Hexaoctagonal tiling

hexaoctagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	(6.8)2
Schläfli symbol	r{8,6} or
Wythoff symbol	2 | 8 6
Coxeter diagram
Symmetry group	[8,6], (*862)
Dual	Order-8-6 quasiregular rhombic tiling
Properties	Vertex-transitive edge-transitive

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

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)

Related polyhedra and tiling

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

Symmetry mutation of quasiregular tilings: 6.n.6.n
Symmetry *6n2 [n,6]	Euclidean	Compact hyperbolic					Paracompact	Noncompact
	*632 [3,6]	*642 [4,6]	*652 [5,6]	*662 [6,6]	*762 [7,6]	*862 [8,6]...	*∞62 [∞,6]	[iπ/λ,6]
Quasiregular figures configuration	6.3.6.3	6.4.6.4	6.5.6.5	6.6.6.6	6.7.6.7	6.8.6.8	6.∞.6.∞	6.∞.6.∞
Dual figures
Rhombic figures configuration	V6.3.6.3	V6.4.6.4	V6.5.6.5	V6.6.6.6	V6.7.6.7	V6.8.6.8	V6.∞.6.∞

Dimensional family of quasiregular polyhedra and tilings: (8.n)2
Symmetry *8n2 [n,8]	Hyperbolic...						Paracompact	Noncompact
	*832 [3,8]	*842 [4,8]	*852 [5,8]	*862 [6,8]	*872 [7,8]	*882 [8,8]...	*∞82 [∞,8]	[iπ/λ,8]
Coxeter
Quasiregular figures configuration	3.8.3.8	4.8.4.8	8.5.8.5	8.6.8.6	8.7.8.7	8.8.8.8	8.∞.8.∞	8.∞.8.∞

See also

Wikimedia Commons has media related to Uniform tiling 6-8-6-8.

• 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

Tessellation
Periodic Pythagorean Rhombille Schwarz triangle Rectangle Domino Uniform tiling & honeycomb Coloring Convex Kisrhombille Wallpaper group Wythoff
Aperiodic Ammann–Beenker Aperiodic set of prototiles List Einstein problem Socolar–Taylor Gilbert Penrose Pentagonal Pinwheel Quaquaversal Rep-tile & Self-tiling Sphinx Truchet
Other Anisohedral & Isohedral Architectonic & catoptric Circle Limit III Computer graphics Honeycomb Isotoxal Packing Problems Domino Wang Heesch's Squaring Prototile Conway criterion Girih Regular Division of the Plane Regular grid Substitution Voronoi Voderberg
By vertex type Spherical 2n 33.n V33.n 42.n V42.n Regular 2∞ 36 44 63 Semi- regular 32.4.3.4 V32.4.3.4 33.42 33.∞ 34.6 V34.6 3.4.6.4 (3.6)2 3.122 42.∞ 4.6.12 4.82 Hyper- bolic 32.4.3.5 32.4.3.6 32.4.3.7 32.4.3.8 32.4.3.∞ 32.5.3.5 32.5.3.6 32.6.3.6 32.6.3.8 32.7.3.7 32.8.3.8 33.4.3.4 32.∞.3.∞ 34.7 34.8 34.∞ 35.4 37 38 3∞ (3.4)3 (3.4)4 3.4.62.4 3.4.7.4 3.4.8.4 3.4.∞.4 3.6.4.6 (3.7)2 (3.8)2 3.142 3.162 (3.∞)2 3.∞2 42.5.4 42.6.4 42.7.4 42.8.4 42.∞.4 45 46 47 48 4∞ (4.5)2 (4.6)2 4.6.12 4.6.14 V4.6.14 4.6.16 V4.6.16 4.6.∞ (4.7)2 (4.8)2 4.8.10 V4.8.10 4.8.12 4.8.14 4.8.16 4.8.∞ 4.102 4.10.12 4.122 4.12.16 4.142 4.162 4.∞2 (4.∞)2 54 55 56 5∞ 5.4.6.4 (5.6)2 5.82 5.102 5.122 (5.∞)2 64 65 66 68 6.4.8.4 (6.8)2 6.82 6.102 6.122 6.162 73 74 77 7.62 7.82 7.142 83 84 86 88 812 8.62 8.162 ∞3 ∞4 ∞5 ∞∞ ∞.62 ∞.82