1.
Truncated icosahedron
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In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose faces are two or more types of regular polygons. It has 12 regular pentagonal faces,20 regular hexagonal faces,60 vertices and 90 edges and it is the Goldberg polyhedron GPV or 1,1, containing pentagonal and hexagonal faces. This geometry is associated with footballs typically patterned with white hexagons, geodesic domes such as those whose architecture Buckminster Fuller pioneered are often based on this structure. It also corresponds to the geometry of the fullerene C60 molecule and it is used in the cell-transitive hyperbolic space-filling tessellation, the bitruncated order-5 dodecahedral honeycomb. This polyhedron can be constructed from an icosahedron with the 12 vertices truncated such that one third of each edge is cut off at each of both ends and 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, cartesian coordinates for the vertices of a truncated icosahedron centered at the origin are all even permutations of, where φ =1 + √5/2 is the golden mean. Using φ2 = φ +1 one verifies that all vertices are on a sphere, centered at the origin, with the radius equal to √9φ +10. Permutations, X axis Y axis Z axis The truncated icosahedron has five special orthogonal projections, centered, on a vertex, the last two correspond to the A2 and H2 Coxeter planes. The truncated icosahedron 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 and this result is easy to get by using one of the three orthogonal golden rectangles drawn into the original icosahedron as the starting point for our considerations. The angle between the segments joining the center and the vertices connected by shared edge is approximately 23. 281446°. The area A and the volume V of the truncated icosahedron of edge length a are, with unit edges, the surface area is 21 for the pentagons and 52 for the hexagons, together 73. The truncated icosahedron easily demonstrates the Euler characteristic,32 +60 −90 =2, 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 due to the pressure of the air inside. This ball type was introduced to the World Cup in 1970, geodesic domes are typically based on triangular facetings of this geometry with example structures found across the world, popularized by Buckminster Fuller. A variation of the icosahedron was used as the basis of the wheels 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 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, or buckyball, molecule, an allotrope of elemental carbon, discovered in 1985

2.
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

3.
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

4.
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

5.
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

6.
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

7.
Truncated tetrahedron
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In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces,4 equilateral triangle faces,12 vertices and 18 edges and it can be constructed by truncating all 4 vertices of a regular tetrahedron at one third of the original edge length. A deeper truncation, removing a tetrahedron of half the edge length from each vertex, is called rectification. The rectification of a tetrahedron produces an octahedron, a truncated tetrahedron is the Goldberg polyhedron GIII, containing triangular and hexagonal faces. A truncated tetrahedron can be called a cube, with Coxeter diagram. There are two positions of this construction, and combining them creates the uniform compound of two truncated tetrahedra. The area A and the volume V of a tetrahedron of edge length a are. The densest packing of the Archimedean truncated tetrahedron is believed to be Φ = 207/208, in fact, if the truncation of the corners is slightly smaller than that of an Archimedean truncated tetrahedron, this new shape can be used to completely fill space. 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 lower symmetry version of the tetrahedron is called a Friauf polyhedron in crystals such as complex metallic alloys. This form fits 5 Friauf polyhedra around an axis, giving a 72-degree dihedral angle on a subset of 6-6 edges, Friauf and his 1927 paper The crystal structure of the intermetallic compound MgCu2. Giant truncated tetrahedra were used for the Man the Explorer and Man the Producer theme pavilions in Expo 67 and they were made of massive girders of steel bolted together in a geometric lattice. The truncated tetrahedra were interconnected with lattice steel platforms, all of these buildings were demolished after the end of Expo 67, as they had not been built to withstand the severity of the Montreal weather over the years. Their only remnants are in the Montreal city archives, the Public Archives Of Canada, the Tetraminx puzzle has a truncated tetrahedral shape. This puzzle shows a dissection of a tetrahedron into 4 octahedra and 6 tetrahedra. It contains 4 central planes of rotations, in the mathematical field of graph theory, a truncated tetrahedral graph is a Archimedean graph, the graph of vertices and edges of the truncated tetrahedron, one of the Archimedean solids. It has 12 vertices and 18 edges and it is a connected cubic graph, and connected cubic transitive graph. It is also a part of a sequence of cantic polyhedra, in this wythoff construction the edges between the hexagons represent degenerate digons

8.
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

9.
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

10.
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

11.
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

12.
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