Order-6 octagonal tiling

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

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

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}^1⁄2	r(8,6,8)
Coxeter diagram		=	=

Related polyhedra and tiling

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	8^3	8^4	8^5	8^6	8^7	8^8	...8^∞

Regular tilings {n,6}
Spherical	Euclidean	Hyperbolic tilings
{2,6}	{3,6}	{4,6}	{5,6}	{6,6}	{7,6}	{8,6}	...	{∞,6}

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
V8^6	V6.16.16	V(6.8)^2	V8.12.12	V6^8	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	V3.4^5	V3.3.6.3.8

See also

• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes

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.

External links

• 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