Snub tetraoctagonal tiling

Snub tetraoctagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex figure	3.3.4.3.8
Schläfli symbol	sr{8,4}
Wythoff symbol	| 8 4 2
Coxeter diagram
Symmetry group	[8,4]+, (842)
Dual	Order-8-4 floret pentagonal tiling
Properties	Vertex-transitive Chiral

In geometry, the snub tetraoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{8,4}.

Images

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

Related polyhedra and tiling

The snub tetraoctagonal tiling is seventh in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n.

Dimensional family of snub polyhedra and tilings: 3.3.4.3.n

Symmetry 4n2 [n,4]+	Spherical		Euclidean	Compact hyperbolic				Paracompact
	242 [2,4]+	342 [3,4]+	442 [4,4]+	542 [5,4]+	642 [6,4]+	742 [7,4]+	842 [8,4]+...	∞42 [∞,4]+
Snub figure	3.3.4.3.2	3.3.4.3.3	3.3.4.3.4	3.3.4.3.5	3.3.4.3.6	3.3.4.3.7	3.3.4.3.8	3.3.4.3.∞
Coxeter Schläfli	sr{2,4}	sr{3,4}	sr{4,4}	sr{5,4}	sr{6,4}	sr{7,4}	sr{8,4}	sr{∞,4}
Snub dual figure	V3.3.4.3.2	V3.3.4.3.3	V3.3.4.3.4	V3.3.4.3.5	V3.3.4.3.6	V3.3.4.3.7	V3.3.4.3.8	V3.3.4.3.∞
Coxeter

Uniform octagonal/square tilings

[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

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 3-3-4-3-8.

• 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

Tessellation & tiling   Periodic Pythagorean Rhombille Schwarz triangle Rectangle Domino Uniform tiling & tessellation Coloring Convex Kisrhombille Wallpaper group Wythoff   Aperiodic Ammann–Beenker Aperiodic set of prototiles List Einstein problem Socolar–Taylor Gilbert Penrose Pentagon 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