Rhombitrioctagonal tiling
| Rhombitrioctagonal tiling | |
|---|---|
| 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 | File:CDel node 1.pngFile:CDel 8.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png or File:CDel node.pngFile:CDel split1-83.pngFile:CDel nodes 11.png File:CDel node 1.pngFile:CDel 8.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.png |
| 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
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 File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 8.pngFile:CDel node 1.png, 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, File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 8.pngFile:CDel node.png.
Related polyhedra and tilings
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.
Symmetry mutations
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 | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 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] | ||
| Figure | File:Spherical triangular prism.svg | File:Uniform tiling 332-t02.png | File:Uniform tiling 432-t02.png | File:Uniform tiling 532-t02.png | File:Uniform polyhedron-63-t02.png | File:Rhombitriheptagonal tiling.svg | File:H2-8-3-cantellated.svg | File:H2 tiling 23i-5.png | File:H2 tiling 23j12-5.png | File:H2 tiling 23j9-5.png | File:H2 tiling 23j6-5.png | |
| 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
- Rhombitrihexagonal tiling
- Order-3 octagonal tiling
- Tilings of regular polygons
- List of uniform tilings
- Kagome lattice
References
- 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.
