1.
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

2.
Infinite-order square tiling
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In geometry, the infinite-order square tiling is a regular tiling of the hyperbolic plane. All vertices are ideal, located at infinity, seen on the boundary of the Poincaré hyperbolic disk projection, there is a half symmetry form, seen with alternating colors, This tiling represents the mirror lines of *∞∞∞∞ symmetry. The dual to this tiling defines the fundamental domains of orbifold symmetry and this tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure. Square tiling Uniform tilings in hyperbolic plane List of regular polytopes John H. Conway, Heidi Burgiel, chapter 19, The Hyperbolic Archimedean Tessellations. Chapter 10, Regular honeycombs in hyperbolic space, the Beauty of Geometry, Twelve Essays. Weisstein, Eric W. Poincaré hyperbolic disk

3.
Octagonal bipyramid
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The octagonal bipyramid is one of the infinite set of bipyramids, dual to the infinite prisms. If an octagonal bipyramid is to be face-transitive, all faces must be isosceles triangles and it can be drawn as a tiling on a sphere which also represents the fundamental domains of, *422 symmetry, Weisstein, Eric W. Dipyramid. Archived from the original on 4 February 2007, virtual Reality Polyhedra The Encyclopedia of Polyhedra VRML models <8> Conway Notation for Polyhedra Try, dP8

4.
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

5.
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

6.
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

7.
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

8.
Snub tetraapeirogonal tiling
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In geometry, the snub tetrapeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr, drawn in chiral pairs, with edges missing between black triangles, The snub tetrapeirogonal tiling is last in an infinite series of snub polyhedra and tilings with vertex figure 3.3.4.3. n. The Beauty of Geometry, Twelve Essays, weisstein, Eric W. Poincaré hyperbolic disk