Tetraapeirogonal tiling

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

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

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
Symmetry *4n2 [n,4]	*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^∞	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.∞

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.∞

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

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