Snub tetraoctagonal tiling

Snub tetraoctagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	3.3.4.3.8
Schläfli symbol	sr{8,4} or $s{\begin{Bmatrix}8\\4\end{Bmatrix}}$
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:

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

4n2 symmetry mutations of snub tilings: 3.3.4.3.n
Symmetry 4n2	Spherical		Euclidean	Compact hyperbolic				Paracomp.
Symmetry 4n2	242	342	442	542	642	742	842	∞42
Snub figures
Config.	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.∞
Gyro figures
Config.	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.∞

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

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.

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

This article is issued from Wikipedia - version of the 2/10/2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.