Tetraapeirogonal tiling

tetraapeirogonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	(4.∞)^2
Schläfli symbol	r{∞,4} or $\begin{Bmatrix} \infin \\ 4 \end{Bmatrix}$ rr{∞,∞} or $r\begin{Bmatrix} \infin \\ \infin \end{Bmatrix}$
Wythoff symbol	2 | ∞ 4 ∞ | ∞ 2
Coxeter diagram	or
Symmetry group	[∞,4], (*∞42) [∞,∞], (*∞∞2)
Dual	Order-4-infinite rhombille tiling
Properties	Vertex-transitive edge-transitive

In geometry, the tetrapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of r{∞,4}.

Uniform constructions

There are 3 lower symmetry uniform construction, one with two colors of apeirogons, one with two colors of squares, and one with two colors of each:

Symmetry	(*∞42) [∞,4]	(*∞33) [1^+,∞,4] = [(∞,4,4)]	(*∞∞2) [∞,4,1^+] = [∞,∞]	(*∞2∞2) [1^+,∞,4,1^+]
Coxeter		=	=	=
Schläfli	r{∞,4}	r{4,∞}^1⁄2	r{∞,4}^1⁄2=rr{∞,∞}	r{∞,4}^1⁄4
Coloring
Dual

Symmetry

The dual to this tiling represents the fundamental domains of *∞2∞2 symmetry group. The symmetry can be doubled by adding mirrors on either diagonal of the rhombic domains, creating *∞∞2 and *∞44 symmetry.

Related polyhedra and tiling

*n42 symmetry mutations of quasiregular tilings: (4.n)^2
Symmetry *4n2 [n,4]	Spherical	Euclidean	Compact hyperbolic				Paracompact	Noncompact
	*342 [3,4]	*442 [4,4]	*542 [5,4]	*642 [6,4]	*742 [7,4]	*842 [8,4]...	*∞42 [∞,4]	[ni,4]
Figures
Config.	(4.3)^2	(4.4)^2	(4.5)^2	(4.6)^2	(4.7)^2	(4.8)^2	(4.∞)^2	(4.ni)^2

Paracompact uniform tilings in [∞,4] family
{∞,4}	t{∞,4}	r{∞,4}	2t{∞,4}=t{4,∞}	2r{∞,4}={4,∞}	rr{∞,4}	tr{∞,4}
Dual figures
V∞^4	V4.∞.∞	V(4.∞)^2	V8.8.∞	V4^∞	V4^3.∞	V4.8.∞
Alternations
[1^+,∞,4] (*44∞)	[∞^+,4] (∞*2)	[∞,1^+,4] (*2∞2∞)	[∞,4^+] (4*∞)	[∞,4,1^+] (*∞∞2)	[(∞,4,2^+)] (2*2∞)	[∞,4]^+ (∞42)
=				=
h{∞,4}	s{∞,4}	hr{∞,4}	s{4,∞}	h{4,∞}	hrr{∞,4}	s{∞,4}
Alternation duals
V(∞.4)^4	V3.(3.∞)^2	V(4.∞.4)^2	V3.∞.(3.4)^2	V∞^∞	V∞.4^4	V3.3.4.3.∞

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.∞

See also

• List of uniform planar tilings
• Tilings of regular polygons
• Uniform tilings in hyperbolic plane

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.