Truncated trioctagonal tiling

Truncated trioctagonal tiling
Truncated trioctagonal tiling
	face-transitive

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

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		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

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

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 v t e
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

This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and

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* v t e
Sym. *n32 [n,3]	Spherical				Euclid.	Compact hyperb.		Paraco.	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