1.
3-7 kisrhombille
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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
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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.
Arrangement of lines
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In geometry an arrangement of lines is the partition of the plane formed by a collection of lines. Bounds on the complexity of arrangements have been studied in discrete geometry and these three types of objects link together to form a cell complex covering the plane. Two arrangements are said to be isomorphic or combinatorially equivalent if there is a one-to-one adjacency-preserving correspondence between the objects in their associated cell complexes. The study of arrangements was begun by Jakob Steiner, who proved the first bounds on the number of features of different types that an arrangement may have. An arrangement with n lines has at most n/2 vertices, one per pair of crossing lines and this maximum is achieved for simple arrangements, those in which each two lines has a distinct pair of crossing points. In any arrangement there will be n infinite-downward rays, one per line, the zone of a line l in a line arrangement is the collection of cells having edges belonging to l. The zone theorem states that the number of edges in the cells of a single zone is linear. Summing the complexities of all zones, one finds that the sum of squares of cell complexities in an arrangement is O. The k-level of an arrangement is the chain formed by the edges that have exactly k other lines directly below them. Finding matching upper and lower bounds for the complexity of a k-level remains an open problem in discrete geometry. In contrast, the complexity of the ≤k-level is known to be Θ. A k-level is a case of a monotone path in an arrangement, that is. Although a single cell in an arrangement may be bounded by all n lines, rather, the total complexity of m cells is at most Θ, almost the same bound as occurs in the Szemerédi–Trotter theorem on point-line incidences in the plane. A simple proof of this follows from the crossing number inequality, if m cells have a total of x + n edges, one can form a graph with m nodes and x edges. However, by the crossing number inequality, there are Ω crossings and it is often convenient to study line arrangements not in the Euclidean plane but in the projective plane, due to the fact that in projective geometry every pair of lines has a crossing point. Due to projective duality, many statements about the properties of points in the plane may be more easily understood in an equivalent dual form about arrangements of lines. The earliest known proof of the Sylvester–Gallai theorem, by Melchior, an arrangement of lines in the projective plane is said to be simplicial if every cell of the arrangement is bounded by exactly three edges, simplicial arrangements were first studied by Melchior. Additionally there are other examples of sporadic simplicial arrangements that do not fit into any known infinite family

4.
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

5.
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

6.
Hexagonal bipyramid
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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

7.
Hexagonal prism
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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

8.
Hexagonal tiling
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In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which three hexagons meet at each vertex. It has Schläfli symbol of or t, english mathematician Conway calls it a hextille. The internal angle of the hexagon is 120 degrees so three hexagons at a point make a full 360 degrees and it is one of three regular tilings of the plane. The other two are the triangular tiling and the square tiling, the hexagonal tiling is the densest way to arrange circles in two dimensions. The Honeycomb conjecture states that the tiling is the best way to divide a surface into regions of equal area with the least total perimeter. The optimal three-dimensional structure for making honeycomb was investigated by Lord Kelvin, however, the less regular Weaire–Phelan structure is slightly better. This structure exists naturally in the form of graphite, where each sheet of graphene resembles chicken wire, tubular graphene sheets have been synthesised, these are known as carbon nanotubes. They have many applications, due to their high tensile strength. Chicken wire consists of a lattice of wires. The hexagonal tiling appears in many crystals, in three dimensions, the face-centered cubic and hexagonal close packing are common crystal structures. They are the densest known sphere packings in three dimensions, and are believed to be optimal, structurally, they comprise parallel layers of hexagonal tilings, similar to the structure of graphite. They differ in the way that the layers are staggered from each other, pure copper, amongst other materials, forms a face-centered cubic lattice. There are three distinct uniform colorings of a tiling, all generated from reflective symmetry of Wythoff constructions. The represent the periodic repeat of one colored tile, counting hexagonal distances as h first, the 3-color tiling is a tessellation generated by the order-3 permutohedrons. A chamferred hexagonal tiling replacing edges with new hexagons and transforms into another hexagonal tiling, in the limit, the original faces disappear, and the new hexagons degenerate into rhombi, and it becomes a rhombic tiling. The hexagons can be dissected into sets of 6 triangles and this is similar to the relation of the rhombic dodecahedron and the rhombo-hexagonal dodecahedron tessellations in 3 dimensions. This tiling is related to regular polyhedra with vertex figure n3. It is similarly related to the uniform truncated polyhedra with vertex figure n.6.6 and this tiling is also a part of a sequence of truncated rhombic polyhedra and tilings with Coxeter group symmetry

9.
Rhombitrihexagonal tiling
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In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex and it has Schläfli symbol of rr. John Conway calls it a rhombihexadeltille and it can be considered a cantellated by Norman Johnsons terminology or an expanded hexagonal tiling by Alicia Boole Stotts operational language. There are 3 regular and 8 semiregular tilings in the plane, there is only one uniform coloring in a rhombitrihexagonal tiling. With edge-colorings there is a half symmetry form orbifold notation, the hexagons can be considered as truncated triangles, t with two types of edges. It has Coxeter diagram, Schläfli symbol s2, the bicolored square can be distorted into isosceles trapezoids. In the limit, where the rectangles degenerate into edges, a triangular tiling results, there is one related 2-uniform tilings, having hexagons dissected into 6 triangles. Every circle is in contact with 4 other circles in the packing, the translational lattice domain contains 6 distinct circles. The gap inside each hexagon allows for one circle, related to a 2-uniform tiling with the hexagons divided into 6 triangles, there are eight uniform tilings that can be based from the regular hexagonal tiling. Drawing the tiles colored as red on the faces, yellow at the original vertices. This tiling is related as a part of sequence of cantellated polyhedra with vertex figure. These vertex-transitive figures have reflectional symmetry, the deltoidal trihexagonal tiling is a dual of the semiregular tiling known as the rhombitrihexagonal tiling. The edges of this tiling can be formed by the overlay of the regular triangular tiling. Each kite face of this tiling has angles 120°, 90°, 60° and 90° and it is one of only eight tilings of the plane in which every edge lies on a line of symmetry of the tiling. The deltoidal trihexagonal tiling is a dual of the semiregular tiling rhombitrihexagonal tiling and its faces are deltoids or kites. It is one of 7 dual uniform tilings in hexagonal symmetry and this tiling has face transitive variations, that can distort the kites into bilateral trapezoids or more general quadrillaterals. Ignoring the face colors below, the symmetry is p6m, and the lower symmetry is p31m with 3 mirrors meeting at a point. This tiling is related to the tiling by dividing the triangles and hexagons into central triangles