Snub apeiroapeirogonal tiling

Snub apeiroapeirogonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	3.3.∞.3.∞
Schläfli symbol	s{∞,4} sr{∞,∞} or $s{\begin{Bmatrix}\infty \\\infty \end{Bmatrix}}$
Wythoff symbol	\| ∞ ∞ 2
Coxeter diagram	or
Symmetry group	, (∞∞2)
Dual	Infinitely-infinite-order floret pentagonal tiling
Properties	Vertex-transitive Chiral

In geometry, the snub apeiroapeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of s{∞,∞}. It has 3 equilateral triangles and 2 apeirogons around every vertex, with vertex figure 3.3.∞.3.∞.

Dual tiling

Paracompact uniform tilings in family v t e
= =	= =	= =	= =	= =	=	=

{∞,∞}	t{∞,∞}	r{∞,∞}	2t{∞,∞}=t{∞,∞}	2r{∞,∞}={∞,∞}	rr{∞,∞}	tr{∞,∞}
Dual tilings


V∞	V∞.∞.∞	V(∞.∞)	V∞.∞.∞	V∞	V4.∞.4.∞	V4.4.∞
Alternations
(*∞∞2)	(∞*∞)	(*∞∞∞∞)	(∞*∞)	(*∞∞2)	(2*∞∞)	(2∞∞)


h{∞,∞}	s{∞,∞}	hr{∞,∞}	s{∞,∞}	h₂{∞,∞}	hrr{∞,∞}	sr{∞,∞}
Alternation duals


V(∞.∞)	V(3.∞)	V(∞.4)	V(3.∞)	V∞	V(4.∞.4)	V3.3.∞.3.∞

The snub tetrapeirogonal tiling is last in an infinite series of snub polyhedra and tilings with vertex figure 3.3.n.3.n.

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.∞

John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, 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 and honeycomb Coloring Convex Kisrhombille Wallpaper group Wythoff

Aperiodic
Ammann–Beenker Aperiodic set of prototiles List Einstein problem Socolar–Taylor Gilbert Penrose Pinwheel Quaquaversal Rep-tile and Self-tiling Sphinx Socolar Truchet

Other
Anisohedral and Isohedral Architectonic and catoptric Circle Limit III Computer graphics Honeycomb Isotoxal List Packing Pentagonal Problems Domino Wang Heesch's Squaring Dividing a square into similar rectangles Prototile Conway criterion Girih Regular Division of the Plane Regular grid Substitution Voronoi Voderberg

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