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
Wythoff symbol 2 ∞ 3 |
Coxeter diagram or
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

edit
 
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)
edit
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

edit

References

edit
  1. ^ Norman W. Johnson and Asia Ivic Weiss, Quadratic Integers and Coxeter Groups, Can. J. Math. Vol. 51 (6), 1999 pp. 1307–1336 [1]
  • 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.
edit