Truncated trioctagonal tiling
| Truncated trioctagonal tiling | |
|---|---|
| Truncated trioctagonal tiling Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 4.6.16 |
| Schläfli symbol | tr{8,3} or |
| Wythoff symbol | 2 8 3 | |
| Coxeter diagram |  File:CDel 8.pngFile:CDel 3.png or File:CDel split1-83.pngFile:CDel nodes 11.png
|
| Symmetry group | [8,3], (*832) |
| Dual | Order 3-8 kisrhombille |
| Properties | Vertex-transitive |
In geometry, the truncated trioctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one hexagon, and one hexadecagon (16-sides) on each vertex. It has Schläfli symbol of tr{8,3}.
Symmetry
The dual of this tiling, the order 3-8 kisrhombille, represents the fundamental domains of [8,3] (*832) symmetry. There are 3 small index subgroups constructed from [8,3] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
A larger index 6 subgroup constructed as [8,3*], becomes [(4,4,4)], (*444). An intermediate index 3 subgroup is constructed as [8,3⅄], with 2/3 of blue mirrors removed.
Order 3-8 kisrhombille
Template:Infobox face-uniform tiling
The order 3-8 kisrhombille is a semiregular dual tiling of the hyperbolic plane. It is constructed by congruent right triangles with 4, 6, and 16 triangles meeting at each vertex.
The image shows a Poincaré disk model projection of the hyperbolic plane.
It is labeled V4.6.16 because each right triangle face has three types of vertices: one with 4 triangles, one with 6 triangles, and one with 16 triangles. It is the dual tessellation of the truncated trioctagonal tiling, described above.
Naming
An alternative name is 3-8 kisrhombille by Conway, seeing it as a 3-8 rhombic tiling, divided by a kis operator, adding a center point to each rhombus, and dividing into four triangles.
Related polyhedra and tilings
This tiling is one of 10 uniform tilings constructed from [8,3] hyperbolic symmetry and three subsymmetries [1+,8,3], [8,3+] and [8,3]+.
Template:Octagonal tiling table
This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram File:CDel p.pngFile:CDel 3.png. For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.
Template:Omnitruncated table
See also
 |
Wikimedia Commons has media related to Uniform tiling 4-6-16. |
|
Wikimedia Commons has media related to Uniform dual tiling V 4-6-16. |
- Tilings of regular polygons
- Hexakis triangular tiling
- List of uniform tilings
- Uniform tilings in hyperbolic plane
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
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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