In geometry, the truncated triapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of tr.
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<sup>+</sup>)], (3*∞).[1] Given [∞,3] with generating mirrors, then its index 4 subgroup has generators .
An index 6 subgroup constructed as [∞,3*], becomes [(∞,∞,∞)], (*∞∞∞).
| Index | 1 | 2 | 3 | 4 | 6 | 8 | 12 | 24 | |||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Diagrams | |||||||||||
| Coxeter (orbifold) | [∞,3] = (*∞32) | [1<sup>+</sup>,∞,3] = (
| [∞,3<sup>+</sup>] (3*∞) | [∞,∞] (
| [(∞,∞,3)] (
| [∞,3*] = (
| [∞,1<sup>+</sup>,∞] (*(∞2)2) | [(∞,1<sup>+</sup>,∞,3)] (*(∞3)2) | [1<sup>+</sup>,∞,∞,1<sup>+</sup>] (*∞4) | [(∞,∞,3*)] (*∞6) | |
| Direct subgroups | |||||||||||
| Index | 2 | 4 | 6 | 8 | 12 | 16 | 24 | 48 | |||
| Diagrams | |||||||||||
| Coxeter (orbifold) | [∞,3]+ = (∞32) | [∞,3<sup>+</sup>]+ = (∞33) | [∞,∞]+ (∞∞2) | [(∞,∞,3)]+ (∞∞3) | [∞,3*]+ = (∞3) | [∞,1<sup>+</sup>,∞]+ (∞2)2 | [(∞,1<sup>+</sup>,∞,3)]+ (∞3)2 | [1<sup>+</sup>,∞,∞,1<sup>+</sup>]+ (∞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.