Order-8 hexagonal tiling

Order-8 hexagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic regular tiling
Vertex configuration	68
Schläfli symbol	{6,8}
Wythoff symbol	8 | 6 2
Coxeter diagram
Symmetry group	[8,6], (*862)
Dual	Order-6 octagonal tiling
Properties	Vertex-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}.

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		=	=

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 v t e
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

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

• Uniform tilings in hyperbolic plane
• List of regular polytopes

• 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.

• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch