Hexaoctagonal tiling

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 $\left\{\begin{array}{c}8\\ 6\end{array}\right\}$
Wythoff symbol	2 | 8 6
Coxeter diagram	Template:CDD
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	File:H2 tiling 268-2.png	File:H2 tiling 388-5.png	File:H2 tiling 466-5.png
Symmetry	[8,6] (*862) Template:CDD	[(8,3,8)] = [8,6,1+] (*883) Template:CDD	[(6,4,6)] = [1+,8,6] (*664) Template:CDD	[1+,8,6,1+] (*4343) Template:CDD
Symbol	r{8,6}	r{(8,3,8)}	r{(6,4,6)}
Coxeter diagram	Template:CDD	Template:CDD = Template:CDD	Template:CDD = Template:CDD	Template:CDD = Template:CDD

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].

File:862 symmetry z0z.png [1+,8,4,1+], (*4343)	File:862 symmetry b0b.png [(8,4,2+)], (2*43)

Related polyhedra and tiling

Symmetry mutation of quasiregular tilings: (6.n)2
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	File:Uniform tiling 63-t1.svg 6.3.6.3	File:H2 tiling 246-2.png 6.4.6.4	File:H2 tiling 256-2.png 6.5.6.5	File:H2 tiling 266-2.png 6.6.6.6	File:H2 tiling 267-2.png 6.7.6.7	File:H2 tiling 268-2.png 6.8.6.8	File:H2 tiling 26i-2.png 6.∞.6.∞	6.∞.6.∞
Dual figures
Rhombic figures configuration	File:Rhombic star tiling.svg V6.3.6.3	File:H2chess 246a.png V6.4.6.4	File:Order-6-5 quasiregular rhombic tiling.png V6.5.6.5	File:H2 tiling 246-4.png V6.6.6.6	V6.7.6.7	File:H2chess 268a.png V6.8.6.8	File:H2chess 26ia.png 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	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Quasiregular figures configuration	File:H2-8-3-rectified.svg 3.8.3.8	File:H2 tiling 248-2.png 4.8.4.8	File:H2 tiling 258-2.png 8.5.8.5	File:H2 tiling 268-2.png 8.6.8.6	File:H2 tiling 278-2.png 8.7.8.7	File:H2 tiling 288-2.png 8.8.8.8	File:H2 tiling 25i-2.png 8.∞.8.∞	8.∞.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-Strauss, The Symmetries of Things 2008, Template:ISBN (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

• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch