Order-6 octagonal tiling

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

Order-6 octagonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex configuration86
Schläfli symbol{8,6}
Wythoff symbol8 2
Coxeter diagram
Symmetry group[8,6], (*862)
DualOrder-8 hexagonal tiling
PropertiesVertex-transitive, edge-transitive, face-transitive

Symmetry

This tiling represents a hyperbolic kaleidoscope of 8 mirrors meeting at a point and bounding regular octagon fundamental domains. This symmetry by orbifold notation is called *33333333 with 8 order-3 mirror intersections. In Coxeter notation can be represented as [8*,6], removing two of three mirrors (passing through the octagon center) in the [8,6] symmetry.

Uniform constructions

There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,6] kaleidoscope. Removing the mirror between the order 2 and 6 points, [8,6,1+], gives [(8,8,3)], (*883). Removing two mirrors as [8,6*], leaves remaining mirrors (*444444).

Four uniform constructions of 8.8.8.8
Uniform
Coloring
Symmetry [8,6]
(*862)
[8,6,1+] = [(8,8,3)]
(*883)
=
[8,1+,6]
(*4232)
=
[8,6*]
(*444444)
Symbol {8,6} {8,6}12 r(8,6,8)
Coxeter
diagram
= =

This tiling is topologically related as a part of sequence of regular tilings with octagonal faces, starting with the octagonal tiling, with Schläfli symbol {8,n}, and Coxeter diagram , progressing to infinity.

n82 symmetry mutations of regular tilings: 8n
Space Spherical Compact hyperbolic Paracompact
Tiling
Config. 8.8 83 84 85 86 87 88 ...8

See also

References

  • 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.
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