1.
Archimedean solid
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In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the semi-regular convex polyhedrons composed of regular meeting in identical vertices, excluding the 5 Platonic solids. They differ from the Johnson solids, whose regular polygonal faces do not meet in identical vertices, identical vertices means that for any two vertices, there is a global isometry of the entire solid that takes one vertex to the other. Excluding these two families, there are 13 Archimedean solids. All the Archimedan solids can be made via Wythoff constructions from the Platonic solids with tetrahedral, octahedral and icosahedral symmetry, the Archimedean solids take their name from Archimedes, who discussed them in a now-lost work. Pappus refers to it, stating that Archimedes listed 13 polyhedra, kepler may have also found the elongated square gyrobicupola, at least, he once stated that there were 14 Archimedean solids. However, his published enumeration only includes the 13 uniform polyhedra, here the vertex configuration refers to the type of regular polygons that meet at any given vertex. For example, a configuration of means that a square, hexagon. Some definitions of semiregular polyhedron include one more figure, the square gyrobicupola or pseudo-rhombicuboctahedron. The number of vertices is 720° divided by the angle defect. The cuboctahedron and icosidodecahedron are edge-uniform and are called quasi-regular, the duals of the Archimedean solids are called the Catalan solids. Together with the bipyramids and trapezohedra, these are the face-uniform solids with regular vertices, the snub cube and snub dodecahedron are known as chiral, as they come in a left-handed form and right-handed form. When something comes in forms which are each others three-dimensional mirror image. The different Archimedean and Platonic solids can be related to each other using a handful of general constructions, starting with a Platonic solid, truncation involves cutting away of corners. To preserve symmetry, the cut is in a perpendicular to the line joining a corner to the center of the polyhedron and is the same for all corners. Depending on how much is truncated, different Platonic and Archimedean solids can be created, expansion or cantellation involves moving each face away from the center and taking the convex hull. Expansion with twisting also involves rotating the faces, thus breaking the rectangles corresponding to edges into triangles, the last construction we use here is truncation of both corners and edges. Ignoring scaling, expansion can also be viewed as truncation of corners and edges, note the duality between the cube and the octahedron, and between the dodecahedron and the icosahedron

2.
Catalan solid
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In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, the Catalan solids are all convex. They are face-transitive but not vertex-transitive and this is because the dual Archimedean solids are vertex-transitive and not face-transitive. Note that unlike Platonic solids and Archimedean solids, the faces of Catalan solids are not regular polygons, however, the vertex figures of Catalan solids are regular, and they have constant dihedral angles. Being face-transitive, Catalan solids are isohedra, additionally, two of the Catalan solids are edge-transitive, the rhombic dodecahedron and the rhombic triacontahedron. These are the duals of the two quasi-regular Archimedean solids, just as prisms and antiprisms are generally not considered Archimedean solids, so bipyramids and trapezohedra are generally not considered Catalan solids, despite being face-transitive. Two of the Catalan solids are chiral, the pentagonal icositetrahedron and these each come in two enantiomorphs. Not counting the enantiomorphs, bipyramids, and trapezohedra, there are a total of 13 Catalan solids, the Catalan solids, along with their dual Archimedean solids, can be grouped by their symmetry, tetrahedral, octahedral, and icosahedral. There are 6 forms per symmetry, while the self-symmetric tetrahedral group only has three forms and two of those are duplicated with octahedral symmetry. J. lÉcole Polytechnique 41, 1-71,1865, alan Holden Shapes, Space, and Symmetry. Wenninger, Magnus, Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR730208 Williams, the Geometrical Foundation of Natural Structure, A Source Book of Design. California, University of California Press Berkeley, chapter 4, Duals of the Archimedean polyhedra, prisma and antiprisms Weisstein, Eric W. Catalan Solids. Archived from the original on 4 February 2007, Archimedean duals – at Virtual Reality Polyhedra Interactive Catalan Solid in Java

3.
Chamfer (geometry)
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In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion, moving faces apart and outward, for polyhedra, this operation adds a new hexagonal face in place of each original edge. In Conway polyhedron notation it is represented by the letter c, a polyhedron with e edges will have a chamfered form containing 2e new vertices, 3e new edges, and e new hexagonal faces. The chamfer operation applied in series creates progressively larger polyhedra with new hexagonal faces replacing edges from the previous one, the chamfer operator transforms GP to GP. A regular polyhedron, GP, create a Goldberg polyhedra sequence, the truncated octahedron or truncated icosahedron, GP creates a Goldberg sequence, GP, GP, GP, GP. A truncated tetrakis hexahedron or pentakis dodecahedron, GP, creates a Goldberg sequence, like the expansion operation, chamfer can be applied to any dimension. For polygons, it triples the number of vertices, for polychora, new cells are created around the original edges, The cells are prisms, containing two copies of the original face, with pyramids augmented onto the prism sides. The chamfered tetrahedron is a convex polyhedron constructed as a truncated cube or chamfer operation on a tetrahedron. It is the Goldberg polyhedron GIII, containing triangular and hexagonal faces and it can look a little like a truncated tetrahedron, which has 4 hexagonal and 4 triangular faces, which is the related Goldberg polyhedron, GIII. Net In geometry, the cube is a convex polyhedron constructed from the rhombic dodecahedron by truncating the 6 vertices. The 6 vertices are truncated such that all edges are equal length, the original 12 rhombic faces become flattened hexagons, and the truncated vertices become squares. The hexagonal faces are equilateral but not regular and they are formed by a truncated rhombus, have 2 internal angles of about 109.47 degrees and 4 internal angles of about 125.26 degrees, while a regular hexagon would have all 120 degree angles. Because all its faces have a number of sides with 180 degree rotation symmetry. It is also the Goldberg polyhedron GIV, containing square and hexagonal faces, the chamfered cube is the Minkowski sum of a rhombic dodecahedron and a cube of side length 1 when eight vertices of the rhombic dodecahedron are at and its six vertices are at the permutations of. The chamfered cube can be constructed with symmetry and rectangular faces. This can be seen as a pyritohedron with 6 axial edges planned and we can construct a truncated octahedron model by twenty four chamfered cube blocks. This polyhedron looks similar to the truncated octahedron, In geometry. The 8 vertices are truncated such that all edges are equal length, the original 12 rhombic faces become flattened hexagons, and the truncated vertices become triangles

4.
Cuboctahedron
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In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, as such, it is a quasiregular polyhedron, i. e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. Its dual polyhedron is the rhombic dodecahedron, the cuboctahedron was probably known to Plato, Herons Definitiones quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares. Heptaparallelohedron Fuller applied the name Dymaxion to this shape, used in a version of the Dymaxion map. He also called it the Vector Equilibrium and he called a cuboctahedron consisting of rigid struts connected by flexible vertices a jitterbug. With Oh symmetry, order 48, it is a cube or rectified octahedron With Td symmetry, order 24. With D3d symmetry, order 12, it is a triangular gyrobicupola. The area A and the volume V of the cuboctahedron of edge length a are, the cuboctahedron has four special orthogonal projections, centered on a vertex, an edge, and the two types of faces, triangular and square. The last two correspond to the B2 and A2 Coxeter planes, the skew projections show a square and hexagon passing through the center of the cuboctahedron. The cuboctahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. The cuboctahedrons 12 vertices can represent the vectors of the simple Lie group A3. With the addition of 6 vertices of the octahedron, these represent the 18 root vectors of the simple Lie group B3. The cuboctahedron can be dissected into two triangular cupolas by a common hexagon passing through the center of the cuboctahedron, if these two triangular cupolas are twisted so triangles and squares line up, Johnson solid J27, the triangular orthobicupola, is created. The cuboctahedron can also be dissected into 6 square pyramids and 8 tetrahedra meeting at a central point and this dissection is expressed in the alternated cubic honeycomb where pairs of square pyramids are combined into octahedra. A cuboctahedron can be obtained by taking a cross section of a four-dimensional 16-cell. Its first stellation is the compound of a cube and its dual octahedron, the cuboctahedron is a rectified cube and also a rectified octahedron. It is also a cantellated tetrahedron, with this construction it is given the Wythoff symbol,33 |2

5.
Deltoidal hexecontahedron
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In geometry, a deltoidal hexecontahedron is a Catalan solid which is the dual polyhedron of the rhombicosidodecahedron, an Archimedean solid. It is one of six Catalan solids to not have a Hamiltonian path among its vertices, the 60 faces are deltoids or kites. The short and long edges of each kite are in the ratio 1,7 + √5/6 ≈1,1.539344663, the angle between two short edges is 118. 22°. The opposite angle, between long edges, is 67. 76°, the other two angles, between a short and a long edge each, are both 87. 01°. The dihedral angle between all faces is 154. 12°, topologically, the deltoidal hexecontahedron is identical to the nonconvex rhombic hexecontahedron. The deltoidal hexecontahedron can be derived from a dodecahedron by pushing the face centers, edge centers, 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. These face-transitive figures have reflectional symmetry, the Geometrical Foundation of Natural Structure, A Source Book of Design. Deltoidal Hexecontahedron —Interactive Polyhedron Model Example in real life—A ball almost 4 meters in diameter, from ripstop nylon and it bounces around on the ground so that kids can play with it at kite festivals

6.
Deltoidal icositetrahedron
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In geometry, a deltoidal icositetrahedron is a Catalan solid which looks a bit like an overinflated cube. Its dual polyhedron is the rhombicuboctahedron, the short and long edges of each kite are in the ratio 1, ≈1,1.292893. The shape is called a trapezohedron in mineral contexts, although in solid geometry that name has another meaning. The deltoidal icositetrahedron has three positions, all centered on vertices, The great triakis octahedron is a stellation of the deltoidal icositetrahedron. The deltoidal icositetrahedron is topologically equivalent to a cube whose faces are divided in quadrants and it can also be projected onto a regular octahedron, with kite faces, or more general quadrilaterals with pyritohedral symmetry. In Conway polyhedron notation, they represent an ortho operation to a cube or octahedron, in crystallography a rotational variation is called a dyakis dodecahedron or diploid. The deltoidal icositetrahedron is one of a family of duals to the uniform polyhedra related to the cube and this polyhedron is topologically related as a part of sequence of deltoidal polyhedra with face figure, and continues as tilings of the hyperbolic plane. These face-transitive figures have reflectional symmetry, deltoidal hexecontahedron Tetrakis hexahedron, another 24-face Catalan solid which looks a bit like an overinflated cube. The Haunter of the Dark, a story by H. P, lovecraft, whose plot involves this figure Williams, Robert. The Geometrical Foundation of Natural Structure, A Source Book of Design, deltoidal Icositetrahedron – Interactive Polyhedron model

7.
Disdyakis dodecahedron
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In geometry, a disdyakis dodecahedron, or hexakis octahedron or kisrhombic dodecahedron), is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons, more formally, the disdyakis dodecahedron is the Kleetope of the rhombic dodecahedron. Its collective edges represent the reflection planes of the symmetry and it can also be seen in the corner and mid-edge triangulation of the regular cube and octahedron, and rhombic dodecahedron. Seen in stereographic projection the edges of the dodecahedron form 9 circles in the plane. Between a polyhedron and its dual, vertices and faces are swapped in positions, the disdyakis dodecahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron. It is a polyhedra in a sequence defined by the face configuration V4.6. 2n, with an even number of faces at every vertex, these polyhedra and tilings can be shown by alternating two colors so all adjacent faces have different colors. Each face on these domains also corresponds to the domain of a symmetry group with order 2,3, n mirrors at each triangle face vertex. First stellation of rhombic dodecahedron Disdyakis triacontahedron Kisrhombille tiling Great rhombihexacron—A uniform dual polyhedron with the surface topology Williams. The Geometrical Foundation of Natural Structure, A Source Book of Design, the Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 Eric W. Weisstein, Disdyakis dodecahedron at MathWorld

8.
Disdyakis triacontahedron
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In geometry, a disdyakis triacontahedron, hexakis icosahedron or kisrhombic triacontahedron is a Catalan solid with 120 faces and the dual to the Archimedean truncated icosidodecahedron. As such it is uniform but with irregular face polygons. That is, the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron and it also has the most faces among the Archimedean and Catalan solids, with the snub dodecahedron, with 92 faces, in second place. If the bipyramids and the trapezohedra are excluded, the disdyakis triacontahedron has the most faces of any strictly convex polyhedron where every face of the polyhedron has the same shape. The edges of the polyhedron projected onto a sphere form 15 great circles, combining pairs of light and dark triangles define the fundamental domains of the nonreflective icosahedral symmetry. The edges of a compound of five octahedra also represent the 10 mirror planes of icosahedral symmetry and this unsolved problem, often called the big chop problem, currently has no satisfactory mechanism. It is the most significant unsolved problem in mechanical puzzles and this shape was used to create d120 dice using 3D printing. More recently, the Dice Lab has used the Disdyakis triacontahedron to mass market an injection moulded 120 sided die. It is claimed that the d120 is the largest number of faces on a fair dice. It is topologically related to a sequence defined by the face configuration V4.6. 2n. With an even number of faces at every vertex, these polyhedra, each face on these domains also corresponds to the fundamental domain of a symmetry group with order 2,3, n mirrors at each triangle face vertex. This is *n32 in orbifold notation, and in Coxeter notation, the Geometrical Foundation of Natural Structure, A Source Book of Design. Disdyakis triacontahedron – Interactive Polyhedron Model

9.
Icosidodecahedron
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In geometry, an icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the Archimedean solids and more particularly and its dual polyhedron is the rhombic triacontahedron. An icosidodecahedron can be split along any of six planes to form a pair of pentagonal rotundae, the icosidodecahedron can be considered a pentagonal gyrobirotunda, as a combination of two rotundae. In this form its symmetry is D5d, order 20, the wire-frame figure of the icosidodecahedron consists of six flat regular decagons, meeting in pairs at each of the 30 vertices. Convenient Cartesian coordinates for the vertices of an icosidodecahedron with unit edges are given by the permutations of. The icosidodecahedron has four special orthogonal projections, centered on a vertex, an edge, a face. The last two correspond to the A2 and H2 Coxeter planes, the icosidodecahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. The icosidodecahedron is a dodecahedron and also a rectified icosahedron. With orbifold notation symmetry of all of these tilings are wythoff construction within a fundamental domain of symmetry. The icosidodecahedron is related to the Johnson solid called a pentagonal orthobirotunda created by two pentagonal rotunda connected as mirror images, the icosidodecahedron can therefore be called a pentagonal gyrobirotunda with the gyration between top and bottom halves. Eight uniform star polyhedra share the same vertex arrangement, of these, two also share the same edge arrangement, the small icosihemidodecahedron, and the small dodecahemidodecahedron. The vertex arrangement is shared with the compounds of five octahedra. In four-dimensional geometry the icosidodecahedron appears in the regular 600-cell as the slice that belongs to the vertex-first passage of the 600-cell through 3D space. In other words, the 30 vertices of the 600-cell which lie at arc distances of 90 degrees on its circumscribed hypersphere from a pair of vertices, are the vertices of an icosidodecahedron. The wire frame figure of the 600-cell consists of 72 flat regular decagons, six of these are the equatorial decagons to a pair of opposite vertices. They are precisely the six decagons which form the wire frame figure of the icosidodecahedron, in the mathematical field of graph theory, a icosidodecahedral graph is the graph of vertices and edges of the icosidodecahedron, one of the Archimedean solids

10.
Pentagonal hexecontahedron
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In geometry, a pentagonal hexecontahedron is a Catalan solid, dual of the snub dodecahedron. It has two forms, which are mirror images of each other. It is also well-known to be the Catalan solid with the most vertices, among the Catalan and Archimedean solids, it has the second largest number of vertices, after the truncated icosidodecahedron, which has 120 vertices. The faces are irregular pentagons with two edges and three short edges. The ratio of lengths is about 1,1.7489525667362. The faces have four angles of about 118.13662 degrees each. The dihedral angle is about 153.178732558 degrees between all faces, consequently, the vertices of the pentagonal hexecontahedron do not all lie on the same sphere and by definition it is not a zonohedron. The pentagonal hexecontahedron can be constructed from a snub dodecahedron without taking the dual, pentagonal pyramids are added to the 12 pentagonal faces of the snub dodecahedron, and triangular pyramids are added to the 20 triangular faces that do not share an edge with a pentagon. The pyramid heights are adjusted to make them coplanar with the other 60 triangular faces of the snub dodecahedron, the result is the pentagonal hexecontahedron. The pentagonal hexecontahedron has three positions, two on vertices, and one mid-edge. This polyhedron is topologically related as a part of sequence of polyhedra and these face-transitive figures have rotational symmetry. The Geometrical Foundation of Natural Structure, A Source Book of Design, pentagonal Hexecontrahedron – Interactive Polyhedron Model

11.
Pentagonal icositetrahedron
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In geometry, a pentagonal icositetrahedron or pentagonal icosikaitetrahedron is a Catalan solid which is the dual of the snub cube. In crystallography it is called a gyroid. It has two forms, which are mirror images of each other. Denote the tribonacci constant by t, approximately 1.8393, then the pentagonal faces have four angles of cos−1 ≈114. 8° and one angle of cos−1 ≈80. 75°. The pentagon has three edges of unit length each, and two long edges of length t + 1/2 ≈1.42. The acute angle is between the two long edges and this polyhedron is topologically related as a part of sequence of polyhedra and tilings of pentagons with face configurations. These face-transitive figures have rotational symmetry, the pentagonal icositetrahedron is second in a series of dual snub polyhedra and tilings with face configuration V3.3.4.3. n. The pentagonal icositetrahedron is one of a family of duals to the uniform polyhedra related to the cube, the Geometrical Foundation of Natural Structure, A Source Book of Design

12.
Pentakis dodecahedron
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In geometry, a pentakis dodecahedron or kisdodecahedron is a dodecahedron with a pentagonal pyramid covering each face, that is, it is the Kleetope of the dodecahedron. This interpretation is expressed in its name, there are in fact several topologically equivalent but geometrically distinct kinds of pentakis dodecahedron, depending on the height of the pentagonal pyramids. These include, The usual Catalan pentakis dodecahedron, a convex hexecontahedron with sixty isosceles triangular faces illustrated in the sidebar figure and it is a Catalan solid, dual to the truncated icosahedron, an Archimedean solid. As the heights of the pyramids are raised, at a certain point adjoining pairs of triangular faces merge to become rhombi. As the height is raised further, the shape becomes non-convex, other more non-convex geometric variants include, The small stellated dodecahedron. Great pentakis dodecahedron Wenningers third stellation of icosahedron, if one affixes pentagrammic pyramids into Wenningers third stellation of icosahedron one obtains the great icosahedron. 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. The pentakis dodecahedron is also a model of some icosahedrally symmetric viruses and these have 60 symmetry related capsid proteins, which combine to make the 60 symmetrical faces of a pentakis dodecahedron. The pentakis dodecahedron has three positions, two on vertices, and one on a midedge, The Spaceship Earth structure at Walt Disney Worlds Epcot is a derivative of a pentakis dodecahedron. The model for a campus arts workshop designed by Jeffrey Lindsay was actually a hemispherical pentakis dodecahedron https, in Doctor Atomic, the shape of the first atomic bomb detonated in New Mexico was a pentakis dodecahedron. In De Blob 2 in the Prison Zoo, domes are made up of parts of a Pentakis Dodecahedron and these Domes also appear whenever the player transforms on a dome in the Hypno Ray level. Some Geodomes in which play on are Pentakis Dodecahedra. The Geometrical Foundation of Natural Structure, A Source Book of Design, the Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 Eric W. Weisstein, Pentakis dodecahedron at MathWorld. Pentakis Dodecahedron – Interactive Polyhedron Model

13.
Rhombic dodecahedron
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In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of two types and it is a Catalan solid, and the dual polyhedron of the cuboctahedron. The rhombic dodecahedron is a zonohedron and its polyhedral dual is the cuboctahedron. The long diagonal of each face is exactly √2 times the length of the diagonal, so that the acute angles on each face measure arccos. Being the dual of an Archimedean polyhedron, the rhombic dodecahedron is face-transitive, meaning the symmetry group of the solid acts transitively on the set of faces. In elementary terms, this means that for any two faces A and B there is a rotation or reflection of the solid that leaves it occupying the region of space while moving face A to face B. The rhombic dodecahedron is one of the nine edge-transitive convex polyhedra, the others being the five Platonic solids, the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron can be used to tessellate three-dimensional space. It can be stacked to fill a space much like hexagons fill a plane and this polyhedron in a space-filling tessellation can be seen as the Voronoi tessellation of the face-centered cubic lattice. It is the Brillouin zone of body centered cubic crystals, some minerals such as garnet form a rhombic dodecahedral crystal habit. Honey bees use the geometry of rhombic dodecahedra to form honeycombs from a tessellation of cells each of which is a hexagonal prism capped with half a rhombic dodecahedron, the rhombic dodecahedron also appears in the unit cells of diamond and diamondoids. In these cases, four vertices are absent, but the chemical bonds lie on the remaining edges, the graph of the rhombic dodecahedron is nonhamiltonian. The last two correspond to the B2 and A2 Coxeter planes, the rhombic dodecahedron is a parallelohedron, a space-filling polyhedron. Other symmetry constructions of the dodecahedron are also space-filling. For example, with 4 square faces, and 60-degree rhombic faces and it be seen as a cuboctahedron with square pyramids augmented on the top and bottom. In 1960 Stanko Bilinski discovered a second rhombic dodecahedron with 12 congruent rhombus faces and it has the same topology but different geometry. The rhombic faces in this form have the golden ratio, another topologically equivalent variation, sometimes called a trapezoidal dodecahedron, is isohedral with tetrahedral symmetry order 24, distorting rhombic faces into kites. It has 8 vertices adjusted in or out in sets of 4. Variations can be parametrized by, where b is determined from a for planar faces and this polyhedron is a part of a sequence of rhombic polyhedra and tilings with Coxeter group symmetry

14.
Rhombic triacontahedron
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In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types and it is a Catalan solid, and the dual polyhedron of the icosidodecahedron. The ratio of the diagonal to the short diagonal of each face is exactly equal to the golden ratio, φ, so that the acute angles on each face measure 2 tan−1 = tan−1. A rhombus so obtained is called a golden rhombus, being the dual of an Archimedean solid, the rhombic triacontahedron is face-transitive, meaning the symmetry group of the solid acts transitively on the set of faces. This means that for any two faces, A and B, there is a rotation or reflection of the solid that leaves it occupying the region of space while moving face A to face B. The rhombic triacontahedron is also interesting in that its vertices include the arrangement of four Platonic solids and it contains ten tetrahedra, five cubes, an icosahedron and a dodecahedron. The centers of the faces contain five octahedra, the plane of each face is perpendicular to the center of the rhombic triacontahedron, and is located at the same distance. Using one of the three golden rectangles drawn into the inscribed icosahedron we can easily deduce the distance between the center of the solid and the center of its rhombic face. The rhombic triacontahedron can be dissected into 20 golden rhombohedra,10 acute ones and 10 flat ones, danish designer Holger Strøm used the rhombic triacontahedron as a basis for the design of his buildable lamp IQ-light. Woodworker Jane Kostick builds boxes in the shape of a rhombic triacontahedron, the simple construction is based on the less than obvious relationship between the rhombic triacontahedron and the cube. Roger von Oechs Ball of Whacks comes in the shape of a rhombic triacontahedron, the rhombic triacontahedron is used as the d30 thirty-sided die, sometimes useful in some roleplaying games or other places. The rhombic triacontahedron has three positions, two centered on vertices, and one mid-edge. Embedded in projection 10 are the fat rhombus and skinny rhombus which tile together to produce the non-periodic tessellation often referred to as Penrose tiling, the rhombic triacontahedron has over 227 stellations. This polyhedron is a part of a sequence of rhombic polyhedra, the cube can be seen as a rhombic hexahedron where the rhombi are also rectangles. The rhombic triacontahedron forms the hull of one projection of a 6-cube to 3 dimensions. Truncated rhombic triacontahedron Rhombille tiling Golden rhombus Williams, Robert, the Geometrical Foundation of Natural Structure, A Source Book of Design

15.
Rhombicosidodecahedron
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It has 20 regular triangular faces,30 square faces,12 regular pentagonal faces,60 vertices and 120 edges. The name rhombicosidodecahedron refers to the fact that the 30 square faces lie in the planes as the 30 faces of the rhombic triacontahedron which is dual to the icosidodecahedron. It can also be called an expanded or cantellated dodecahedron or icosahedron, therefore, it has the same number of triangles as an icosahedron and the same number of pentagons as a dodecahedron, with a square for each edge of either. The rhombicosidodecahedron shares the vertex arrangement with the stellated truncated dodecahedron. The Zometool kits for making geodesic domes and other polyhedra use slotted balls as connectors, the balls are expanded rhombicosidodecahedra, with the squares replaced by rectangles. The expansion is chosen so that the resulting rectangles are golden rectangles, eight more can be constructed by removing up to three cupolae, sometimes also rotating one or more of the other cupolae. Cartesian coordinates for the vertices of a rhombicosidodecahedron with edge length 2 centered at the origin are all permutations of. The rhombicosidodecahedron has five special orthogonal projections, centered, on a vertex, the last two correspond to the A2 and H2 Coxeter planes. The rhombicosidodecahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. This polyhedron is topologically related as a part of a sequence of cantellated polyhedra with vertex figure and these vertex-transitive figures have reflectional symmetry. It also shares its vertex arrangement with the compounds of six or twelve pentagrammic prisms. In the mathematical field of theory, a rhombicosidodecahedral graph is the graph of vertices and edges of the rhombicosidodecahedron. It has 60 vertices and 120 edges, and is a quartic graph Archimedean graph, the Geometrical Foundation of Natural Structure, A Source Book of Design. Eric W. Weisstein, Small Rhombicosidodecahedron at MathWorld, 3D convex uniform polyhedra x3o5x - srid

16.
Rhombicuboctahedron
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In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. There are 24 identical vertices, with one triangle and three meeting at each. The polyhedron has octahedral symmetry, like the cube and octahedron and its dual is called the deltoidal icositetrahedron or trapezoidal icositetrahedron, although its faces are not really true trapezoids. Johannes Kepler in Harmonices Mundi named this polyhedron a rhombicuboctahedron, being short for truncated cuboctahedral rhombus and this truncation creates new vertices mid-edge to the rhombic dodecahedron, creating rectangular faces inside the original rhombic faces, and new square and triangle faces at the original vertices. The semiregular form here requires the geometry be adjusted so the rectangles become squares and it can also be called an expanded cube or cantellated cube or a cantellated octahedron from truncation operations of the uniform polyhedron. There are distortions of the rhombicuboctahedron that, while some of the faces are not regular polygons, are still vertex-uniform. Some of these can be made by taking a cube or octahedron and cutting off the edges, then trimming the corners, so the resulting polyhedron has six square and twelve rectangular faces. The lines along which a Rubiks Cube can be turned are, projected onto a sphere, similar, topologically identical, in fact, variants using the Rubiks Cube mechanism have been produced which closely resemble the rhombicuboctahedron. The rhombicuboctahedron is used in three uniform space-filling tessellations, the cubic honeycomb, the runcitruncated cubic honeycomb, and the runcinated alternated cubic honeycomb. The rhombicuboctahedron can be dissected into two square cupolae and an octagonal prism. A rotation of one cupola by 45 degrees creates the pseudorhombicuboctahedron, both of these polyhedra have the same vertex figure,3.4.4.4. There are three pairs of parallel planes that each intersect the rhombicuboctahedron in a regular octagon and these pieces can be reassembled to give a new solid called the elongated square gyrobicupola or pseudorhombicuboctahedron, with the symmetry of a square antiprism. The rhombicuboctahedron has six special orthogonal projections, centered, on a vertex, the last two correspond to the B2 and A2 Coxeter planes. The rhombicuboctahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. A half symmetry form of the rhombicuboctahedron, exists with pyritohedral symmetry, as Coxeter diagram, Schläfli symbol s2 and this form can be visualized by alternatingly coloring the edges of the 6 squares. These squares can then be distorted into rectangles, while the 8 triangles remain equilateral, the 12 diagonal square faces will become isosceles trapezoids. Cartesian coordinates for the vertices of a rhombicuboctahedron centred at the origin, if the original rhombicuboctahedron has unit edge length, its dual strombic icositetrahedron has edge lengths 2710 −2 and 4 −22

17.
Snub cube
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In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces,6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices and it is a chiral polyhedron, that is, it has two distinct forms, which are mirror images of each other. The union of both forms is a compound of two cubes, and the convex hull of both sets of vertices is a truncated cuboctahedron. Kepler first named it in Latin as cubus simus in 1619 in his Harmonices Mundi. If the original cube has edge length 1, its dual pentagonal icositetrahedron has side lengths 1 t +1 ≈0.593465. Taking the even permutations with an odd number of signs, and the odd permutations with an even number of plus signs, gives a different snub cube. Taking all of them yields the compound of two snub cubes. The snub cube has two orthogonal projections, centered, on two types of faces, triangles, and squares, correspond to the A2 and B2 Coxeter planes. The snub cube can also be represented as a spherical tiling and this projection is conformal, preserving angles but not areas or lengths. Great circle arcs on the sphere are projected as circular arcs on the plane and it can also be constructed as an alternation of a nonuniform omnitruncated cube, deleting every other vertex and creating new triangles at the deleted vertices. A properly proportioned great rhombicuboctahedron will create equilateral triangles at the deleted vertices, depending on which set of vertices are alternated, the resulting snub cube can have a clockwise or counterclockwise twist. An improved snub cube, with a smaller square face. The snub cube is one of a family of uniform polyhedra related to the cube and this semiregular polyhedron is a member of a sequence of snubbed polyhedra and tilings with vertex figure and Coxeter–Dynkin diagram. These figures and their duals have rotational symmetry, being in the Euclidean plane for n =6, the series can be considered to begin with n=2, with one set of faces degenerated into digons. The snub cube is second in a series of snub polyhedra, in the mathematical field of graph theory, a snub cubical graph is the graph of vertices and edges of the snub cube, one of the Archimedean solids. It has 24 vertices and 60 edges, and is an Archimedean graph, truncated cube Compound of two snub cubes Snub square tiling Jayatilake, Udaya. Calculations on face and vertex regular polyhedra, the Geometrical Foundation of Natural Structure, A Source Book of Design. Eric W. Weisstein, Snub cube at MathWorld, Weisstein, Eric W. Snub cubic graph

18.
Snub dodecahedron
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In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces. The snub dodecahedron has 92 faces,12 are pentagons and the other 80 are equilateral triangles and it also has 150 edges, and 60 vertices. It has two forms, which are mirror images of each other. The union of both forms is a compound of two snub dodecahedra, and the hull of both forms is a truncated icosidodecahedron. Kepler first named it in Latin as dodecahedron simum in 1619 in his Harmonices Mundi and this snub dodecahedron has an edge length of approximately 6.0437380841. Taking the odd permutations of the coordinates with an even number of plus signs gives another form. Similarly, the image has either an odd permutation with an even number of plus signs or an even permutation with an odd number of plus signs. The snub dodecahedron has the highest sphericity of all Archimedean solids, the snub dodecahedron has two special orthogonal projections, centered on two types of faces, triangles and pentagons, corresponding to the A2 and H2 Coxeter planes. The snub dodecahedron can be generated by taking the twelve pentagonal faces of the dodecahedron, at a proper distance this can create the rhombicosidodecahedron by filling in square faces between the divided edges and triangle faces between the divided vertices. But for the form, only add the triangle faces. Then apply a rotation to the centers of the pentagons and triangles. The snub dodecahedron can also be derived from the truncated icosidodecahedron by the process of alternation, sixty of the vertices of the truncated icosidodecahedron form a polyhedron topologically equivalent to one snub dodecahedron, the remaining sixty form its mirror-image. The resulting polyhedron is vertex-transitive but not uniform, because its edges are of unequal lengths and this semiregular polyhedron is a member of a sequence of snubbed polyhedra and tilings with vertex figure and Coxeter–Dynkin diagram. These figures and their duals have rotational symmetry, being in the Euclidean plane for n =6, the series can be considered to begin with n =2, with one set of faces degenerated into digons. In the mathematical field of theory, a snub dodecahedral graph is the graph of vertices and edges of the snub dodecahedron. It has 60 vertices and 150 edges, and is an Archimedean graph, planar polygon to polyhedron transformation animation ccw and cw spinning snub dodecahedron Jayatilake, Udaya. Calculations on face and vertex regular polyhedra, the Geometrical Foundation of Natural Structure, A Source Book of Design. Eric W. Weisstein, Snub dodecahedron at MathWorld, Weisstein, Eric W. Snub dodecahedral graph

19.
Tetrakis hexahedron
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In geometry, a tetrakis hexahedron is a Catalan solid. Its dual is the octahedron, an Archimedean solid. It also can be called a disdyakis hexahedron as the dual of an omnitruncated tetrahedron, the tetrakis hexahedron, dual of the truncated octahedron has 3 symmetry positions, two located on vertices and one mid-edge. Naturally occurring formations of tetrahexahedra are observed in copper and fluorite systems, polyhedral dice shaped like the tetrakis hexahedron are occasionally used by gamers. The tetrakis hexahedron appears as one of the simplest examples in building theory, consider the Riemannian symmetric space associated to the group SL4. Its Tits boundary has the structure of a building whose apartments are 2-dimensional spheres. The partition of this sphere into spherical simplices can be obtained by taking the radial projection of a tetrakis hexahedron, with Td, tetrahedral symmetry, the triangular faces represent the 24 fundamental domains of tetrahedral symmetry. This polyhedron can be constructed from 6 great circles on a sphere and it can also be seen by a cube with its square faces triangulated by their vertices and face centers and a tetrahedron with its faces divided by vertices, mid-edges, and a central point. The edges of the tetrakis hexahedron form 6 circles in the plane, each of these 6 circles represent a mirror line in tetrahedral symmetry. The 6 circles can be grouped into 3 sets of 2 pairs of orthogonal circles and these edges can also be seen as a compound of 3 orthogonal square hosohedrons. If we denote the length of the base cube by a. The inclination of each face of the pyramid versus the cube face is arctan. One edge of the triangles has length a, the other two have length 3a/4, which follows by applying the Pythagorean theorem to height and base length. This yields an altitude of √5a/4 in the triangle and its area is √5a/8, and the internal angles are arccos and the complementary 180° −2 arccos. The volume of the pyramid is a3/12, so the volume of the six pyramids. It can be seen as a cube with square pyramids covering each square face and it is a polyhedra in a sequence defined by the face configuration V4.6. 2n. With an even number of faces at every vertex, these polyhedra, each face on these domains also corresponds to the fundamental domain of a symmetry group with order 2,3, n mirrors at each triangle face vertex. Disdyakis triacontahedron Disdyakis dodecahedron Kisrhombille tiling Compound of three octahedra Deltoidal icositetrahedron, another 24-face Catalan solid, the Geometrical Foundation of Natural Structure, A Source Book of Design

20.
Triakis icosahedron
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In geometry, the triakis icosahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated dodecahedron, the last two correspond to the A2 and H2 Coxeter planes. It can be seen as an icosahedron with triangular pyramids augmented to each face and this interpretation is expressed in the name, triakis. The triakis icosahedron is a part of a sequence of polyhedra and tilings and these face-transitive figures have reflectional symmetry. Triakis triangular tiling for other triakis polyhedral forms, the Geometrical Foundation of Natural Structure, A Source Book of Design. The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 Eric W. Weisstein, Triakis Icosahedron – Interactive Polyhedron Model

21.
Triakis octahedron
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In geometry, a triakis octahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube and it can be seen as an octahedron with triangular pyramids added to each face, that is, it is the Kleetope of the octahedron. It is also called a trisoctahedron, or, more fully. Both names reflect the fact that it has three triangular faces for every face of an octahedron, the tetragonal trisoctahedron is another name for the deltoidal icositetrahedron, a different polyhedron with three quadrilateral faces for every face of an octahedron. This convex polyhedron is topologically similar to the concave stellated octahedron and they have the same face connectivity, but the vertices are in different relative distances from the center. The triakis octahedron is one of a family of duals to the uniform polyhedra related to the cube, the triakis octahedron is a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have reflectional symmetry, the triakis octahedron is also a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have reflectional symmetry, the Geometrical Foundation of Natural Structure, A Source Book of Design. Triakis Octahedron – Interactive Polyhedron Model Virtual Reality Polyhedra www. georgehart. com, The Encyclopedia of Polyhedra VRML model Conway Notation for Polyhedra Try, dtC

22.
Triakis tetrahedron
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In geometry, a triakis tetrahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated tetrahedron and it can be seen as a tetrahedron with triangular pyramids added to each face, that is, it is the Kleetope of the tetrahedron. This interpretation is expressed in the name, the length of the shorter edges is 3/5 that of the longer edges. If the triakis tetrahedron has shorter edge length 1, it has area 5/3√11, a triakis tetrahedron with equilateral triangle faces represents a net of the four-dimensional regular polytope known as the 5-cell. If the triangles are right-angled isosceles, the faces will be coplanar and this can be seen by adding the 6 edges of tetrahedron inside of a cube. This chiral figure is one of thirteen stellations allowed by Millers rules, the triakis tetrahedron is a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have reflectional symmetry, the Geometrical Foundation of Natural Structure, A Source Book of Design

23.
Truncated cube
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In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces,36 edges, and 24 vertices, if the truncated cube has unit edge length, its dual triakis octahedron has edges of lengths 2 and 2 + √2. The area A and the volume V of a cube of edge length a are. The truncated cube has five special orthogonal projections, centered, on a vertex, the last two correspond to the B2 and A2 Coxeter planes. The truncated cube can also be represented as a spherical tiling and this projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane, the following Cartesian coordinates define the vertices of a truncated hexahedron centered at the origin with edge length 2ξ, where ξ = √2 −1. The parameter ξ can be varied between ±1, a value of 1 produces a cube,0 produces a cuboctahedron, and negative values produces self-intersecting octagrammic faces. The truncated cube can be dissected into a cube, with six square cupola around each of the cubes faces. This dissection can also be seen within the cubic honeycomb, with cube, tetrahedron. This dissection can be used to create a Stewart toroid with all regular faces by removing two square cupola and the central cube and this excavated cube has 16 triangles,12 squares, and 4 octagons. It shares the vertex arrangement with three nonconvex uniform polyhedra, The truncated cube is related to polyhedra and tlings in symmetry. The truncated cube is one of a family of uniform polyhedra related to the cube and this polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations, and Coxeter group symmetry, and a series of polyhedra and tilings n.8.8. A cube can be alternately truncated producing tetrahedral symmetry, with six hexagonal faces and it is one of a sequence of alternate truncations of polyhedra and tiling. It has 24 vertices and 36 edges, and is a cubic Archimedean graph, spinning truncated cube Cube-connected cycles, a family of graphs that includes the skeleton of the truncated cube Williams, Robert. The Geometrical Foundation of Natural Structure, A Source Book of Design, cromwell, P. Polyhedra, CUP hbk, pbk. Ch.2 p. 79-86 Archimedean solids Eric W. Weisstein, Weisstein, Eric W. Truncated cubical graph. 3D convex uniform polyhedra o3x4x - tic

24.
Truncated cuboctahedron
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In geometry, the truncated cuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces,8 regular hexagonal faces,6 regular octagonal faces,48 vertices and 72 edges, since each of its faces has point symmetry, the truncated cuboctahedron is a zonohedron. If you truncate a cuboctahedron by cutting the corners off, you do not get this uniform figure, however, the resulting figure is topologically equivalent to a truncated cuboctahedron and can always be deformed until the faces are regular. The alternative name great rhombicuboctahedron refers to the fact that the 12 square faces lie in the planes as the 12 faces of the rhombic dodecahedron which is dual to the cuboctahedron. One unfortunate point of confusion, There is a uniform polyhedron by the same name. See nonconvex great rhombicuboctahedron.7551724 a 2 V = a 3 ≈41.7989899 a 3, many other lower symmetry toroids can also be constructed by removing a subset of these dissected components. For example, removing half of the triangular cupolas creates a genus 3 torus, There is only one uniform coloring of the faces of this polyhedron, one color for each face type. A 2-uniform coloring, with symmetry, exists with alternately colored hexagons. The truncated cuboctahedron can also be represented as a spherical tiling and this projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane, the truncated cuboctahedron is one of a family of uniform polyhedra related to the cube and regular octahedron. This polyhedron can be considered a member of a sequence of patterns with vertex configuration. For p <6, the members of the sequence are omnitruncated polyhedra, for p <6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling. In the mathematical field of theory, a truncated cuboctahedral graph is the graph of vertices and edges of the truncated cuboctahedron. It has 48 vertices and 72 edges, and is a zero-symmetric and cubic Archimedean graph, cube Cuboctahedron Octahedron Truncated icosidodecahedron Truncated octahedron – truncated tetratetrahedron Cromwell, P. Polyhedra. Eric W. Weisstein, Great rhombicuboctahedron at MathWorld, 3D convex uniform polyhedra x3x4x - girco

25.
Truncated dodecahedron
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In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces,20 regular triangular faces,60 vertices and 90 edges and this polyhedron can be formed from a dodecahedron by truncating the corners so the pentagon faces become decagons and the corners become triangles. It is used in the cell-transitive hyperbolic space-filling tessellation, the bitruncated icosahedral honeycomb, the truncated dodecahedron has five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces, hexagonal and pentagonal. The last two correspond to the A2 and H2 Coxeter planes, the truncated dodecahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. Schlegel diagrams are similar, with a projection and straight edges. In the mathematical field of theory, a truncated dodecahedral graph is the graph of vertices and edges of the truncated dodecahedron. It has 60 vertices and 90 edges, and is a cubic Archimedean graph, spinning truncated cube Cube-connected cycles, a family of graphs that includes the skeleton of the truncated cube Williams, Robert. The Geometrical Foundation of Natural Structure, A Source Book of Design, Eric W. Weisstein, Truncated dodecahedron at MathWorld. Weisstein, Eric W. Truncated dodecahedral graph, 3D convex uniform polyhedra o3x5x - tid. Editable printable net of a dodecahedron with interactive 3D view The Uniform Polyhedra Virtual Reality Polyhedra The Encyclopedia of Polyhedra