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1. 3-7 kisrhombille – In geometry, the 3-7 kisrhombille tiling is a semiregular dual tiling of the hyperbolic plane. It is constructed by congruent right triangles with 4,6, the image shows a Poincaré disk model projection of the hyperbolic plane. It is labeled V4.6.14 because each right triangle face has three types of vertices, one with 4 triangles, one with 6 triangles, and one with 14 triangles. It is the tessellation of the truncated triheptagonal tiling which has one square and one heptagon. The name 3-7 kisrhombille is given by Conway, seeing it as a 3-7 rhombic tiling, divided by a kis operator, adding a point to each rhombus. There are no mirror removal subgroups of, the only small index subgroup is the alternation, +. Three isohedral tilings can be constructed from this tiling by combining triangles, It is topologically related to a polyhedra sequence, see also the uniform tilings of the hyperbolic plane with symmetry. The kisrhombille tilings can be seen as from the sequence of rhombille tilings, starting with the cube, just as the triangle group is a quotient of the modular group, the associated tiling is the quotient of the modular tiling, as depicted in the video at right. Hexakis triangular tiling Tilings of regular polygons List of uniform tilings Uniform tilings in hyperbolic plane
2. Apeirogon – In geometry, an apeirogon is a generalized polygon with a countably infinite number of sides. It can be considered as the limit of a polygon as n approaches infinity. The interior of an apeirogon can be defined by a direction order of vertices. This article describes an apeirogon in its form as a tessellation or partition of a line. A regular apeirogon has equal edge lengths, just like any regular polygon and its Schläfli symbol is, and its Coxeter-Dynkin diagram is. It is the first in the family of regular hypercubic honeycombs. This line may be considered as a circle of radius, by analogy with regular polygons with great number of edges. In two dimensions, a regular apeirogon divides the plane into two half-planes as a regular apeirogonal dihedron, the interior of an apeirogon can be defined by its orientation, filling one half plane. Dually the apeirogonal hosohedron has digon faces and a vertex figure. A truncated apeirogonal hosohedron becomes a apeirogonal prism, with each vertex bounded by two squares and an apeirogon, an alternated apeirogonal prism is a apeirogonal antiprism, with each vertex bounded by three triangles and an apeirogon. The regular apeirogon can also be seen as linear sets within 4 of the regular, uniform tilings, an isogonal apeirogon has a single type of vertex and alternates two types of edges. A quasiregular apeirogon is an isogonal apeirogon with equal edge lengths, an isotoxal apeirogon, being the dual of an isogonal one, has one type of edge, and two types of vertices, and is therefore geometrically identical to the regular apeirogon. It can be seen by drawing vertices in alternate colors. All of these will have half the symmetry of the regular apeirogon, Regular apeirogons that are scaled to converge at infinity have the symbol and exist on horocycles, while more generally they can exist on hypercycles. The regular tiling has regular apeirogon faces, hypercyclic apeirogons can also be isogonal or quasiregular, with truncated apeirogon faces, t, like the tiling tr, with two types of edges, alternately connecting to triangles or other apeirogons. Apeirogonal tiling Apeirogonal prism Apeirogonal antiprism Apeirohedron Circle Coxeter, H. S. M. Regular Polytopes, Regular polyhedra - old and new, Aequationes Math. 16 p. 1-20 Coxeter, H. S. M. & Moser, W. O. J. Generators, archived from the original on 4 February 2007
3. Disdyakis dodecahedron – 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
4. Disdyakis triacontahedron – 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
5. Hexagonal bipyramid – A hexagonal bipyramid is a polyhedron formed from two hexagonal pyramids joined at their bases. The resulting solid has 12 triangular faces,8 vertices and 18 edges, the 12 faces are identical isosceles triangles. Although it is face-transitive, it is not a Platonic solid because some vertices have four faces meeting and others have six faces and it is one of an infinite set of bipyramids. Having twelve faces, it is a type of dodecahedron, although that name is associated with the regular polyhedral form with pentagonal faces. The term dodecadeltahedron is sometimes used to distinguish the bipyramid from the Platonic solid, the hexagonal bipyramid has a plane of symmetry where the bases of the two pyramids are joined. This plane is a regular hexagon, there are also six planes of symmetry crossing through the two apices. These planes are rhombic and lie at 30° angles to each other, 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. Hexagonal trapezohedron A similar 12-sided polyhedron with a twist and kite faces, snub disphenoid Another 12-sided polyhedron with 2-fold symmetry and only triangular faces. Archived from the original on 4 February 2007, virtual Reality Polyhedra The Encyclopedia of Polyhedra VRML model hexagonal dipyramid Conway Notation for Polyhedra Try, dP6
6. Hexagonal prism – In geometry, the hexagonal prism is a prism with hexagonal base. This polyhedron has 8 faces,18 edges, and 12 vertices, since it has eight faces, it is an octahedron. However, the octahedron is primarily used to refer to the regular octahedron. Because of the ambiguity of the octahedron and the dissimilarity of the various eight-sided figures. Before sharpening, many take the shape of a long hexagonal prism. It can be seen as a truncated hexagonal hosohedron, represented by Schläfli symbol t, alternately it can be seen as the Cartesian product of a regular hexagon and a line segment, and represented by the product ×. The dual of a prism is a hexagonal bipyramid. The symmetry group of a hexagonal prism is D6h of order 24. The rotation group is D6 of order 12, for p <6, the members of the sequence are omnitruncated polyhedra, shown below as spherical tilings. For p >6, they are tilings of the hyperbolic plane, Uniform Honeycombs in 3-Space VRML models The Uniform Polyhedra Virtual Reality Polyhedra The Encyclopedia of Polyhedra Prisms and antiprisms Weisstein, Eric W. Hexagonal prism. Hexagonal Prism Interactive Model -- works in your web browser
7. Tetrakis hexahedron – 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
8. Truncated cuboctahedron – 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
9. Truncated icosidodecahedron – In geometry, the truncated icosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces. It has 30 square faces,20 regular hexagonal faces,12 regular decagonal faces,120 vertices and 180 edges – more than any other convex nonprismatic uniform polyhedron, since each of its faces has point symmetry, the truncated icosidodecahedron is a zonohedron. If one truncates an icosidodecahedron by cutting the corners off, one does not get this uniform figure, however, the resulting figure is topologically equivalent to this and can always be deformed until the faces are regular. One unfortunate point of confusion is there is a nonconvex uniform polyhedron of the same name. The surface area A and the volume V of the truncated icosidodecahedron of edge length a are, V = a 3 ≈206.803399 a 3. If a set of all 13 Archimedean solids were constructed with all edge lengths equal, the truncated icosidodecahedron would be the largest. Cartesian coordinates for the vertices of a truncated icosidodecahedron with edge length 2φ −2, centered at the origin, are all the permutations of, and. The truncated icosidodecahedron has seven special orthogonal projections, centered on a vertex, the last two correspond to the A2 and H2 Coxeter planes. The truncated icosidodecahedron 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, schlegel diagrams are similar, with a perspective projection and straight edges. Within Icosahedral symmetry there are unlimited geometric variations of the truncated icosidodecahedron with isogonal faces, the truncated dodecahedron, rhombicosidodecahedron, and truncated icosahedron as degenerate limiting cases. In the mathematical field of theory, a truncated icosidodecahedral graph is the graph of vertices and edges of the truncated icosidodecahedron. It has 120 vertices and 180 edges, and is a zero-symmetric and cubic Archimedean graph and this polyhedron can be considered a member of a sequence of uniform patterns with vertex figure and Coxeter-Dynkin diagram. 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. Wenninger, Magnus, Polyhedron Models, Cambridge University Press, ISBN 978-0-521-09859-5, MR0467493 Cromwell, the Geometrical Foundation of Natural Structure, A Source Book of Design. Cromwell, P. Polyhedra, CUP hbk, pbk, eric W. Weisstein, GreatRhombicosidodecahedron at MathWorld. 3D convex uniform polyhedra x3x5x - grid, editable printable net of a truncated icosidodecahedron with interactive 3D view The Uniform Polyhedra Virtual Reality Polyhedra The Encyclopedia of Polyhedra
10. Truncated octahedron – 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