In geometry , the truncated order-6 hexagonal tiling is a uniform tiling of the hyperbolic plane . It has Schläfli symbol of t{6,6}. It can also be identically constructed as a cantic order-6 square tiling , h2 {4,6}
Uniform colorings By *663 symmetry, this tiling can be constructed as an omnitruncation , t{(6,6,3)}:
File:H2 tiling 366-7.png Symmetry File:Truncated order-6 hexagonal tiling with mirrors.png Truncated order-6 hexagonal tiling with *663 mirror lines The dual to this tiling represent the fundamental domains of [(6,6,3)] (*663) symmetry. There are 3 small index subgroup symmetries constructed from [(6,6,3)] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
The symmetry can be doubled as 662 symmetry by adding a mirror bisecting the fundamental domain.
Small index subgroups of [(6,6,3)] (*663)
Index
1
2
6
Diagram
File:663 symmetry 000.png File:663 symmetry 0a0.png File:663 symmetry a0a.png File:663 symmetry z0z.png
Coxeter orbifold )
[(6,6,3)] = File:CDel node c1.png File:CDel split1-66.png File:CDel branch c2.png
[(6,1+ ,6,3)] = File:CDel labelh.png File:CDel node.png File:CDel split1-66.png File:CDel branch c2.png File:CDel branch c2.png File:CDel 3a3b-cross.png File:CDel branch c2.png *3333 )
[(6,6,3+ )] = File:CDel node c1.png File:CDel split1-66.png File:CDel branch h2h2.png
[(6,6,3*)] = File:CDel node c1.png File:CDel split1-66.png File:CDel branch.png File:CDel labels.png *333333 )
Direct subgroups
Index
2
4
12
Diagram
File:663 symmetry aaa.png File:663 symmetry abc.png File:663 symmetry zaz.png
Coxeter
[(6,6,3)]+ = File:CDel node h2.png File:CDel split1-66.png File:CDel branch h2h2.png
[(6,6,3+ )]+ = File:CDel labelh.png File:CDel node.png File:CDel split1-66.png File:CDel branch h2h2.png File:CDel branch h2h2.png File:CDel 3a3b-cross.png File:CDel branch h2h2.png
[(6,6,3*)]+ = File:CDel node h2.png File:CDel split1-66.png File:CDel branch.png File:CDel labels.png
Related polyhedra and tiling
Uniform hexahexagonal tilings
Symmetry: [6,6], (*662)
File:CDel node 1.png File:CDel 6.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel nodes 10ru.png File:CDel split2-66.png File:CDel node.png File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel node 1.png File:CDel 6.png File:CDel node 1.png File:CDel 6.png File:CDel node.png File:CDel nodes 10ru.png File:CDel split2-66.png File:CDel node 1.png File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 6.png File:CDel node 1.png File:CDel node.png File:CDel 6.png File:CDel node 1.png File:CDel 6.png File:CDel node.png File:CDel nodes.png File:CDel split2-66.png File:CDel node 1.png File:CDel node h0.png File:CDel 4.png File:CDel node.png File:CDel 6.png File:CDel node 1.png File:CDel node.png File:CDel 6.png File:CDel node 1.png File:CDel 6.png File:CDel node 1.png File:CDel nodes 01rd.png File:CDel split2-66.png File:CDel node 1.png File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 6.png File:CDel node 1.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel 6.png File:CDel node 1.png File:CDel nodes 01rd.png File:CDel split2-66.png File:CDel node.png File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel node 1.png File:CDel 6.png File:CDel node.png File:CDel 6.png File:CDel node 1.png File:CDel nodes 11.png File:CDel split2-66.png File:CDel node.png File:CDel node h0.png File:CDel 4.png File:CDel node 1.png File:CDel 6.png File:CDel node.png File:CDel node 1.png File:CDel 6.png File:CDel node 1.png File:CDel 6.png File:CDel node 1.png File:CDel nodes 11.png File:CDel split2-66.png File:CDel node 1.png File:CDel node h0.png File:CDel 4.png File:CDel node 1.png File:CDel 6.png File:CDel node 1.png
File:H2 tiling 266-1.png File:H2 tiling 266-3.png File:H2 tiling 266-2.png File:H2 tiling 266-6.png File:H2 tiling 266-4.png File:H2 tiling 266-5.png File:H2 tiling 266-7.png
{6,6} t{6,6} 2 {4,6}
r{6,6} t{6,6} 2 {4,6}
{6,6} rr{6,6} tr{6,6}
Uniform duals
File:CDel node f1.png File:CDel 6.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel node f1.png File:CDel 6.png File:CDel node f1.png File:CDel 6.png File:CDel node.png File:CDel node.png File:CDel 6.png File:CDel node f1.png File:CDel 6.png File:CDel node.png File:CDel node.png File:CDel 6.png File:CDel node f1.png File:CDel 6.png File:CDel node f1.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel 6.png File:CDel node f1.png File:CDel node f1.png File:CDel 6.png File:CDel node.png File:CDel 6.png File:CDel node f1.png File:CDel node f1.png File:CDel 6.png File:CDel node f1.png File:CDel 6.png File:CDel node f1.png
File:H2chess 266b.png File:H2chess 266f.png File:H2chess 266a.png File:H2chess 266e.png File:H2chess 266c.png File:H2chess 266d.png File:H2checkers 266.png
V66
V6.12.12
V6.6.6.6
V6.12.12
V66
V4.6.4.6
V4.12.12
Alternations
[1+ ,6,6]
[6+ ,6]
[6,1+ ,6]
[6,6+ ]
[6,6,1+ ]
[(6,6,2+ )]
[6,6]+
File:CDel node h1.png File:CDel 6.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel branch 10ru.png File:CDel split2-66.png File:CDel node.png File:CDel node h.png File:CDel 6.png File:CDel node h.png File:CDel 6.png File:CDel node.png File:CDel node.png File:CDel 6.png File:CDel node h1.png File:CDel 6.png File:CDel node.png File:CDel nodes 11.png File:CDel 3a3b-cross.png File:CDel nodes.png File:CDel node.png File:CDel 6.png File:CDel node h.png File:CDel 6.png File:CDel node h.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel 6.png File:CDel node h1.png File:CDel node.png File:CDel split1-66.png File:CDel branch 01ld.png File:CDel node h.png File:CDel 6.png File:CDel node.png File:CDel 6.png File:CDel node h.png File:CDel node h.png File:CDel 6.png File:CDel node h.png File:CDel 6.png File:CDel node h.png
File:CDel node h1.png File:CDel 6.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel node h.png File:CDel 6.png File:CDel node h.png File:CDel 6.png File:CDel node.png File:CDel node.png File:CDel 6.png File:CDel node h1.png File:CDel 6.png File:CDel node.png File:CDel node.png File:CDel 6.png File:CDel node h.png File:CDel 6.png File:CDel node h.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel 6.png File:CDel node h1.png File:CDel node h.png File:CDel 6.png File:CDel node.png File:CDel 6.png File:CDel node h.png File:CDel node h.png File:CDel 6.png File:CDel node h.png File:CDel 6.png File:CDel node h.png
File:Uniform tiling 66-h0.png
File:Uniform tiling verf 34343434.png
File:Uniform tiling 66-h0.png File:Uniform tiling 64-h1.png File:Uniform tiling 66-snub.png
h{6,6}
s{6,6}
hr{6,6}
s{6,6}
h{6,6}
hrr{6,6}
sr{6,6}
References See also External links
