# Truncated triapeirogonal tiling

Truncated triapeirogonal tiling :
| Poincaré disk model of the hyperbolic plane ||
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 4.6.∞ |
| Schläfli symbol | tr{∞,3} or $t{\begin{Bmatrix}\infty \\3\end{Bmatrix}}$ |
| Wythoff symbol | 2 ∞ 3 | |
| Coxeter diagram | or |
| 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

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 | 2 | 3 | 4 | 6 | 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 | 4 | 6 | 8 | 12 | 12 | 16 | 24 | 48 |
| Diagrams | | | | | | | | | | |
| Coxeter (orbifold) | [∞,3]$^+$ = (∞32) | [∞,3$^+$]$^+$ = (∞33) | [∞,3$^+$]$^+$ = (∞33) | [∞,∞]$^+$ (∞∞2) | [(∞,∞,3)]$^+$ (∞∞3) | [∞,3*]$^+$ = (∞$^3$) | [∞,1$^+$,∞]$^+$ (∞2)$^2$ | [(∞,1$^+$,∞,3)]$^+$ (∞3)$^2$ | [1$^+$,∞,∞,1$^+$]$^+$ (∞$^4$) | [(∞,∞,3*)]$^+$ (∞$^6$) |

## Related polyhedra and tiling

Paracompact uniform tilings in [∞,3] family vte :
| Symmetry: [∞,3], (*∞32) | Symmetry: [∞,3], (*∞32) | Symmetry: [∞,3], (*∞32) | Symmetry: [∞,3], (*∞32) | Symmetry: [∞,3], (*∞32) | Symmetry: [∞,3], (*∞32) | Symmetry: [∞,3], (*∞32) | [∞,3]$^+$ (∞32) | [1$^+$,∞,3] (*∞33) | [1$^+$,∞,3] (*∞33) | [∞,3$^+$] (3*∞) |
| | | = | = | = | | | | = or | = or | = |
| {∞,3} | t{∞,3} | r{∞,3} | t{3,∞} | {3,∞} | rr{∞,3} | tr{∞,3} | sr{∞,3} | h{∞,3} | h$_2${∞,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.
*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 |