Truncated triapeirogonal tiling

File:Truncated_triapeirogonal_tiling_with_mirrors.png

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*∞).Norman W. Johnson and Asia Ivic Weiss, Quadratic Integers and Coxeter Groups, Can. J. Math. Vol. 51 (6), 1999 pp. 1307–1336 [http://cms.math.ca/cjm/v51/weisscox8.pdf] 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 [(∞,∞,∞)], (*∞∞∞).

class=wikitable |+ Small index subgroups of [∞,3], (*∞32)

align=center

align=center !Index !1 !colspan=2|2 !3 !4 !colspan=2|6 !8 !12 !24

align=center !Diagrams |80px |80px |80px |80px |80px |80px |80px |80px |80px |80px

align=center !Coxeter (orbifold) |[∞,3] {{CDD|node_c1|infin|node_c2|3|node_c2}} = {{CDD|node_c2|split1-i3|branch_c1-2|label2}} (*∞32) |[1+,∞,3] {{CDD|node_h0|infin|node_c2|3|node_c2}} = {{CDD|labelinfin|branch_c2|split2|node_c2}} (*∞33) |[∞,3+] {{CDD|node_c1|infin|node_h2|3|node_h2}} (3*∞) |[∞,∞] (*∞∞2) |[(∞,∞,3)] (*∞∞3) |[∞,3*] {{CDD|node_c1|infin|node_g|3sg|node_g}} = {{CDD|labelinfin|branch_c1|split2-ii|node_c1}} (*∞3) |[∞,1+,∞] (*(∞2)2) |[(∞,1+,∞,3)] (*(∞3)2) |[1+,∞,∞,1+] (*∞4) |[(∞,∞,3*)] (*∞6)

align=center !colspan=11|Direct subgroups

align=center !Index !2 !colspan=2|4 !6 !8 !colspan=2|12 !16 !24 !48

align=center !Diagrams |80px |colspan=2|80px |80px |80px |80px |80px |80px |80px |80px

align=center !Coxeter (orbifold) |[∞,3]+ {{CDD|node_h2|infin|node_h2|3|node_h2}} = {{CDD|node_h2|split1-i3|branch_h2h2|label2}} (∞32) |colspan=2|[∞,3+]+ {{CDD|node_h0|infin|node_h2|3|node_h2}} = {{CDD|labelinfin|branch_h2h2|split2|node_h2}} (∞33) |[∞,∞]+ (∞∞2) |[(∞,∞,3)]+ (∞∞3) |[∞,3*]+ {{CDD|node_h2|infin|node_g|3sg|node_g}} = {{CDD|labelinfin|branch_h2h2|split2-ii|node_h2}} (∞3) |[∞,1+,∞]+ (∞2)2 |[(∞,1+,∞,3)]+ (∞3)2 |[1+,∞,∞,1+]+ (∞4) |[(∞,∞,3*)]+ (∞6)

{{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}}

{{Commons category|Uniform tiling 4-6-i}}

• List of uniform planar tilings
• Tilings of regular polygons
• Uniform tilings in hyperbolic plane

{{reflist}}

{{refbegin}}

• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, {{ISBN|978-1-56881-220-5}} (Chapter 19, The Hyperbolic Archimedean Tessellations)
• {{Cite book|title=The Beauty of Geometry: Twelve Essays|year=1999|publisher=Dover Publications|lccn=99035678|isbn=0-486-40919-8|chapter=Chapter 10: Regular honeycombs in hyperbolic space}}

{{refend}}

• {{MathWorld | urlname= HyperbolicTiling | title = Hyperbolic tiling}}
• {{MathWorld | urlname=PoincareHyperbolicDisk | title = Poincaré hyperbolic disk }}

{{Tessellation}}

Category:Apeirogonal tilings

Category:Hyperbolic tilings

Category:Isogonal tilings

Category:Truncated tilings

Category:Uniform tilings