Truncated tetrapentagonal tiling
In geometry, the truncated tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1,2{4,5} or tr{4,5}.
| Truncated tetrapentagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 4.8.10 |
| Schläfli symbol | tr{5,4} or |
| Wythoff symbol | |
| Coxeter diagram |    or   |
| Symmetry group | [5,4], (*542) |
| Dual | Order-4-5 kisrhombille tiling |
| Properties | Vertex-transitive |
Symmetry
Truncated tetrapentagonal tiling with mirror lines. 
There are four small index subgroup constructed from [5,4] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
A radical subgroup is constructed [5*,4], index 10, as [5+,4], (5*2) with gyration points removed, becoming orbifold (*22222), and its direct subgroup [5*,4]+, index 20, becomes orbifold (22222).
| Small index subgroups of [5,4] | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Index | 1 | 2 | 10 | ||||||||
| Diagram |  |
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| Coxeter (orbifold) |
[5,4] = (*542) |
[5,4,1+] =  =   (*552) |
[5+,4] =  (5*2) |
[5*,4] =    (*22222) | |||||||
| Direct subgroups | |||||||||||
| Index | 2 | 4 | 20 | ||||||||
| Diagram |  |
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| Coxeter (orbifold) |
[5,4]+ = (542) |
[5+,4]+ = =   (552) |
[5*,4]+ = (22222) | ||||||||
Related polyhedra and tiling
| *n42 symmetry mutation of omnitruncated tilings: 4.8.2n | ||||||||
|---|---|---|---|---|---|---|---|---|
| Symmetry *n42 [n,4] |
Spherical | Euclidean | Compact hyperbolic | Paracomp. | ||||
| *242 [2,4] |
*342 [3,4] |
*442 [4,4] |
*542 [5,4] |
*642 [6,4] |
*742 [7,4] |
*842 [8,4]... |
*∞42 [∞,4] | |
| Omnitruncated figure |
 4.8.4 |
 4.8.6 |
 4.8.8 |
4.8.10 |
 4.8.12 |
 4.8.14 |
 4.8.16 |
 4.8.∞ |
| Omnitruncated duals |
 V4.8.4 |
 V4.8.6 |
 V4.8.8 |
 V4.8.10 |
 V4.8.12 |
 V4.8.14 |
 V4.8.16 |
 V4.8.∞ |
| *nn2 symmetry mutations of omnitruncated tilings: 4.2n.2n | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry *nn2 [n,n] |
Spherical | Euclidean | Compact hyperbolic | Paracomp. | ||||||||||
| *222 [2,2] |
*332 [3,3] |
*442 [4,4] |
*552 [5,5] |
*662 [6,6] |
*772 [7,7] |
*882 [8,8]... |
*∞∞2 [∞,∞] | |||||||
| Figure |  |
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| Config. | 4.4.4 | 4.6.6 | 4.8.8 | 4.10.10 | 4.12.12 | 4.14.14 | 4.16.16 | 4.∞.∞ | ||||||
| Dual |  |
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| Config. | V4.4.4 | V4.6.6 | V4.8.8 | V4.10.10 | V4.12.12 | V4.14.14 | V4.16.16 | V4.∞.∞ | ||||||
| Uniform pentagonal/square tilings | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 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 | ||
See also
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Wikimedia Commons has media related to Uniform tiling 4-8-10. |
References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
- Coxeter, H. S. M. (1999). "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. ISBN 0-486-40919-8. LCCN 99035678.
External links
- Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
- Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
- Hyperbolic and Spherical Tiling Gallery
- KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
- Hyperbolic Planar Tessellations, Don Hatch
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