Triheptagonal tiling
| Triheptagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | (3.7)2 |
| Schläfli symbol | r{7,3} |
| Wythoff symbol | 2 | 7 3 |
| Coxeter diagram |     |
| Symmetry group | [7,3], (*732) |
| Dual | Order-7-3 rhombille tiling |
| Properties | Vertex-transitive edge-transitive |
In geometry, the triheptagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 heptagonal tiling. There are two triangles and two heptagons alternating on each vertex. It has Schläfli symbol of r{7,3}.
Compare to trihexagonal tiling with vertex configuration 3.6.3.6.
Images
 Klein disk model of this tiling preserves straight lines, but distorts angles |
 The dual tiling is called an Order-7-3 rhombille tiling, made from rhombic faces, alternating 3 and 7 per vertex. |
Related polyhedra and tilings
The triheptagonal tiling can be seen in a sequence of quasiregular polyhedrons and tilings:
*n32 orbifold symmetries of quasiregular tilings: (3.n)2
| Sym. *n32 [n,3] |
Spherical | Euclid. | Compact hyperb. | Paraco. | Noncompact hyperbolic | |||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| *332 [3,3] Td |
*432 [4,3] Oh |
*532 [5,3] Ih |
*632 [6,3] p6m |
*732 [7,3] |
*832 [8,3]... |
*∞32 [∞,3] |
[12i,3] | [9i,3] | [6i,3] | |||
Figure |
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| Vertex | (3.3)2 | (3.4)2 | (3.6)2 | (3.6)2 | (3.7)2 | (3.8)2 | (3.∞)2 | (3.12i)2 | (3.9i)2 | (3.6i)2 | ||
| Schläfli | r{3,3} | r{4,3} | r{5,3} | r{6,3} | r{7,3} | r{8,3} | r{∞,3} | r{12i,3} | r{9i,3} | r{6i,3} | ||
| Coxeter | |
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| Dual uniform figures | ||||||||||||
| Dual conf. |
 V(3.3)2 |
 V(3.4)2 |
 V(3.5)2 |
 V(3.6)2 |
V(3.7)2 |
 V(3.8)2 |
 V(3.∞)2 |
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From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.
Dimensional family of quasiregular polyhedra and tilings: 7.n.7.n
| Symmetry *7n2 [n,7] |
Hyperbolic... | Paracompact | Noncompact | |||||
|---|---|---|---|---|---|---|---|---|
| *732 [3,7] |
*742 [4,7] |
*752 [5,7] |
*762 [6,7] |
*772 [7,7] |
*872 [8,7]... |
*∞72 [∞,7] |
[iπ/λ,7] | |
| Coxeter | |
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| Quasiregular figures configuration |
3.7.3.7 |
 4.7.4.7 |
 7.5.7.5 |
 7.6.7.6 |
 7.7.7.7 |
 7.8.7.8 |
 7.∞.7.∞ |
7.∞.7.∞ |
See also
 |
Wikimedia Commons has media related to Uniform tiling 3-7-3-7. |
- Trihexagonal tiling - 3.6.3.6 tiling
- Rhombille tiling - dual V3.6.3.6 tiling
- Tilings of regular polygons
- List of uniform tilings
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.
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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