| Snub trioctagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 3.3.3.3.8 |
| Schläfli symbol | sr{8,3} or |
| Wythoff symbol | | 8 3 2 |
| Coxeter diagram | |
| Symmetry group | [8,3]+, (832) |
| Dual | Order-8-3 floret pentagonal tiling |
| Properties | Vertex-transitive Chiral |
In geometry, the order-3 snub octagonal tiling is a semiregular tiling of the hyperbolic plane. There are four triangles, one octagon on each vertex. It has Schläfli symbol of sr{8,3}.
Images
editDrawn in chiral pairs, with edges missing between black triangles:
Related polyhedra and tilings
editThis semiregular tiling is a member of a sequence of snubbed polyhedra and tilings with vertex figure (3.3.3.3.n) and Coxeter–Dynkin diagram
. These figures and their duals have (n32) rotational symmetry, being in the Euclidean plane for n=6, and hyperbolic plane for any higher n. The series can be considered to begin with n=2, with one set of faces degenerated into digons.
| n32 symmetry mutations of snub tilings: 3.3.3.3.n | ||||||||
|---|---|---|---|---|---|---|---|---|
| Symmetry n32 |
Spherical | Euclidean | Compact hyperbolic | Paracomp. | ||||
| 232 | 332 | 432 | 532 | 632 | 732 | 832 | ∞32 | |
| Snub figures |
||||||||
| Config. | 3.3.3.3.2 | 3.3.3.3.3 | 3.3.3.3.4 | 3.3.3.3.5 | 3.3.3.3.6 | 3.3.3.3.7 | 3.3.3.3.8 | 3.3.3.3.∞ |
| Gyro figures |
||||||||
| Config. | V3.3.3.3.2 | V3.3.3.3.3 | V3.3.3.3.4 | V3.3.3.3.5 | V3.3.3.3.6 | V3.3.3.3.7 | V3.3.3.3.8 | V3.3.3.3.∞ |
From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 10 forms.
| Uniform octagonal/triangular tilings | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry: [8,3], (*832) | [8,3]+ (832) |
[1+,8,3] (*443) |
[8,3+] (3*4) | ||||||||||
| {8,3} | t{8,3} | r{8,3} | t{3,8} | {3,8} | rr{8,3} s2{3,8} |
tr{8,3} | sr{8,3} | h{8,3} | h2{8,3} | s{3,8} | |||
| Uniform duals | |||||||||||||
| V83 | V3.16.16 | V3.8.3.8 | V6.6.8 | V38 | V3.4.8.4 | V4.6.16 | V34.8 | V(3.4)3 | V8.6.6 | V35.4 | |||
References
edit- 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)
- "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.