# Tetrapentagonal tiling

Tetrapentagonal tiling :
| Poincaré disk model of the hyperbolic plane ||
| Type | Hyperbolic uniform tiling |
| Vertex configuration | (4.5)$^2$ |
| Schläfli symbol | r{5,4} or ${\begin{Bmatrix}5\\4\end{Bmatrix}}$ rr{5,5} or $r{\begin{Bmatrix}5\\5\end{Bmatrix}}$ |
| Wythoff symbol | 2 | 5 4 5 5 | 2 |
| Coxeter diagram | or or |
| Symmetry group | [5,4], (*542) [5,5], (*552) |
| Dual | Order-5-4 rhombille tiling |
| Properties | Vertex-transitive edge-transitive |
In geometry, the tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t$_1${4,5} or r{4,5}.

## Symmetry

A half symmetry [1$^+$,4,5] = [5,5] construction exists, which can be seen as two colors of pentagons. This coloring can be called a rhombipentapentagonal tiling.

## Dual tiling

The dual tiling is made of rhombic faces and has a face configuration V4.5.4.5:

## 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 |
*n42 symmetry mutations of quasiregular tilings: (4.n)$^2$ vte :
| Symmetry *4n2 [n,4] | Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracompact | Noncompact |
| Symmetry *4n2 [n,4] | *342 [3,4] | *442 [4,4] | *542 [5,4] | *642 [6,4] | *742 [7,4] | *842 [8,4]... | *∞42 [∞,4] | [ni,4] |
| Figures | | | | | | | | |
| Config. | (4.3)$^2$ | (4.4)$^2$ | (4.5)$^2$ | (4.6)$^2$ | (4.7)$^2$ | (4.8)$^2$ | (4.∞)$^2$ | (4.ni)$^2$ |
*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.∞)$^2$ |