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
V3 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

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.
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