Snub heptaheptagonal tiling

Snub heptaheptagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	3.3.7.3.7
Schläfli symbol	sr{7,7} or $s{\begin{Bmatrix}7\\7\end{Bmatrix}}$
Wythoff symbol	\| 7 7 2
Coxeter diagram
Symmetry group	[7,7]⁺, (772) [7⁺,4], (7*2)
Dual	Order-7-7 floret pentagonal tiling
Properties	Vertex-transitive

In geometry, the snub heptaheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{7,7}, constructed from two regular heptagons 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 [7,4] symmetry with only one color heptagon.

Related tilings

Uniform heptaheptagonal tilings
Symmetry: [7,7], (*772)							[7,7]⁺, (772)
= =	= =	= =	= =	= =	= =	= =	= =

{7,7}	t{7,7}	r{7,7}	2t{7,7}=t{7,7}	2r{7,7}={7,7}	rr{7,7}	tr{7,7}	sr{7,7}
Uniform duals


V7⁷	V7.14.14	V7.7.7.7	V7.14.14	V7⁷	V4.7.4.7	V4.14.14	V3.3.7.3.7

Uniform heptagonal/square tilings
Symmetry: [7,4], (*742)							[7,4]⁺, (742)	[7⁺,4], (7*2)	[7,4,1⁺], (*772)


{7,4}	t{7,4}	r{7,4}	2t{7,4}=t{4,7}	2r{7,4}={4,7}	rr{7,4}	tr{7,4}	sr{7,4}	s{7,4}	h{4,7}
Uniform duals


V7⁴	V4.14.14	V4.7.4.7	V7.8.8	V4⁷	V4.4.7.4	V4.8.14	V3.3.4.3.7	V3.3.7.3.7	V7⁷

4n2 symmetry mutations of snub tilings: 3.3.n.3.n
Symmetry 4n2	Spherical		Euclidean	Compact hyperbolic				Paracompact
Symmetry 4n2	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.

External links

Weisstein, Eric W. "Hyperbolic tiling". MathWorld.

Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.

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

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