# Truncated triapeirogonal tiling

Truncated triapeirogonal tiling Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.6.
Schläfli symboltr{,3} or Wythoff symbol2 3 |
Coxeter diagram     or   Symmetry group[,3], (*32)
DualOrder 3-infinite kisrhombille
PropertiesVertex-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 special index 4 reflective subgroup, is [(∞,∞,3)], (*∞∞3), and its direct subgroup [(∞,∞,3)]+, (∞∞3), and semidirect subgroup [(∞,∞,3+)], (3*∞). Given [∞,3] with generating mirrors {0,1,2}, then its index 4 subgroup has generators {0,121,212}.

An index 6 subgroup constructed as [∞,3*], becomes [(∞,∞,∞)], (*∞∞∞).

Small index subgroups of [∞,3], (*∞32)
Index 1 2 3 4 6 8 12 24
Diagrams          Coxeter
(orbifold)
[∞,3]     =    (*∞32)
[1+,∞,3]     =    (*∞33)
[∞,3+]     (3*∞)
[∞,∞]

(*∞∞2)
[(∞,∞,3)]

(*∞∞3)
[∞,3*]     =    (*∞3)
[∞,1+,∞]

(*(∞2)2)
[(∞,1+,∞,3)]

(*(∞3)2)
[1+,∞,∞,1+]

(*∞4)
[(∞,∞,3*)]

(*∞6)
Direct subgroups
Index 2 4 6 8 12 16 24 48
Diagrams         Coxeter
(orbifold)
[∞,3]+     =    (∞32)
[∞,3+]+     =    (∞33)
[∞,∞]+

(∞∞2)
[(∞,∞,3)]+

(∞∞3)
[∞,3*]+     =    (∞3)
[∞,1+,∞]+

(∞2)2
[(∞,1+,∞,3)]+

(∞3)2
[1+,∞,∞,1+]+

(∞4)
[(∞,∞,3*)]+

(∞6)

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. Wikimedia Commons has media related to Uniform tiling 4-6-i.