Truncated order-6 hexagonal tiling

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Truncated order-6 hexagonal tiling
Truncated order-6 hexagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 6.12.12
Schläfli symbol t{6,6} or h2{4,6}
t(6,6,3)
Wythoff symbol 2 6 | 6
3 6 6 |
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 6.pngCDel node.png = CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h1.png
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 6.pngCDel node h0.png = CDel node 1.pngCDel split1-66.pngCDel branch 11.png
Symmetry group [6,6], (*662)
[(6,6,3)], (*663)
Dual Order-6 hexakis hexagonal tiling
Properties Vertex-transitive

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[edit]

By *663 symmetry, this tiling can be constructed as an omnitruncation, t{(6,6,3)}:

H2 tiling 366-7.png

Symmetry[edit]

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 663 symmetry 000.png 663 symmetry 0a0.png 663 symmetry a0a.png 663 symmetry z0z.png
Coxeter
(orbifold)
[(6,6,3)] = CDel node c1.pngCDel split1-66.pngCDel branch c2.png
(*663)
[(6,1+,6,3)] = CDel labelh.pngCDel node.pngCDel split1-66.pngCDel branch c2.png = CDel branch c2.pngCDel 3a3b-cross.pngCDel branch c2.png
(*3333)
[(6,6,3+)] = CDel node c1.pngCDel split1-66.pngCDel branch h2h2.png
(3*33)
[(6,6,3*)] = CDel node c1.pngCDel split1-66.pngCDel branch.pngCDel labels.png
(*333333)
Direct subgroups
Index 2 4 12
Diagram 663 symmetry aaa.png 663 symmetry abc.png 663 symmetry zaz.png
Coxeter
(orbifold)
[(6,6,3)]+ = CDel node h2.pngCDel split1-66.pngCDel branch h2h2.png
(663)
[(6,6,3+)]+ = CDel labelh.pngCDel node.pngCDel split1-66.pngCDel branch h2h2.png = CDel branch h2h2.pngCDel 3a3b-cross.pngCDel branch h2h2.png
(3333)
[(6,6,3*)]+ = CDel node h2.pngCDel split1-66.pngCDel branch.pngCDel labels.png
(333333)

Related polyhedra and tiling[edit]

References[edit]

  • 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. 

See also[edit]

External links[edit]