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In geometry, the truncated triapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of tr{∞,3}.
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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 [(∞,∞,∞)], (*∞∞∞).
Related polyhedra and tiling
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.