Snub octaoctagonal tiling

Snub octaoctagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	3.3.8.3.8
Schläfli symbol	s{8,4} sr{8,8}
Wythoff symbol	| 8 8 2
Coxeter diagram	or
Symmetry group	[8,8]+, (882) [8+,4], (8*2)
Dual
Properties	Vertex-transitive

In geometry, the snub octaoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{8,8}.

Images[]

Drawn in chiral pairs, with edges missing between black triangles:

Symmetry[]

A higher symmetry coloring can be constructed from [8,4] symmetry as s{8,4}, . In this construction there is only one color of octagon.

Related polyhedra and tiling[]

Uniform octaoctagonal tilings v
Symmetry: [8,8], (*882)
= =	= =	= =	= =	= =	= =	= =
{8,8}	t{8,8}	r{8,8}	2t{8,8}=t{8,8}	2r{8,8}={8,8}	rr{8,8}	tr{8,8}
Uniform duals
V88	V8.16.16	V8.8.8.8	V8.16.16	V88	V4.8.4.8	V4.16.16
Alternations
[1+,8,8] (*884)	[8+,8] (8*4)	[8,1+,8] (*4242)	[8,8+] (8*4)	[8,8,1+] (*884)	[(8,8,2+)] (2*44)	[8,8]+ (882)
=		=		=	= =	= =
h{8,8}	s{8,8}	hr{8,8}	s{8,8}	h{8,8}	hrr{8,8}	sr{8,8}
Alternation duals
V(4.8)8	V3.4.3.8.3.8	V(4.4)4	V3.4.3.8.3.8	V(4.8)8	V46	V3.3.8.3.8

Uniform octagonal/square tilings v
[8,4], (*842) (with [8,8] (*882), [(4,4,4)] (*444) , [∞,4,∞] (*4222) index 2 subsymmetries) (And [(∞,4,∞,4)] (*4242) index 4 subsymmetry)
= = =	=	= = =	=	= =	=
{8,4}	t{8,4}	r{8,4}	2t{8,4}=t{4,8}	2r{8,4}={4,8}	rr{8,4}	tr{8,4}
Uniform duals
V84	V4.16.16	V(4.8)2	V8.8.8	V48	V4.4.4.8	V4.8.16
Alternations
[1+,8,4] (*444)	[8+,4] (8*2)	[8,1+,4] (*4222)	[8,4+] (4*4)	[8,4,1+] (*882)	[(8,4,2+)] (2*42)	[8,4]+ (842)
=	=	=	=	=	=
h{8,4}	s{8,4}	hr{8,4}	s{4,8}	h{4,8}	hrr{8,4}	sr{8,4}
Alternation duals
V(4.4)4	V3.(3.8)2	V(4.4.4)2	V(3.4)3	V88	V4.44	V3.3.4.3.8

4n2 symmetry mutations of snub tilings: 3.3.n.3.n
Symmetry 4n2	Spherical		Euclidean	Compact hyperbolic				Paracompact
	222	322	442	552	662	772	882	∞∞2
Snub figures
Config.	3.3.2.3.2	3.3.3.3.3	3.3.4.3.4	3.3.5.3.5	3.3.6.3.6	3.3.7.3.7	3.3.8.3.8	3.3.∞.3.∞
Gyro figures
Config.	V3.3.2.3.2	V3.3.3.3.3	V3.3.4.3.4	V3.3.5.3.5	V3.3.6.3.6	V3.3.7.3.7	V3.3.8.3.8	V3.3.∞.3.∞

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.

See also[]

Wikimedia Commons has media related to Uniform tiling 3-3-8-3-8.

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

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