Truncated triapeirogonal tiling

Truncated triapeirogonal tiling
Truncated triapeirogonal tiling Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	4.6.∞
Schläfli symbol	tr{∞,3} or $t\left\{\begin{array}{c}\mathrm{\infty }\\ 3\end{array}\right\}$
Wythoff symbol	2 ∞ 3 |
Coxeter diagram	File:CDel node 1.pngFile:CDel infin.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png or File:CDel node 1.pngFile:CDel split1-i3.pngFile:CDel nodes 11.png
Symmetry group	[∞,3], (*∞32)
Dual	Order 3-infinite kisrhombille
Properties	Vertex-transitive

In geometry, the truncated triapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of tr{∞,3}.

Symmetry

File:Truncated triapeirogonal tiling with mirrors.png

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*∞). 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	File:I32 symmetry mirrors.png	File:I32 symmetry a00.png	File:I32 symmetry 0bb.png	File:I32 symmetry mirrors-index3.png	File:I32 symmetry mirrors-index4a.png	File:I32 symmetry 0zz.png	File:I32 symmetry mirrors-index6-i2i2.png	File:I32 symmetry mirrors-index8a.png	File:I32 symmetry mirrors-index12a.png	File:I32 symmetry mirrors-index24a.png
Coxeter (orbifold)	[∞,3] File:CDel node c1.pngFile:CDel infin.pngFile:CDel node c2.pngFile:CDel 3.pngFile:CDel node c2.png = File:CDel node c2.pngFile:CDel split1-i3.pngFile:CDel branch c1-2.pngFile:CDel label2.png (*∞32)	[1+,∞,3] File:CDel node h0.pngFile:CDel infin.pngFile:CDel node c2.pngFile:CDel 3.pngFile:CDel node c2.png = File:CDel labelinfin.pngFile:CDel branch c2.pngFile:CDel split2.pngFile:CDel node c2.png (*∞33)	[∞,3+] File:CDel node c1.pngFile:CDel infin.pngFile:CDel node h2.pngFile:CDel 3.pngFile:CDel node h2.png (3*∞)	[∞,∞] (*∞∞2)	[(∞,∞,3)] (*∞∞3)	[∞,3*] File:CDel node c1.pngFile:CDel infin.pngFile:CDel node g.pngFile:CDel 3sg.pngFile:CDel node g.png = File:CDel labelinfin.pngFile:CDel branch c1.pngFile:CDel split2-ii.pngFile:CDel 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	File:I32 symmetry aaa.png	File:I32 symmetry abb.png		File:Ii2 symmetry aaa.png	File:I32 symmetry mirrors-index4.png	File:I32 symmetry azz.png	File:Ii2 symmetry bab.png	File:H2chess 26ia.png	File:Ii2 symmetry abc.png	File:H2chess 26ib.png
Coxeter (orbifold)	[∞,3]+ File:CDel node h2.pngFile:CDel infin.pngFile:CDel node h2.pngFile:CDel 3.pngFile:CDel node h2.png = File:CDel node h2.pngFile:CDel split1-i3.pngFile:CDel branch h2h2.pngFile:CDel label2.png (∞32)	[∞,3+]+ File:CDel node h0.pngFile:CDel infin.pngFile:CDel node h2.pngFile:CDel 3.pngFile:CDel node h2.png = File:CDel labelinfin.pngFile:CDel branch h2h2.pngFile:CDel split2.pngFile:CDel node h2.png (∞33)		[∞,∞]+ (∞∞2)	[(∞,∞,3)]+ (∞∞3)	[∞,3*]+ File:CDel node h2.pngFile:CDel infin.pngFile:CDel node g.pngFile:CDel 3sg.pngFile:CDel node g.png = File:CDel labelinfin.pngFile:CDel branch h2h2.pngFile:CDel split2-ii.pngFile:CDel 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 File:CDel node 1.pngFile:CDel p.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel 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

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

References

• 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. LCCN 99035678.

External links

• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.