| 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 t { 5 4 } {\displaystyle t{\begin{Bmatrix}5\\4\end{Bmatrix}}} |
| Wythoff symbol | 2 5 4 | |
| Coxeter diagram | or |
| Symmetry group | [5,4], (*542) |
| Dual | Order-4-5 kisrhombille tiling |
| Properties | Vertex-transitive |
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}.
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Symmetry
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 | |||||||||||
| Coxeter(orbifold) | [5,4] = (*542) | [5,4,1+] = = (*552) | [5+,4] = (5*2) | [5*,4] = (*22222) | |||||||
| Direct subgroups | |||||||||||
| Index | 2 | 4 | 20 | ||||||||
| Diagram | |||||||||||
| 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] | |
| Omnitruncatedfigure | 4.8.4 | 4.8.6 | 4.8.8 | 4.8.10 | 4.8.12 | 4.8.14 | 4.8.16 | 4.8.∞ |
| Omnitruncatedduals | 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 | ||||||||||||||
| Config. | 4.4.4 | 4.6.6 | 4.8.8 | 4.10.10 | 4.12.12 | 4.14.14 | 4.16.16 | 4.∞.∞ | ||||||
| Dual | ||||||||||||||
| 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) | ||||||||
| {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 | ||
See also
Wikimedia Commons has media related to Uniform tiling 4-8-10.- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, 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.