Truncated order-6 octagonal tiling

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Truncated order-6 octagonal tiling

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

Truncated order-6 octagonal tiling
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Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration6.16.16
Schläfli symbolt{8,6}
Wythoff symbol2 6 | 8
Coxeter diagram
Symmetry group[8,6], (*862)
DualOrder-8 hexakis hexagonal tiling
PropertiesVertex-transitive

Uniform colorings

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

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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.

More information Index, Diagram ...
Small index subgroups of [(8,8,3)] (*883)
Index 1 2 6
Diagram Thumb Thumb Thumb Thumb
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 Thumb Thumb Thumb
Coxeter
(orbifold)
[(8,8,3)]+ =
(883)
[(8,8,3+)]+ = =
(4343)
[(8,8,3*)]+ =
(444444)
Close
More information Symmetry: [8,6], (*862), Uniform duals ...
Uniform octagonal/hexagonal tilings
Symmetry: [8,6], (*862)
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{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
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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)
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h{8,6} s{8,6} hr{8,6} s{6,8} h{6,8} hrr{8,6} sr{8,6}
Alternation duals
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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
Close

References

  • 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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