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Truncated triapeirogonal tiling
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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}.
Truncated triapeirogonal tiling | |
---|---|
![]() Poincaré disk model of the hyperbolic plane | |
Type | Hyperbolic uniform tiling |
Vertex configuration | 4.6.∞ |
Schläfli symbol | tr{∞,3} or |
Wythoff symbol | 2 ∞ 3 | |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Symmetry group | [∞,3], (*∞32) |
Dual | Order 3-infinite kisrhombille |
Properties | Vertex-transitive |
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Symmetry
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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*∞).[1] 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 [(∞,∞,∞)], (*∞∞∞).
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Related polyhedra and tiling
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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.
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See also
Wikimedia Commons has media related to Uniform tiling 4-6-i.
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