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 symbol
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.

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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