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}.

Symmetry

Truncated triapeirogonal tiling with mirrors

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 [(∞,∞,∞)], (*∞∞∞).

Small index subgroups of [∞,3], (*∞32)
Index 1 2 3 4 6 8 12 24
Diagrams I32 symmetry mirrors.png I32 symmetry a00.png I32 symmetry 0bb.png I32 symmetry mirrors-index3.png I32 symmetry mirrors-index4a.png I32 symmetry 0zz.png I32 symmetry mirrors-index6-i2i2.png I32 symmetry mirrors-index8a.png I32 symmetry mirrors-index12a.png I32 symmetry mirrors-index24a.png
Coxeter
(orbifold) 		[∞,3]
CDel node c1.pngCDel infin.pngCDel node c2.pngCDel 3.pngCDel node c2.png = CDel node c2.pngCDel split1-i3.pngCDel branch c1-2.pngCDel label2.png
(*∞32) 		[1+,∞,3]
CDel node h0.pngCDel infin.pngCDel node c2.pngCDel 3.pngCDel node c2.png = CDel labelinfin.pngCDel branch c2.pngCDel split2.pngCDel node c2.png
(*∞33) 		[∞,3+]
CDel node c1.pngCDel infin.pngCDel node h2.pngCDel 3.pngCDel node h2.png
(3*∞) 		[∞,∞]

(*∞∞2) 		[(∞,∞,3)]

(*∞∞3) 		[∞,3*]
CDel node c1.pngCDel infin.pngCDel node g.pngCDel 3sg.pngCDel node g.png = CDel labelinfin.pngCDel branch c1.pngCDel split2-ii.pngCDel node c1.png
(*∞3) 		[∞,1+,∞]

(*(∞2)2) 		[(∞,1+,∞,3)]

(*(∞3)2) 		[1+,∞,∞,1+]

(*∞4) 		[(∞,∞,3*)]

(*∞6)
Direct subgroups
Index 2 4 6 8 12 16 24 48
Diagrams I32 symmetry aaa.png I32 symmetry abb.png Ii2 symmetry aaa.png I32 symmetry mirrors-index4.png I32 symmetry azz.png Ii2 symmetry bab.png H2chess 26ia.png Ii2 symmetry abc.png H2chess 26ib.png
Coxeter
(orbifold) 		[∞,3]+
CDel node h2.pngCDel infin.pngCDel node h2.pngCDel 3.pngCDel node h2.png = CDel node h2.pngCDel split1-i3.pngCDel branch h2h2.pngCDel label2.png
(∞32) 		[∞,3+]+
CDel node h0.pngCDel infin.pngCDel node h2.pngCDel 3.pngCDel node h2.png = CDel labelinfin.pngCDel branch h2h2.pngCDel split2.pngCDel node h2.png
(∞33) 		[∞,∞]+

(∞∞2) 		[(∞,∞,3)]+

(∞∞3) 		[∞,3*]+
CDel node h2.pngCDel infin.pngCDel node g.pngCDel 3sg.pngCDel node g.png = CDel labelinfin.pngCDel branch h2h2.pngCDel split2-ii.pngCDel node h2.png
(∞3) 		[∞,1+,∞]+

(∞2)2 		[(∞,1+,∞,3)]+

(∞3)2 		[1+,∞,∞,1+]+

(∞4) 		[(∞,∞,3*)]+

(∞6)

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 CDel node 1.pngCDel p.pngCDel node 1.pngCDel 3.pngCDel node 1.png. 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.

See also

References

  1. ↑ Norman W. Johnson and Asia Ivic Weiss, Quadratic Integers and Coxeter Groups, Can. J. Math. Vol. 51 (6), 1999 pp. 1307–1336 [1]
  • 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)
  • "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. 

External links



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