Order-4 pentagonal tiling (original) (raw)

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Regular tiling of the hyperbolic plane

Order-4 pentagonal tiling
Order-4 pentagonal tilingPoincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 54
Schläfli symbol {5,4}r{5,5} or { 5 5 } {\displaystyle {\begin{Bmatrix}5\\5\end{Bmatrix}}} {\displaystyle {\begin{Bmatrix}5\\5\end{Bmatrix}}}
Wythoff symbol 4 | 5 22 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

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.

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 t1{5,5} and as a quasiregular tiling is called a pentapentagonal tiling.

Uniform pentagonal/square tilings vte
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
V54 V4.10.10 V4.5.4.5 V5.8.8 V45 V4.4.5.4 V4.8.10 V3.3.4.3.5 V3.3.5.3.5 V55
Uniform pentapentagonal tilings vte
Symmetry: [5,5], (*552) [5,5]+, (552)
= = = = = = = =
Order-5 pentagonal tiling {5,5} Truncated order-5 pentagonal tiling t{5,5} Order-4 pentagonal tiling r{5,5} Truncated order-5 pentagonal tiling 2t{5,5} = t{5,5} Order-5 pentagonal tiling 2r{5,5} = {5,5} Tetrapentagonal tiling rr{5,5} Truncated order-4 pentagonal tiling tr{5,5} Snub pentapentagonal tiling sr{5,5}
Uniform duals
Order-5 pentagonal tiling V5.5.5.5.5 V5.10.10 Order-5 square tiling V5.5.5.5 V5.10.10 Order-5 pentagonal tiling V5.5.5.5.5 V4.5.4.5 V4.10.10 V3.3.5.3.5

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.

{5,n} tilings
{5,3} {5,4} {5,5} {5,6} {5,7}

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.

*_n_42 symmetry mutation of regular tilings: {n,4} vte
Spherical Euclidean Hyperbolic tilings
24 34 44 54 64 74 84 ...∞4

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n).

*_n_42 symmetry mutation of regular tilings: {4,n} vte
Spherical Euclidean Compact hyperbolic Paracompact
{4,3} {4,4} {4,5} {4,6} {4,7} {4,8}... {4,∞}
*5_n_2 symmetry mutations of quasiregular tilings: (5.n)2 vte
Symmetry*5_n_2[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] [_n_i,5]
Figures
Config. (5.3)2 (5.4)2 (5.5)2 (5.6)2 (5.7)2 (5.8)2 (5.∞)2 (5._n_i)2
Rhombicfigures
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