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 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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
[edit]![](http://upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Truncated_trioctagonal_tiling_with_mirrors.png/220px-Truncated_trioctagonal_tiling_with_mirrors.png)
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.
Index | 1 | 2 | 3 | 6 | |
---|---|---|---|---|---|
Diagrams | ![]() |
![]() |
![]() |
![]() |
![]() |
Coxeter (orbifold) |
[8,3] = ![]() ![]() ![]() ![]() ![]() (*832) |
[1+,8,3] = ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() (*433) |
[8,3+] = ![]() ![]() ![]() ![]() ![]() (3*4) |
[8,3⅄] = ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() (*842) |
[8,3*] = ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() (*444) |
Direct subgroups | |||||
Index | 2 | 4 | 6 | 12 | |
Diagrams | ![]() |
![]() |
![]() |
![]() | |
Coxeter (orbifold) |
[8,3]+ = ![]() ![]() ![]() ![]() ![]() (832) |
[8,3+]+ = ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() (433) |
[8,3⅄]+ = ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() (842) |
[8,3*]+ = ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() (444) |
Order 3-8 kisrhombille
[edit]Truncated trioctagonal tiling | |
---|---|
![]() | |
Type | Dual semiregular hyperbolic tiling |
Faces | Right triangle |
Edges | Infinite |
Vertices | Infinite |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() |
Symmetry group | [8,3], (*832) |
Rotation group | [8,3]+, (832) |
Dual polyhedron | Truncated trioctagonal tiling |
Face configuration | V4.6.16 |
Properties | face-transitive |
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 which has one square and one octagon and one hexakaidecagon at each vertex.
Naming
[edit]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
[edit]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]+.
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} | |||
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() | |||||
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |||||||
![]() |
![]() |
![]() ![]() |
![]() ![]() |
![]() ![]() |
![]() |
![]() |
![]() |
![]() ![]() |
![]() ![]() |
![]() ![]() | |||
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 | |||
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() | |||
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . 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.
*n32 symmetry mutation of omnitruncated tilings: 4.6.2n | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Sym. *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] |
[3i,3] | |
Figures | ![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
Config. | 4.6.4 | 4.6.6 | 4.6.8 | 4.6.10 | 4.6.12 | 4.6.14 | 4.6.16 | 4.6.∞ | 4.6.24i | 4.6.18i | 4.6.12i | 4.6.6i |
Duals | ![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
Config. | V4.6.4 | V4.6.6 | V4.6.8 | V4.6.10 | V4.6.12 | V4.6.14 | V4.6.16 | V4.6.∞ | V4.6.24i | V4.6.18i | V4.6.12i | V4.6.6i |
See also
[edit]![](http://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png)
![](http://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png)
- Tilings of regular polygons
- Hexakis triangular tiling
- List of uniform tilings
- Uniform tilings in hyperbolic plane
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.