Truncated triapeirogonal tiling

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

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

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)

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.

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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
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, 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.
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