Hexaoctagonal tiling

In geometry, the hexaoctagonal tiling is a uniform tiling of the hyperbolic plane.

hexaoctagonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration(6.8)2
Schläfli symbolr{8,6} or
Wythoff symbol8 6
Coxeter diagram
Symmetry group[8,6], (*862)
DualOrder-8-6 quasiregular rhombic tiling
PropertiesVertex-transitive edge-transitive

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 4 points, [8,6,1+], gives [(8,8,3)], (*883). Removing the mirror between the order 2 and 8 points, [1+,8,6], gives [(4,6,6)], (*664). Removing two mirrors as [8,1+,6,1+], leaves remaining mirrors (*4343).

Four uniform constructions of 6.8.6.8
Uniform
Coloring
Symmetry [8,6]
(*862)
[(8,3,8)] = [8,6,1+]
(*883)
[(6,4,6)] = [1+,8,6]
(*664)
[1+,8,6,1+]
(*4343)
Symbol r{8,6} r{(8,3,8)} r{(6,4,6)}
Coxeter
diagram
= = =

Symmetry

The dual tiling has face configuration V6.8.6.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4343), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*43) orbifold. These are subsymmetries of [8,6].


[1+,8,4,1+], (*4343)

[(8,4,2+)], (2*43)

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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