Truncated octagonal tiling
| Truncated octagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 3.16.16 |
| Schläfli symbol | t{8,3} |
| Wythoff symbol | 2 3 | 8 |
| Coxeter diagram |    
|
| Symmetry group | [8,3], (*832) |
| Dual | Order-8 triakis triangular tiling |
| Properties | Vertex-transitive
|
In
hexakaidecagons on each vertex. It has Schläfli symbol
of t{8,3}.
Dual tiling
The dual tiling has face configuration V3.16.16.
Related polyhedra and tilings
This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry.
| *n32 symmetry mutation of truncated tilings: t{n,3} | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 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] | |
| Truncated figures |
|
|
|
|
|
|
|
|
|
|
|
| Symbol | t{2,3} | t{3,3} | t{4,3} | t{5,3} | t{6,3} | t{7,3} | t{8,3} | t{∞,3} | t{12i,3} | t{9i,3} | t{6i,3} |
| Triakis figures |
|
|
|
|
|
|
|
|
|||
Config.
|
V3.4.4
|
V3.6.6 | V3.8.8 | V3.10.10 | V3.12.12
|
V3.14.14 | V3.16.16 | V3.∞.∞ | |||
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 | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 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} | |||
|
|
|
|
|
|
|
|
|
| |||||
   
|
|
|
|
  or 
|
or
|
| |||||||
|
|
 
|
 
|
 
|
|
|
|
 
|
 
|
 
| |||
| 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 | |||
|
|
|
|
|
|
|
|
|
|
| |||
|
|
|
|
|
|
|
|
|
|
|
| |||
See also
Wikimedia Commons has media related to Uniform tiling 3-16-16.
- Truncated hexagonal tiling
- Octagonal tiling
- Tilings of regular polygons
- List of uniform tilings
References
- 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. LCCN 99035678.
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
  | ||||||
| ||||||
 | This hyperbolic geometry-related article is a stub. You can help Wikipedia by expanding it. |