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Truncated order-6 octagonal tiling

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Truncated order-6 octagonal tiling
Truncated order-6 octagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 6.16.16
Schläfli symbol t{8,6}
Wythoff symbol 2 6 | 8
Coxeter diagram
Symmetry group [8,6], (*862)
Dual Order-8 hexakis hexagonal tiling
Properties Vertex-transitive

In geometry, the truncated order-6 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{8,6}.

Uniform colorings

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A secondary construction t{(8,8,3)} is called a truncated trioctaoctagonal tiling:

Symmetry

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Truncated order-6 octagonal tiling with mirror lines,

The dual to this tiling represent the fundamental domains of [(8,8,3)] (*883) symmetry. There are 3 small index subgroup symmetries constructed from [(8,8,3)] 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 862 symmetry by adding a mirror bisecting the fundamental domain.

Small index subgroups of [(8,8,3)] (*883)
Index 1 2 6
Diagram
Coxeter
(orbifold) 		[(8,8,3)] =
(*883) 		[(8,1+,8,3)] = =
(*4343) 		[(8,8,3+)] =
(3*44) 		[(8,8,3*)] =
(*444444)
Direct subgroups
Index 2 4 12
Diagram
Coxeter
(orbifold) 		[(8,8,3)]+ =
(883) 		[(8,8,3+)]+ = =
(4343) 		[(8,8,3*)]+ =
(444444)
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Uniform octagonal/hexagonal tilings
Symmetry: [8,6], (*862)
{8,6} t{8,6}
 r{8,6} 2t{8,6}=t{6,8} 2r{8,6}={6,8} rr{8,6} tr{8,6}
Uniform duals
V86 V6.16.16 V(6.8)2 V8.12.12 V68 V4.6.4.8 V4.12.16
Alternations
[1+,8,6]
(*466) 		[8+,6]
(8*3) 		[8,1+,6]
(*4232) 		[8,6+]
(6*4) 		[8,6,1+]
(*883) 		[(8,6,2+)]
(2*43) 		[8,6]+
(862)
h{8,6} s{8,6} hr{8,6} s{6,8} h{6,8} hrr{8,6} sr{8,6}
Alternation duals
V(4.6)6 V3.3.8.3.8.3 V(3.4.4.4)2 V3.4.3.4.3.6 V(3.8)8 V3.45 V3.3.6.3.8

References

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  • 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.

See also

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Wikimedia Commons has media related to Uniform tiling 6-16-16.
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