Truncated triapeirogonal tiling
Truncated triapeirogonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.6.∞
Schläfli symboltr{∞,3} or
Wythoff symbol2 ∞ 3 |
Coxeter diagram or
Symmetry group[∞,3], (*∞32)
DualOrder 3-infinite kisrhombille
PropertiesVertex-transitive

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
Coxeter
(orbifold)		 [∞,3]
=
(*∞32)		 [1+,∞,3]
=
(*∞33)		 [∞,3+]

(3*∞)		 [∞,∞]

(*∞∞2)		 [(∞,∞,3)]

(*∞∞3)		 [∞,3*]
=
(*∞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
Coxeter
(orbifold)		 [∞,3]+
=
(∞32)		 [∞,3+]+
=
(∞33)		 [∞,∞]+

(∞∞2)		 [(∞,∞,3)]+

(∞∞3)		 [∞,3*]+
=
(∞3)		 [∞,1+,∞]+

(∞2)2		 [(∞,1+,∞,3)]+

(∞3)2		 [1+,∞,∞,1+]+

(∞4)		 [(∞,∞,3*)]+

(∞6)
Paracompact uniform tilings in [∞,3] family
Symmetry: [∞,3], (*∞32) [∞,3]+
(∞32)		 [1+,∞,3]
(*∞33)		 [∞,3+]
(3*∞)

=
=
= 		=
or 		=
or
=
{∞,3} t{∞,3} r{∞,3} t{3,∞} {3,∞} rr{∞,3} tr{∞,3} sr{∞,3} h{∞,3} h2{∞,3} s{3,∞}
Uniform duals
V∞3 V3.∞.∞ V(3.∞)2 V6.6.∞ V3∞ V4.3.4.∞ V4.6.∞ V3.3.3.3.∞ V(3.∞)3 V3.3.3.3.3.∞

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.

*n32 symmetry mutation of omnitruncated tilings: 4.6.2n
Sym.
*n32
[n,3]		 Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*232
[2,3]		 *332
[3,3]		 *432
[4,3]		 *532
[5,3]		 *632
[6,3]		 *732
[7,3]		 *832
[8,3]		 *∞32
[∞,3]		  
[12i,3]		  
[9i,3]		  
[6i,3]		  
[3i,3]
Figures
Config. 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞ 4.6.24i 4.6.18i 4.6.12i 4.6.6i
Duals
Config. V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞ V4.6.24i V4.6.18i V4.6.12i V4.6.6i

See also

Wikimedia Commons has media related to Uniform tiling 4-6-i.

References

  1. ↑ Norman W. Johnson and Asia Ivic Weiss, Quadratic Integers and Coxeter Groups, Can. J. Math. Vol. 51 (6), 1999 pp. 1307–1336
  • 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.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.