# Truncated trioctagonal tiling

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}.

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

## Symmetry

Truncated trioctagonal tiling with mirror lines

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 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

Truncated trioctagonal tiling
TypeDual semiregular hyperbolic tiling
FacesRight triangle
EdgesInfinite
VerticesInfinite
Coxeter diagram     Symmetry group[8,3], (*832)
Rotation group[8,3]+, (832)
Dual polyhedronTruncated trioctagonal tiling
Face configurationV4.6.16
Propertiesface-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

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]+.

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.