Order-8 hexagonal tiling
Order-8 hexagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex configuration68
Schläfli symbol{6,8}
Wythoff symbol8 | 6 2
Coxeter diagram
Symmetry group[8,6], (*862)
DualOrder-6 octagonal tiling
PropertiesVertex-transitive, edge-transitive, face-transitive

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

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.

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

See also

Wikimedia Commons has media related to Order-8 hexagonal tiling.

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.
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