# 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 $t{\begin{Bmatrix}8\\3\end{Bmatrix}}$ |
| Wythoff symbol | 2 8 3 | |
| Coxeter diagram | or |
| 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.
Small index subgroups of [8,3], (*832) :
| Index | 1 | 2 | 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 | 4 | 6 | 12 |
| Diagrams | | | | | |
| Coxeter (orbifold) | [8,3]$^+$ = (832) | [8,3$^+$]$^+$ = = (433) | [8,3$^+$]$^+$ = = (433) | [8,3$^⅄$]$^+$ = = (842) | [8,3$^*$]$^+$ = = (444) |

## Order 3-8 kisrhombille

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

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]$^+$.
Uniform octagonal/triangular tilings vte :
| Symmetry: [8,3], (*832) | Symmetry: [8,3], (*832) | Symmetry: [8,3], (*832) | Symmetry: [8,3], (*832) | Symmetry: [8,3], (*832) | Symmetry: [8,3], (*832) | Symmetry: [8,3], (*832) | [8,3]$^+$ (832) | [1$^+$,8,3] (*443) | [1$^+$,8,3] (*443) | [8,3$^+$] (3*4) |
| {8,3} | t{8,3} | r{8,3} | t{3,8} | {3,8} | rr{8,3} s$_2${3,8} | tr{8,3} | sr{8,3} | h{8,3} | h$_2${8,3} | s{3,8} |
| | | | | | | | | or | or | |
| Uniform duals |||||||||||
| V8$^3$ | V3.16.16 | V3.8.3.8 | V6.6.8 | V3$^8$ | V3.4.8.4 | V4.6.16 | V3$^4$.8 | V(3.4)$^3$ | V8.6.6 | V3$^5$.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 vte :
| Sym. *n32 [n,3] | Spherical | Spherical | Spherical | Spherical | Euclid. | Compact hyperb. | Compact hyperb. | Paraco. | Noncompact hyperbolic | Noncompact hyperbolic | Noncompact hyperbolic | Noncompact hyperbolic |
| Sym. *n32 [n,3] | *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 |