1.
Octagonal prism
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In geometry, the octagonal prism is the sixth in an infinite set of prisms, formed by square sides and two regular octagon caps. If faces are all regular, it is a semiregular polyhedron, the octagonal prism can also be seen as a tiling on a sphere, In optics, octagonal prisms are used to generate flicker-free images in movie projectors. It is an element of three uniform honeycombs, It is also an element of two four-dimensional uniform 4-polytopes, Weisstein, Eric W. Octagonal prism, interactive model of an Octagonal Prism

2.
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

3.
Orbifold notation
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Groups representable in this notation include the point groups on the sphere, the frieze groups and wallpaper groups of the Euclidean plane, and their analogues on the hyperbolic plane. e. All translations which occur are assumed to form a subgroup of the group symmetries being described. The symbol ×, which is called a miracle and represents a topological crosscap where a pattern repeats as an image without crossing a mirror line. A string written in boldface represents a group of symmetries of Euclidean 3-space, a string not written in boldface represents a group of symmetries of the Euclidean plane, which is assumed to contain two independent translations. By abuse of language, we say that such a group is a subgroup of symmetries of the Euclidean plane with only one independent translation. The frieze groups occur in this way, the exceptional symbol o indicates that there are precisely two linearly independent translations. An orbifold symbol is called if it is not one of the following, p, pq, *p, *pq, for p, q>=2. An object is chiral if its symmetry group contains no reflections, the corresponding orbifold is orientable in the chiral case and non-orientable otherwise. The Euler characteristic of an orbifold can be read from its Conway symbol, as follows. Each feature has a value, n without or before an asterisk counts as n −1 n n after an asterisk counts as n −12 n asterisk, subtracting the sum of these values from 2 gives the Euler characteristic. If the sum of the values is 2, the order is infinite. Indeed, Conways Magic Theorem indicates that the 17 wallpaper groups are exactly those with the sum of the feature values equal to 2, otherwise, the order is 2 divided by the Euler characteristic. The following groups are isomorphic, 1* and *1122 and 221 *22 and *221 2* and this is because 1-fold rotation is the empty rotation. The symmetry of a 2D object without translational symmetry can be described by the 3D symmetry type by adding a dimension to the object which does not add or spoil symmetry. The bullet is added on one- and two-dimensional groups to imply the existence of a fixed point, thus the discrete symmetry groups in one dimension are *•, *1•, ∞• and *∞•. Another way of constructing a 3D object from a 1D or 2D object for describing the symmetry is taking the Cartesian product of the object, on Three-dimensional Orbifolds and Space Groups. Contributions to Algebra and Geometry,42, 475-507,2001, J. H. Conway, D. H. Huson. The Orbifold Notation for Two-Dimensional Groups, structural Chemistry,13, 247-257, August 2002

4.
Order-4 apeirogonal tiling
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In geometry, the order-4 apeirogonal tiling is a regular tiling of the hyperbolic plane. This tiling represents the mirror lines of *2∞ symmetry and it dual to this tiling represents the fundamental domains of orbifold notation *∞∞∞∞ symmetry, a square domain with four ideal vertices. Like the Euclidean square tiling there are 9 uniform colorings for this tiling, a fourth can be constructed from an infinite square symmetry with 4 colors around a vertex. The checker board, r, coloring defines the fundamental domains of, symmetry, usually shown as black, the Beauty of Geometry, Twelve Essays. Weisstein, Eric W. Poincaré hyperbolic disk, Hyperbolic and Spherical Tiling Gallery KaleidoTile 3, Educational software to create spherical, planar and hyperbolic tilings Hyperbolic Planar Tessellations, Don Hatch

5.
Rhombitetraapeirogonal tiling
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In geometry, the rhombitetraapeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr, there are two uniform constructions of this tiling, one from or symmetry, and secondly removing the mirror middle, gives a rectangular fundamental domain. The dual of this tiling, called a deltoidal tetraapeirogonal tiling represents the fundamental domains of orbifold symmetry and its fundamental domain is a Lambert quadrilateral, with 3 right angles. The Beauty of Geometry, Twelve Essays, weisstein, Eric W. Poincaré hyperbolic disk. Hyperbolic and Spherical Tiling Gallery KaleidoTile 3, Educational software to create spherical, planar and hyperbolic tilings Hyperbolic Planar Tessellations, Don Hatch

6.
Tetrakis square tiling
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In geometry, the tetrakis square tiling is a tiling of the Euclidean plane. It is a square tiling with each divided into four isosceles right triangles from the center point. Conway calls it a kisquadrille, represented by a kis operation that adds a center point and it is also called the Union Jack lattice because of the resemblance to the UK flag of the triangles surrounding its degree-8 vertices. It is labeled V4.8.8 because each isosceles triangle face has two types of vertices, one with 4 triangles, and two with 8 triangles and it is the dual tessellation of the truncated square tiling which has one square and two octagons at each vertex. A5 ×9 portion of the square tiling is used to form the board for the Malagasy board game Fanorona. In this game, pieces are placed on the vertices of the tiling, a similar board is also used for the Brazilian game Adugo, and for the game of Hare and Hounds. The tetrakis square tiling was used for a set of postage stamps issued by the United States Postal Service in 1997. This tiling also forms the basis for a commonly used pinwheel, windmill and these lines form the axes of symmetry of a reflection group, which has the triangles of the tiling as its fundamental domains. Rotational symmetry is shown by white and blue colored areas with a single fundamental domain for each subgroup is filled in yellow. Glide reflections are given with dashed lines, subgroups can be expressed as Coxeter diagrams, along with fundamental domain diagrams. It is topologically related to a series of polyhedra and tilings with face configuration Vn.6.6, tilings of regular polygons List of uniform tilings Percolation threshold Grünbaum, Branko, and Shephard, G. C. CS1 maint, Multiple names, authors list Williams, Robert, the Geometrical Foundation of Natural Structure, A Source Book of Design. Keith Critchlow, Order in Space, A design source book,1970, p. 77-76, pattern 8

7.
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

8.
Truncated infinite-order square tiling
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In geometry, the truncated infinite-order square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t, in symmetry this tiling has 3 colors. Bisecting the isosceles triangle domains can double the symmetry to *∞42 symmetry, the dual of the tiling represents the fundamental domains of orbifold symmetry. From symmetry, there are 15 small index subgroup by mirror removal, mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met, in these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The symmetry can be doubled to *∞42 by adding a mirror across the fundamental domains. The subgroup index-8 group, is the subgroup of. The Beauty of Geometry, Twelve Essays, weisstein, Eric W. Poincaré hyperbolic disk

9.
Truncated octahedron
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In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces,36 edges, and 24 vertices, since each of its faces has point symmetry the truncated octahedron is a zonohedron. It is also the Goldberg polyhedron GIV, containing square and hexagonal faces, like the cube, it can tessellate 3-dimensional space, as a permutohedron. Its dual polyhedron is the tetrakis hexahedron, if the original truncated octahedron has unit edge length, its dual tetrakis cube has edge lengths 9/8√2 and 3/2√2. A truncated octahedron is constructed from an octahedron with side length 3a by the removal of six right square pyramids. These pyramids have both base side length and lateral side length of a, to form equilateral triangles, the base area is then a2. Note that this shape is similar to half an octahedron or Johnson solid J1. The truncated octahedron has five special orthogonal projections, centered, on a vertex, the last two correspond to the B2 and A2 Coxeter planes. The truncated octahedron 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, all permutations of are Cartesian coordinates of the vertices of a truncated octahedron of edge length a = √2 centered at the origin. The vertices are also the corners of 12 rectangles whose long edges are parallel to the coordinate axes. The edge vectors have Cartesian coordinates and permutations of these, the face normals of the 6 square faces are, and. The face normals of the 8 hexagonal faces are, the dot product between pairs of two face normals is the cosine of the dihedral angle between adjacent faces, either −1/3 or −1/√3. The dihedral angle is approximately 1.910633 radians at edges shared by two hexagons or 2.186276 radians at edges shared by a hexagon and a square. The truncated octahedron can be dissected into an octahedron, surrounded by 8 triangular cupola on each face. Therefore, the octahedron is the permutohedron of order 4, each vertex corresponds to a permutation of. The area A and the volume V of an octahedron of edge length a are. There are two uniform colorings, with symmetry and octahedral symmetry, and two 2-uniform coloring with dihedral symmetry as a truncated triangular antiprism