Snub apeiroapeirogonal tiling

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.∞.

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 ${\displaystyle s{\begin{Bmatrix}\infty \\\infty \end{Bmatrix}}}$
Wythoff symbol	| ∞ ∞ 2
Coxeter diagram	or
Symmetry group	[∞,∞]^+, (∞∞2)
Dual	Infinitely-infinite-order floret pentagonal tiling
Properties	Vertex-transitive Chiral

Dual tiling

Related polyhedra and tiling

Paracompact uniform tilings in [∞,∞] family
= =	= =	= =	= =	= =	=	=
{∞,∞}	t{∞,∞}	r{∞,∞}	2t{∞,∞}=t{∞,∞}	2r{∞,∞}={∞,∞}	rr{∞,∞}	tr{∞,∞}
Dual tilings
V∞^∞	V∞.∞.∞	V(∞.∞)^2	V∞.∞.∞	V∞^∞	V4.∞.4.∞	V4.4.∞
Alternations
[1^+,∞,∞] (*∞∞2)	[∞^+,∞] (∞*∞)	[∞,1^+,∞] (*∞∞∞∞)	[∞,∞^+] (∞*∞)	[∞,∞,1^+] (*∞∞2)	[(∞,∞,2^+)] (2*∞∞)	[∞,∞]^+ (2∞∞)
h{∞,∞}	s{∞,∞}	hr{∞,∞}	s{∞,∞}	h2{∞,∞}	hrr{∞,∞}	sr{∞,∞}
Alternation duals
V(∞.∞)^∞	V(3.∞)^3	V(∞.4)^4	V(3.∞)^3	V∞^∞	V(4.∞.4)^2	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
	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

• Tilings of regular polygons
• List of uniform planar tilings
• 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