In geometry, the order-4 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,4}. It can also be called a pentapentagonal tiling in a bicolored quasiregular form.
Order-4 pentagonal tiling | |
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Poincaré disk model of the hyperbolic plane | |
Type | Hyperbolic regular tiling |
Vertex configuration | 5^{4} |
Schläfli symbol | {5,4} r{5,5} or |
Wythoff symbol | 4 | 5 2 2 | 5 5 |
Coxeter diagram | or |
Symmetry group | [5,4], (*542) [5,5], (*552) |
Dual | Order-5 square tiling |
Properties | Vertex-transitive, edge-transitive, face-transitive |
This tiling represents a hyperbolic kaleidoscope of 5 mirrors meeting as edges of a regular pentagon. This symmetry by orbifold notation is called *22222 with 5 order-2 mirror intersections. In Coxeter notation can be represented as [5^{*},4], removing two of three mirrors (passing through the pentagon center) in the [5,4] symmetry.
The kaleidoscopic domains can be seen as bicolored pentagons, representing mirror images of the fundamental domain. This coloring represents the uniform tiling t_{1}{5,5} and as a quasiregular tiling is called a pentapentagonal tiling.
This tiling is topologically related as a part of sequence of regular polyhedra and tilings with pentagonal faces, starting with the dodecahedron, with Schläfli symbol {5,n}, and Coxeter diagram , progressing to infinity.
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.
This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4^{n}).
In geometry, the dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U36. It is the rectification of the great dodecahedron (and that of its dual, the small stellated dodecahedron). It was discovered independently by Hess (1878), Badoureau (1881) and Pitsch (1882).
The edges of this model form 10 central hexagons, and projected onto a sphere, represent 10 great circles. These 10, along with the great circles from projections of two other polyhedra, form the 31 great circles of the spherical icosahedron used in construction of geodesic domes.
Medial rhombic triacontahedronIn geometry, the medial rhombic triacontahedron is a nonconvex isohedral polyhedron. It is a stellation of the rhombic triacontahedron, and can also be called small stellated triacontahedron. Its dual is the dodecadodecahedron.
Its 24 vertices are all on the 12 axes with 5-fold symmetry (i.e. each corresponds to one of the 12 vertices of the icosahedron). This means that on each axis there is an inner and an outer vertex. The ratio of outer to inner vertex radius is , the golden ratio.
It has 30 intersecting rhombic faces, which correspond to the faces of the convex rhombic triacontahedron. The diagonals in the rhombs of the convex solid have a ratio of 1 to . The medial solid can be generated from the convex one by stretching the shorter diagonal from length 1 to . So the ratio of rhomb diagonals in the medial solid is 1 to .
This solid is to the compound of small stellated dodecahedron and great dodecahedron what the convex one is to the compound of dodecahedron and icosahedron: The crossing edges in the dual compound are the diagonals of the rhombs.
Order-4-3 pentagonal honeycombIn the geometry of hyperbolic 3-space, the order-4-3 pentagonal honeycomb or 5,4,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell is an order-4 pentagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
Order-4 120-cell honeycombIn the geometry of hyperbolic 4-space, the order-4 120-cell honeycomb is one of five compact regular space-filling tessellations (or honeycombs). With Schläfli symbol {5,3,3,4}, it has four 120-cells around each face. Its dual is the order-5 tesseractic honeycomb, {4,3,3,5}.
Order-4 dodecahedral honeycombIn the geometry of hyperbolic 3-space, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs). With Schläfli symbol {5,3,4}, it has four dodecahedra around each edge, and 8 dodecahedra around each vertex in an octahedral arrangement. Its vertices are constructed from 3 orthogonal axes. Its dual is the order-5 cubic honeycomb.
A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
Order-4 icosahedral honeycombIn the geometry of hyperbolic 3-space, the order-5 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,4}.
Order-5-4 square honeycombIn the geometry of hyperbolic 3-space, the order-5-4 square honeycomb (or 4,5,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,5,4}.
Order-5 dodecahedral honeycombThe order-5 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol {5,3,5}, it has five dodecahedral cells around each edge, and each vertex is surrounded by twenty dodecahedra. Its vertex figure is a regular icosahedron.
A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
Order-5 square tilingIn geometry, the order-5 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,5}.
Pentagonal tilingIn geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon.
A regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a regular pentagon, 108°, is not a divisor of 360°, the angle measure of a whole turn. However, regular pentagons can tile the hyperbolic plane and the sphere; the latter produces a tiling topologically equivalent to the dodecahedron.
Rectification (geometry)In Euclidean geometry, rectification or complete-truncation is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the rectified facets of the original polytope.
A rectification operator is sometimes denoted by the letter r with a Schläfli symbol. For example, r{4,3} is the rectified cube, also called a cuboctahedron, and also represented as . And a rectified cuboctahedron rr{4,3} is a rhombicuboctahedron, and also represented as .
Conway polyhedron notation uses a for ambo as this operator. In graph theory this operation creates a medial graph.
The rectification of any regular self-dual polyhedron or tiling will result in another regular polyhedron or tiling with a tiling order of 4, for example the tetrahedron {3,3} becoming an octahedron {3,4}. As a special case, a square tiling {4,4} will turn into another square tiling {4,4} under a rectification operation.
Truncated order-4 pentagonal tilingIn geometry, the truncated order-4 pentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{5,4}.
Uniform pentagonal/square tilings | |||||||||||
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Symmetry: [5,4], (*542) | [5,4]^{+}, (542) | [5^{+},4], (5*2) | [5,4,1^{+}], (*552) | ||||||||
{5,4} | t{5,4} | r{5,4} | 2t{5,4}=t{4,5} | 2r{5,4}={4,5} | rr{5,4} | tr{5,4} | sr{5,4} | s{5,4} | h{4,5} | ||
Uniform duals | |||||||||||
V5^{4} | V4.10.10 | V4.5.4.5 | V5.8.8 | V4^{5} | V4.4.5.4 | V4.8.10 | V3.3.4.3.5 | V3.3.5.3.5 | V5^{5} |
Uniform pentapentagonal tilings | |||||||||||
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Symmetry: [5,5], (*552) | [5,5]^{+}, (552) | ||||||||||
= |
= |
= |
= |
= |
= |
= |
= | ||||
{5,5} | t{5,5} |
r{5,5} | 2t{5,5}=t{5,5} | 2r{5,5}={5,5} | rr{5,5} | tr{5,5} | sr{5,5} | ||||
Uniform duals | |||||||||||
V5.5.5.5.5 | V5.10.10 | V5.5.5.5 | V5.10.10 | V5.5.5.5.5 | V4.5.4.5 | V4.10.10 | V3.3.5.3.5 |
{5,n} tilings | ||||
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{5,3} |
{5,4} |
{5,5} |
{5,6} |
{5,7} |
*n42 symmetry mutation of regular tilings: {n,4} | |||||||
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Spherical | Euclidean | Hyperbolic tilings | |||||
2^{4} | 3^{4} | 4^{4} | 54 | 6^{4} | 7^{4} | 8^{4} | ...∞^{4} |
*n42 symmetry mutation of regular tilings: {4,n} | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Spherical | Euclidean | Compact hyperbolic | Paracompact | ||||||||
{4,3} |
{4,4} |
{4,5} |
{4,6} |
{4,7} |
{4,8}... |
{4,∞} |
*5n2 symmetry mutations of quasiregular tilings: (5.n)^{2} | ||||||||
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Symmetry *5n2 [n,5] |
Spherical | Hyperbolic | Paracompact | Noncompact | ||||
*352 [3,5] |
*452 [4,5] |
*552 [5,5] |
*652 [6,5] |
*752 [7,5] |
*852 [8,5]... |
*∞52 [∞,5] |
[ni,5] | |
Figures | ||||||||
Config. | (5.3)^{2} | (5.4)^{2} | (5.5)^{2} | (5.6)^{2} | (5.7)2 | (5.8)2 | (5.∞)^{2} | (5.ni)^{2} |
Rhombic figures |
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Config. | V(5.3)^{2} | V(5.4)^{2} | V(5.5)^{2} | V(5.6)^{2} | V(5.7)^{2} | V(5.8)^{2} | V(5.∞)^{2} | V(5.∞)^{2} |
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