| 词条 | Truncated triapeirogonal tiling | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 释义 |
In geometry, the truncated triapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of tr{∞,3}. SymmetryThe 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 [(∞,∞,∞)], (*∞∞∞).
Related polyhedra and tiling{{Order i-3 tiling table}}This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram {{CDD|node_1|p|node_1|3|node_1}}. 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. {{Omnitruncated table}}See also{{Commons category|Uniform tiling 4-6-i}}
References1. ^Norman W. Johnson and Asia Ivic Weiss, Quadratic Integers and Coxeter Groups, Can. J. Math. Vol. 51 (6), 1999 pp. 1307–1336 {{refbegin}}
External links
5 : Apeirogonal tilings|Hyperbolic tilings|Isogonal tilings|Truncated tilings|Uniform tilings |
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