# Order-4 pentagonal tiling

Order-4 pentagonal tiling :
| Poincaré disk model of the hyperbolic plane ||
| Type | Hyperbolic regular tiling |
| Vertex configuration | 5$^4$ |
| Schläfli symbol | {5,4} r{5,5} or ${\begin{Bmatrix}5\\5\end{Bmatrix}}$ |
| 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 |
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.

## Symmetry

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.

## Related polyhedra and tiling

Uniform pentagonal/square tilings vte :
| Symmetry: [5,4], (*542) | Symmetry: [5,4], (*542) | Symmetry: [5,4], (*542) | Symmetry: [5,4], (*542) | Symmetry: [5,4], (*542) | Symmetry: [5,4], (*542) | 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 vte :
| Symmetry: [5,5], (*552) | Symmetry: [5,5], (*552) | Symmetry: [5,5], (*552) | Symmetry: [5,5], (*552) | Symmetry: [5,5], (*552) | Symmetry: [5,5], (*552) | 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.
*n42 symmetry mutation of regular tilings: {n,4} vte :
| Spherical | Spherical | Euclidean | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings |
| 2$^4$ | 3$^4$ | 4$^4$ | 5$^4$ | 6$^4$ | 7$^4$ | 8$^4$ | ...∞$^4$ |
This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4$^n$).
*n42 symmetry mutation of regular tilings: {4,n} vte :
| Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracompact |
| {4,3} | {4,4} | {4,5} | {4,6} | {4,7} | {4,8}... | {4,∞} |
*5n2 symmetry mutations of quasiregular tilings: (5.n)$^2$ vte :
| Symmetry *5n2 [n,5] | Spherical | Hyperbolic | Hyperbolic | Hyperbolic | Hyperbolic | Hyperbolic | Paracompact | Noncompact |
| Symmetry *5n2 [n,5] | *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 | | | | | | | | |
| 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.∞)$^2$ |