Summary
| Rhombitrioctagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 3.4.8.4 |
| Schläfli symbol | rr{8,3} or s2{3,8} |
| Wythoff symbol | 3 | 8 2 |
| Coxeter diagram |     or   
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| Symmetry group | [8,3], (*832) [8,3+], (3*4) |
| Dual | Deltoidal trioctagonal tiling |
| Properties | Vertex-transitive |
In geometry, the rhombitrioctagonal tiling is a semiregular tiling of the hyperbolic plane. At each vertex of the tiling there is one triangle and one octagon, alternating between two squares. The tiling has Schläfli symbol rr{8,3}. It can be seen as constructed as a rectified trioctagonal tiling, r{8,3}, as well as an expanded octagonal tiling or expanded order-8 triangular tiling.
Symmetry edit
This tiling has [8,3], (*832) symmetry. There is only one uniform coloring.
Similar to the Euclidean rhombitrihexagonal tiling, by edge-coloring there is a half symmetry form (3*4) orbifold notation. The octagons can be considered as truncated squares, t{4} with two types of edges. It has Coxeter diagram , Schläfli symbol s2{3,8}. The squares can be distorted into isosceles trapezoids. In the limit, where the rectangles degenerate into edges, an order-8 triangular tiling results, constructed as a snub tritetratrigonal tiling, .
Related polyhedra and tilings edit
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 8 forms.
| Uniform octagonal/triangular tilings | |||||||||||||
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| 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} | |||
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| 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 | |||
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Symmetry mutations edit
This tiling is topologically related as a part of sequence of cantellated polyhedra with vertex figure (3.4.n.4), and continues as tilings of the hyperbolic plane. These vertex-transitive figures have (*n32) reflectional symmetry.
| *n32 symmetry mutation of expanded tilings: 3.4.n.4 | ||||||||||||
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| Symmetry *n32 [n,3] |
Spherical | Euclid. | Compact hyperb. | Paraco. | Noncompact hyperbolic | |||||||
| *232 [2,3] |
*332 [3,3] |
*432 [4,3] |
*532 [5,3] |
*632 [6,3] |
*732 [7,3] |
*832 [8,3]... |
*∞32 [∞,3] |
[12i,3] |
[9i,3] |
[6i,3] | ||
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| Config. | 3.4.2.4 | 3.4.3.4 | 3.4.4.4 | 3.4.5.4 | 3.4.6.4 | 3.4.7.4 | 3.4.8.4 | 3.4.∞.4 | 3.4.12i.4 | 3.4.9i.4 | 3.4.6i.4 | |
See also edit
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.
External links edit
- 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