Tetraapeirogonal tiling

tetraapeirogonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex figure	4.∞.4.∞
Schläfli symbol	r{∞,4} rr{∞,∞}
Wythoff symbol	2 | ∞ 4 ∞ | ∞ 2
Coxeter diagram
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}.

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

Dimensional family of quasiregular polyhedra and tilings: 4.n.4.n

Symmetry *4n2 [n,4]	Spherical	Euclidean	Hyperbolic...
	*342 [3,4]	*442 [4,4]	*542 [5,4]	*642 [6,4]	*742 [7,4]	*842 [8,4]	*∞42 [∞,4]
Coxeter
Quasiregular figures configuration	4.3.4.3	4.4.4.4	4.5.4.5	4.6.4.6	4.7.4.7	4.8.4.8	4.∞.4.∞
Dual figures
Coxeter
Dual (rhombic) figures configuration	V4.3.4.3	V4.4.4.4	V4.5.4.5	V4.6.4.6	V4.7.4.7	V4.8.4.8	V4.∞.4.∞

Noncompact hyperbolic uniform tilings in [∞,4] family

Symmetry: [∞,4], (*∞42)
{∞,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∞	V43.∞	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∞.44	V3.3.4.3.∞

Noncompact hyperbolic uniform tilings in [∞,∞] family

Symmetry: [∞,∞], (*∞∞2)
{∞,∞}	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∞∞)
h0{∞,∞}	h0,1{∞,∞}	h1{∞,∞}	h1,2{∞,∞}	h2{∞,∞}	h0,2{∞,∞}	s{∞,∞}
Alternation duals
V(∞.∞)∞	V(3.∞)3	V(∞.4)4	V(3.∞)3	V∞∞	V(4.∞.4)2	V3.3.∞.3.∞

See also

Wikimedia Commons has media related to Uniform tiling 4-i-4-i.

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

References