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In geometry, the tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t1{4,5} or r{4,5}.
Tetrapentagonal tiling | |
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![]() Poincaré disk model of the hyperbolic plane | |
Type | Hyperbolic uniform tiling |
Vertex configuration | (4.5)2 |
Schläfli symbol | r{5,4} or rr{5,5} or |
Wythoff symbol | 2 | 5 4 5 5 | 2 |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Symmetry group | [5,4], (*542) [5,5], (*552) |
Dual | Order-5-4 rhombille tiling |
Properties | Vertex-transitive edge-transitive |
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.
The dual tiling is made of rhombic faces and has a face configuration V4.5.4.5:
Uniform pentagonal/square tilings | |||||||||||
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Symmetry: [5,4], (*542) | [5,4]+, (542) | [5+,4], (5*2) | [5,4,1+], (*552) | ||||||||
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{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 | |||||||||||
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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 | |||||||||||
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Symmetry: [5,5], (*552) | [5,5]+, (552) | ||||||||||
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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 | |||||||||||
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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 |
*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] | |
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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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