Order-8 hexagonal tiling

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

Order-8 hexagonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex configuration68
Schläfli symbol{6,8}
Wythoff symbol6 2
Coxeter diagram
Symmetry group[8,6], (*862)
DualOrder-6 octagonal tiling
PropertiesVertex-transitive, edge-transitive, face-transitive

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, [6,8,1+], gives [(6,6,4)], (*664). Removing the mirror between the order 8 and 6 points, [6,1+,8], gives (*4232). Removing two mirrors as [6,8*], leaves remaining mirrors (*33333333).

Four uniform constructions of 6.6.6.6.6.6.6.6
Uniform
Coloring
Symmetry [6,8]
(*862)
 [6,8,1+] = [(6,6,4)]
(*664)
= 		[6,1+,8]
(*4232)
= 		[6,8*]
(*33333333)
Symbol {6,8} {6,8}​1⁄2 r(8,6,8) {6,8}​1⁄8
Coxeter
diagram		 = =

Symmetry

This tiling represents a hyperbolic kaleidoscope of 4 mirrors meeting as edges of a square, with eight squares around every vertex. This symmetry by orbifold notation is called (*444444) with 6 order-4 mirror intersections. In Coxeter notation can be represented as [8,6*], removing two of three mirrors (passing through the square center) in the [8,6] symmetry.

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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