# Truncated triapeirogonal tiling

Truncated triapeirogonal tiling

Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex figure 4.6.∞
Schläfli symbol tr{∞,3}
Wythoff symbol 2 ∞ 3 |
Coxeter diagram
Symmetry group [∞,3], (*∞32)
Dual Order 3-infinite kisrhombille
Properties Vertex-transitive

In geometry, the truncated triapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of tr{∞,3}.

## Symmetry

Truncated triapeirogonal tiling with mirrors

The dual of this tiling represents the fundamental domains of [∞,3], *∞32 symmetry. There are 3 small index subgroup constructed from [∞,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 [∞,3*], becomes [(∞,∞,∞)], (*∞∞∞).

Small index subgroups of [∞,3], (*∞32)
Index 1 2 6
Diagrams
Coxeter
(orbifold)
[∞,3] = =
(*∞32)
[1+,∞,3] = =
(*∞33)
[∞,3+] =
(3*∞)
[∞,3*] = =
(*∞∞∞)
Direct subgroups
Index 2 4 12
Diagrams
Coxeter
(orbifold)
[∞,3]+ = =
(∞32)
[∞,3+]+ = =
(∞33)
[∞,3*]+ = =
(∞∞∞)

## Related polyhedra and tiling

Paracompact hyperbolic uniform tilings in [∞,3] family
Symmetry: [∞,3], (*∞32) [∞,3]+
(∞32)
[1+,∞,3]
(*∞33)
[∞,3+]
(3*∞)

=

=

=
=
or
=
or

=
{∞,3} t{∞,3} r{∞,3} t{3,∞} {3,∞} rr{∞,3} tr{∞,3} sr{∞,3} h{∞,3} h2{∞,3} s{3,∞}
Uniform duals
V∞3 V3.∞.∞ V(3.∞)2 V6.6.∞ V3 V4.3.4.∞ V4.6.∞ V3.3.3.3.∞ V(3.∞)3 V3.3.3.3.3.∞

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.

Dimensional family of omnitruncated polyhedra and tilings: 4.6.2n
Symmetry
*n32
[n,3]
Spherical Euclidean Compact hyperbolic Paracompact
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
P6m
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
Coxeter
Schläfli

tr{2,3}

tr{3,3}

tr{4,3}

tr{5,3}

tr{6,3}

tr{7,3}

tr{8,3}

tr{∞,3}
Omnitruncated
figure
Vertex figure 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞
Dual figures
Coxeter
Omnitruncated
duals
Face
configuration
V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞