Snub pentapentagonal tiling

Snub pentapentagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	3.3.5.3.5
Schläfli symbol	s{5,4} sr{5,5}
Wythoff symbol	| 5 5 2
Coxeter diagram	or
Symmetry group	[5+,4], (5*2) [5,5]+, (552)
Dual	Order-5-5 floret pentagonal tiling
Properties	Vertex-transitive

In geometry, the snub pentapentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of sr{5,5}, constructed from two regular pentagons and three equilateral triangles around every vertex.

Images

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

Symmetry

A double symmetry coloring can be constructed from [5,4] symmetry with only one color pentagon. It has Schläfli symbol s{5,4}, and Coxeter diagram .

Related tilings

Uniform pentapentagonal tilings
Symmetry: [5,5], (*552)							[5,5]+, (552)
=	=	=	=	=	=	=	=
{5,5}	t{5,5}	r{5,5}	2t{5,5}=t{5,5}	2r{5,5}={5,5}	rr{5,5}	tr{5,5}	sr{5,5}
Uniform duals
V5.5.5.5.5	V5.10.10	V5.5.5.5	V5.10.10	V5.5.5.5.5	V4.5.4.5	V4.10.10	V3.3.5.3.5

Uniform pentagonal/square tilings
Symmetry: [5,4], (*542)							[5,4]+, (542)	[5+,4], (5*2)	[5,4,1+], (*552)
{5,4}	t{5,4}	r{5,4}	2t{5,4}=t{4,5}	2r{5,4}={4,5}	rr{5,4}	tr{5,4}	sr{5,4}	s{5,4}	h{4,5}
Uniform duals
V54	V4.10.10	V4.5.4.5	V5.8.8	V45	V4.4.5.4	V4.8.10	V3.3.4.3.5	V3.3.5.3.5	V55

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.∞

See also

Wikimedia Commons has media related to Uniform tiling 3-3-5-3-5.

• Square tiling
• Uniform tilings in hyperbolic plane
• 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