Truncated tetraheptagonal tiling
Truncated tetraheptagonal tiling | |
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![]() Poincaré disk model of the hyperbolic plane | |
Type | Hyperbolic uniform tiling |
Vertex figure | 4.8.14 |
Schläfli symbol | tr{7,4} |
Wythoff symbol | 2 7 4 | |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() |
Symmetry group | [7,4], (*742) |
Dual | Order-4-7 kisrhombille tiling |
Properties | Vertex-transitive |
In geometry, the truncated tetraheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of tr{4,7}.
Images
Poincaré disk projection, centered on 14-gon:
Symmetry
![](../I/m/Truncated_tetraheptagonal_tiling_with_mirrors.png)
Truncated tetraheptagonal tiling with mirror lines
The dual to this tiling represents the fundamental domains of [7,4] (*742) symmetry. There are 3 small index subgroups constructed from [7,4] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
Index | 1 | 2 | 14 | |
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Diagram | ![]() |
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Coxeter (orbifold) |
[7,4] = ![]() ![]() ![]() ![]() ![]() (*742) |
[7,4,1+] = ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() (*772) |
[7+,4] = ![]() ![]() ![]() ![]() ![]() (7*2) |
[7*,4] = ![]() ![]() ![]() ![]() ![]() ![]() (*2222222) |
Index | 2 | 4 | 28 | |
Diagram | ![]() |
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Coxeter (orbifold) |
[7,4]+ = ![]() ![]() ![]() ![]() ![]() (742) |
[7+,4]+ = ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() (772) |
[7*,4]+ = ![]() ![]() ![]() ![]() ![]() ![]() (2222222) |
Related polyhedra and tiling
Symmetry: [7,4], (*742) | [7,4]+, (742) | [7+,4], (7*2) | [7,4,1+], (*772) | |||||||
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{7,4} | t{7,4} | r{7,4} | 2t{7,4}=t{4,7} | 2r{7,4}={4,7} | rr{7,4} | tr{7,4} | sr{7,4} | s{7,4} | h{4,7} | |
Uniform duals | ||||||||||
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V74 | V4.14.14 | V4.7.4.7 | V7.8.8 | V47 | V4.4.7.4 | V4.8.14 | V3.3.4.3.7 | V3.3.7.3.7 | V77 |
Symmetry *n42 [n,4] |
Spherical | Euclidean | Compact hyperbolic | Paracompact | ||||
---|---|---|---|---|---|---|---|---|
*242 [2,4] D4h |
*342 [3,4] Oh |
*442 [4,4] P4m |
*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.∞ |
Coxeter Schläfli |
![]() ![]() ![]() ![]() ![]() tr{2,4} |
![]() ![]() ![]() ![]() ![]() tr{3,4} |
![]() ![]() ![]() ![]() ![]() tr{4,4} |
![]() ![]() ![]() ![]() ![]() tr{5,4} |
![]() ![]() ![]() ![]() ![]() tr{6,4} |
![]() ![]() ![]() ![]() ![]() tr{7,4} |
![]() ![]() ![]() ![]() ![]() tr{8,4} |
![]() ![]() ![]() ![]() ![]() tr{∞,4} |
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.∞ |
Coxeter | ![]() ![]() ![]() ![]() ![]() |
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Symmetry *nn2 [n,n] |
Spherical | Euclidean | Compact hyperbolic | Paracompact | ||||
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*222 [2,2] D2h |
*332 [3,3] Td |
*442 [4,4] P4m |
*552 [5,5] |
*662 [6,6] |
*772 [7,7] |
*882 [8,8]... |
*∞∞2 [∞,∞] | |
Figure | ![]() 4.4.4 |
![]() 4.6.6 |
![]() 4.8.8 |
![]() 4.10.10 |
![]() 4.12.12 |
![]() 4.14.14 |
![]() 4.16.16 |
![]() 4.∞.∞ |
Coxeter Schläfli |
![]() ![]() ![]() ![]() ![]() tr{2,2} |
![]() ![]() ![]() ![]() ![]() tr{3,3} |
![]() ![]() ![]() ![]() ![]() tr{4,4} |
![]() ![]() ![]() ![]() ![]() tr{5,5} |
![]() ![]() ![]() ![]() ![]() tr{6,6} |
![]() ![]() ![]() ![]() ![]() tr{7,7} |
![]() ![]() ![]() ![]() ![]() tr{8,8} |
![]() ![]() ![]() ![]() ![]() tr{∞,∞} |
Dual | ![]() V4.4.4 |
![]() V4.6.6 |
![]() V4.8.8 |
![]() V4.10.10 |
![]() V4.12.12 |
![]() V4.14.14 |
![]() V4.16.16 |
![]() V4.∞.∞ |
Coxeter | ![]() ![]() ![]() ![]() ![]() |
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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)
- "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
See also
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Wikimedia Commons has media related to Uniform tiling 4-8-14. |
- Uniform tilings in hyperbolic plane
- List of regular polytopes
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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