ready to Build The Earth, 1:1 scale in Minecraft, in which he detailed the process undergone to create it. This map projection provides an extremely low amount of distortion of both shapes and sizes on land, at the cost of heavily distorting the oceans. Unlike a typical Airocean projection, Modified Airocean is not intended to be unfolded into a 3D object like an icosahedron, and has its continents placed such that it looks somewhat similar to an equirectangular projection.
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Over the years, Zuni has been invited to more than 30 cities in Europe, Asia, and America for cultural exchange and performances. In 2000, Zuni co-presented “Festival of Vision: Berlin / Hong Kong”, the first ever large-scale cultural exchange between Asia and Europe. In 2009, Zuni’s artistic director Danny Yung was awarded the Merit Cross of the Order of Merit by the German Federal Government in recognition of his contribution to arts and cultural exchange between Germany and Hong Kong. The Forbidden City Exhibition presented and organised by Zuni Icosahedron at the Hong Kong Cultural Centre in 2009, as the kicking-off programme of its venue partnership Since 2009, Zuni has been a venue partner of the Hong Kong Cultural Centre, producing a series of theatre works and outreach education programmes.
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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,.
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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.
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About 1835, Jerrard demonstrated that quintics can be solved by using ultraradicals (also known as Bring radicals), the unique real root of for real numbers . In 1858 Charles Hermite showed that the Bring radical could be characterized in terms of the Jacobi theta functions and their associated elliptic modular functions, using an approach similar to the more familiar approach of solving cubic equations by means of trigonometric functions. At around the same time, Leopold Kronecker, using group theory, developed a simpler way of deriving Hermite's result, as had Francesco Brioschi. Later, Felix Klein came up with a method that relates the symmetries of the icosahedron, Galois theory, and the elliptic modular functions that are featured in Hermite's solution, giving an explanation for why they should appear at all, and developed his own solution in terms of generalized hypergeometric functions.
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In the year 2036, a fourth dimensional axis called Dimension W is proven to exist. Cross-dimensional electromagnetic induction devices, known as Coils, were developed to draw out the inexhaustible supply of energy that exists in Dimension W. New Tesla Energy and governments built sixty giant towers around the world in the pattern of a truncated icosahedron to stabilize the energy from Dimension W and supply power to the entire world. This "world system" is nearing its tenth year of operation as the story begins in 2072, and Coils of various sizes provide remote electrical power to everything from cellphones to vehicles and robots. However, dangerous unregistered Coils that do not send information back to New Tesla Energy are being used for illicit purposes, and bounty hunters known as "Collectors" are tasked with confiscating the illegal Coils.
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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.
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Salts of B12H undergo hydroxylation with hydrogen peroxide to give salts of [B12(OH)12]2−. The hydrogen atoms in the ion [B12H12]2- can be replaced by the halogens with various degrees of substitution. The following numbering scheme is used to identify the products. The first boron atom is numbered 1, then the closest ring of five atoms around it is numbered anticlockwise from 2 to 6. The next ring of boron atoms is started from 7 for the atoms closest to number 2 and 3, and counts anticlockwise to 11. The atom opposite the original is numbered 12. A related derivative is [B12(CH3)12]2−. The icosahedron of boron atoms is aromatic in nature. Under kilobar pressure of carbon monoxide [B12H12]2− reacts to form the carbonyl derivatives [B12H11CO]− and the 1,12- and 1,7-isomers of B12H10(CO)2.
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357; . If a polyhedron with vertices is formed by repeating the Kleetope construction some number of times, starting from a tetrahedron, then its longest path has length ; that is, the shortness exponent of these graphs is , approximately 0.630930. The same technique shows that in any higher dimension , there exist simplicial polytopes with shortness exponent .. Similarly, used the Kleetope construction to provide an infinite family of examples of simplicial polyhedra with an even number of vertices that have no perfect matching. Kleetopes also have some extreme properties related to their vertex degrees: if each edge in a planar graph is incident to at least seven other edges, then there must exist a vertex of degree at most five all but one of whose neighbors have degree 20 or more, and the Kleetope of the Kleetope of the icosahedron provides an example in which the high-degree vertices have degree exactly 20..
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During the 14-year run of AADT's New York Season, the dance company performed at various sites and events throughout New York including Riverside Dance Festival, Marymount Manhattan College Theater, Pace University Schimmel Center, Dance Theater Workshop, Open Eye Theater, Clark Center NYC, and Synod House. For its touring performances, AADT performed at colleges and sites in Texas, Ohio, New Mexico, North Carolina, Pennsylvania, and numerous other states throughout the country. AADT collaborated with notable choreographers Saeko Ichinohe, Sun Ock Lee, and Reynaldo Alejandro in the Asian New Dance Coalition; the company also invited guest choreographers, dancers, performers, and artists such as Yung Yung Tsuai, Muna Tseng, Zuni Icosahedron, Sin Cha Hong, poets Kimiko Hahn & Shu Ting, playwright David Henry Hwang, and artist Zhang Hongtu. Several dance company performances are currently archived in video format at the New York Public Library for the Performing Arts at Lincoln Center.
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Tetrahedral packing: The dihedral angle of a tetrahedron is not commensurable with 2; consequently, a hole remains between two faces of a packing of five tetrahedra with a common edge. A packing of twenty tetrahedra with a common vertex in such a way that the twelve outer vertices form an irregular icosahedron The stability of metals is a longstanding question of solid state physics, which can only be understood in the quantum mechanical framework by properly taking into account the interaction between the positively charged ions and the valence and conduction electrons. It is nevertheless possible to use a very simplified picture of metallic bonding and only keeps an isotropic type of interactions, leading to structures which can be represented as densely packed spheres. And indeed the crystalline simple metal structures are often either close packed face-centered cubic (fcc) or hexagonal close packing (hcp) lattices.
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The vertices of a snub 24-cell centered at the origin of 4-space, with edges of length 2, are obtained by taking even permutations of :(0, ±1, ±φ, ±φ2) (where φ = (1+)/2 is the golden ratio). These 96 vertices can be found by partitioning each of the 96 edges of a 24-cell into the golden ratio in a consistent manner, in much the same way that the 12 vertices of an icosahedron or "snub octahedron" can be produced by partitioning the 12 edges of an octahedron in the golden ratio. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector.Coxeter, Regular polytopes, 1973 The 96 vertices of the snub 24-cell, together with the 24 vertices of a 24-cell, form the 120 vertices of the 600-cell.
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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.
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The boron framework of YAlB14 needs donation of four electrons from metal elements: two electrons for a B12 icosahedron and one electron for each of the two bridging boron atoms – to support their tetrahedral coordination. The actual chemical composition of YAlB14, determined by the structure analysis, is Y0.62Al0.71B14 as described in table I. If both metal elements are trivalent ions then 3.99 electrons can be transferred to the boron framework, which is very close to the required value of 4. However, because the bonding between the bridging boron atoms is weaker than in a typical B-B covalent bond, less than 2 electrons are donated to this bond, and metal atoms need not be trivalent. On the other hand, the electron transfer from metal atoms to the boron framework implies that not only strong covalent B-B bonding within the framework but also ionic interaction between metal atoms and the framework contribute to the YAlB14 phase stabilization.
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In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one. It can be described as the symmetry group of a non-square rectangle (with the three non-identity elements being horizontal and vertical reflection and 180-degree rotation), as the group of bitwise exclusive or operations on two-bit binary values, or more abstractly as , the direct product of two copies of the cyclic group of order 2. It was named Vierergruppe (meaning four-group) by Felix Klein in 1884.Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade (Lectures on the icosahedron and the solution of equations of the fifth degree) It is also called the Klein group, and is often symbolized by the letter V or as K4.
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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).
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A polyhedron formed by replacing each of the faces of an icosahedron by a mesh of 100 triangles, an example of the lower bound construction of In a √n × √n grid graph, a set S of s < √n points can enclose a subset of at most s(s − 1)/2 grid points, where the maximum is achieved by arranging S in a diagonal line near a corner of the grid. Therefore, in order to form a separator that separates at least n/3 of the points from the remaining grid, s needs to be at least √(2n/3), approximately 0.82√n. There exist n-vertex planar graphs (for arbitrarily large values of n) such that, for every separator S that partitions the remaining graph into subgraphs of at most 2n/3 vertices, S has at least √(4π√3)√n vertices, approximately 1.56√n. The construction involves approximating a sphere by a convex polyhedron, replacing each of the faces of the polyhedron by a triangular mesh, and applying isoperimetric theorems for the surface of the sphere.
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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.
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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).
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Its dihedral angle is approximately 138.19°. Thus, just as hexagons have angles not less than 120° and cannot be used as the faces of a convex regular polyhedron because such a construction would not meet the requirement that at least three faces meet at a vertex and leave a positive defect for folding in three dimensions, icosahedra cannot be used as the cells of a convex regular polychoron because, similarly, at least three cells must meet at an edge and leave a positive defect for folding in four dimensions (in general for a convex polytope in n dimensions, at least three facets must meet at a peak and leave a positive defect for folding in n-space). However, when combined with suitable cells having smaller dihedral angles, icosahedra can be used as cells in semi-regular polychora (for example the snub 24-cell), just as hexagons can be used as faces in semi-regular polyhedra (for example the truncated icosahedron). Finally, non-convex polytopes do not carry the same strict requirements as convex polytopes, and icosahedra are indeed the cells of the icosahedral 120-cell, one of the ten non-convex regular polychora.
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