Rhombitetraoctagonal tiling
| Rhombitetraoctagonal tiling | |
|---|---|
| Rhombitetraoctagonal tiling Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 4.4.8.4 |
| Schläfli symbol | rr{8,4} or |
| Wythoff symbol | 4 | 8 2 |
| Coxeter diagram |  File:CDel 8.png File:CDel 4.png or File:CDel split1-84.pngFile:CDel nodes 11.png
|
| Symmetry group | [8,4], (*842) |
| Dual | Deltoidal tetraoctagonal tiling |
| Properties | Vertex-transitive |
In geometry, the rhombitetraoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{8,4}. It can be seen as constructed as a rectified tetraoctagonal tiling, r{8,4}, as well as an expanded order-4 octagonal tiling or expanded order-8 square tiling.
Constructions
There are two uniform constructions of this tiling, one from [8,4] or (*842) symmetry, and secondly removing the mirror middle, [8,1+,4], gives a rectangular fundamental domain [∞,4,∞], (*4222).
| Name | Rhombitetraoctagonal tiling | |
|---|---|---|
| Image | File:Uniform tiling 84-t02.png | File:Uniform tiling 4.4.4.8.png |
| Symmetry | [8,4] (*842) File:CDel node c1.pngFile:CDel 8.pngFile:CDel node c3.pngFile:CDel 4.pngFile:CDel node c2.png |
[8,1+,4] = [∞,4,∞] (*4222) File:CDel node c1.pngFile:CDel 8.pngFile:CDel node h0.pngFile:CDel 4.pngFile:CDel node c2.png = File:CDel label4.pngFile:CDel branch c1.pngFile:CDel 2a2b-cross.pngFile:CDel nodeab c2.png |
| Schläfli symbol | rr{8,4} | t0,1,2,3{∞,4,∞} |
| Coxeter diagram | File:CDel 8.pngFile:CDel 4.png
|
File:CDel 8.pngFile:CDel node h0.pngFile:CDel 4.png = File:CDel label4.pngFile:CDel branch 11.pngFile:CDel 2a2b-cross.pngFile:CDel nodes 11.png
|
Symmetry
A lower symmetry construction exists, with (*4222) orbifold symmetry. This symmetry can be seen in the dual tiling, called a deltoidal tetraoctagonal tiling, alternately colored here. Its fundamental domain is a Lambert quadrilateral, with 3 right angles.
| File:Deltoidal tetraoctagonal til.png | File:H2chess 248d.png |
| The dual tiling, called a deltoidal tetraoctagonal tiling, represents the fundamental domains of the *4222 orbifold. | |
With edge-colorings there is a half symmetry form (4*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 4.pngFile:CDel node h.pngFile:CDel 8.png, Schläfli symbol s2{4,8}. The squares can be distorted into isosceles trapezoids. In the limit, where the rectangles degenerate into edges, an order-8 square tiling results, constructed as a snub tetraoctagonal tiling, File:CDel node h.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel 8.png.
Related polyhedra and tiling
Template:Order 8-4 tiling table
References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Search this book on
. (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. Search this book on
See also
 |
Wikimedia Commons has media related to Uniform tiling 4-4-4-8. |
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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