Rhombitetraapeirogonal tiling

(Redirected from Deltoidal tetraapeirogonal tiling) Uniform tiling of the hyperbolic plane

Rhombitetraapeirogonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	4.4.∞.4
Schläfli symbol	rr{∞,4} or $r{\begin{Bmatrix}\infty \\4\end{Bmatrix}}$
Wythoff symbol	4 \| ∞ 2
Coxeter diagram	or
Symmetry group	, (*∞42)
Dual	Deltoidal tetraapeirogonal tiling
Properties	Vertex-transitive

In geometry, the rhombitetraapeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{∞,4}.

Constructions

There are two uniform constructions of this tiling, one from or (*∞42) symmetry, and secondly removing the mirror middle, , gives a rectangular fundamental domain , (*∞222).

Two uniform constructions of 4.4.4.∞
Name	Rhombitetrahexagonal tiling
Image
Symmetry	(*∞42)	= (*∞222)
Schläfli symbol	rr{∞,4}	t_0,1,2,3{∞,∞,∞}
Coxeter diagram

Symmetry

The dual of this tiling, called a deltoidal tetraapeirogonal tiling represents the fundamental domains of (*∞222) orbifold symmetry. Its fundamental domain is a Lambert quadrilateral, with 3 right angles.

Related polyhedra and tiling

*n42 symmetry mutation of expanded tilings: n.4.4.4 v t e
Symmetry , (*n42)	Spherical	Euclidean	Compact hyperbolic				Paracomp.
Symmetry , (*n42)	*342	*442	*542	*642	*742	*842	*∞42
Expanded figures
Config.	3.4.4.4	4.4.4.4	5.4.4.4	6.4.4.4	7.4.4.4	8.4.4.4	∞.4.4.4
Rhombic figures config.	V3.4.4.4	V4.4.4.4	V5.4.4.4	V6.4.4.4	V7.4.4.4	V8.4.4.4	V∞.4.4.4

Paracompact uniform tilings in family v t e


{∞,4}	t{∞,4}	r{∞,4}	2t{∞,4}=t{4,∞}	2r{∞,4}={4,∞}	rr{∞,4}	tr{∞,4}
Dual figures


V∞	V4.∞.∞	V(4.∞)	V8.8.∞	V4	V4.∞	V4.8.∞
Alternations
(*44∞)	(∞*2)	(*2∞2∞)	(4*∞)	(*∞∞2)	(2*2∞)	(∞42)
=				=
h{∞,4}	s{∞,4}	hr{∞,4}	s{4,∞}	h{4,∞}	hrr{∞,4}	s{∞,4}

Alternation duals


V(∞.4)	V3.(3.∞)	V(4.∞.4)	V3.∞.(3.4)	V∞	V∞.4	V3.3.4.3.∞

References

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.

External links

Tessellation

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

By vertex type

Spherical	2 3.n V3.n 4.n V4.n
Regular	2 3 4 6
Semi- regular	3.4.3.4 V3.4.3.4 3.4 3.∞ 3.6 V3.6 3.4.6.4 (3.6) 3.12 4.∞ 4.6.12 4.8
Hyper- bolic	3.4.3.5 3.4.3.6 3.4.3.7 3.4.3.8 3.4.3.∞ 3.5.3.5 3.5.3.6 3.6.3.6 3.6.3.8 3.7.3.7 3.8.3.8 3.4.3.4 3.∞.3.∞ 3.7 3.8 3.∞ 3.4 3 3 3 (3.4) (3.4) 3.4.6.4 3.4.7.4 3.4.8.4 3.4.∞.4 3.6.4.6 (3.7) (3.8) 3.14 3.16 3.∞ 4.5.4 4.6.4 4.7.4 4.8.4 4.∞.4 4 4 4 4 4 (4.5) (4.6) 4.6.12 4.6.14 V4.6.14 4.6.16 V4.6.16 4.6.∞ (4.7) (4.8) 4.8.10 V4.8.10 4.8.12 4.8.14 4.8.16 4.8.∞ 4.10 4.10.12 4.12 4.12.16 4.14 4.16 4.∞ 5 5 5 5 5.4.6.4 (5.6) 5.8 5.10 5.12 6 6 6 6 6.4.8.4 (6.8) 6.8 6.10 6.12 6.16 7 7 7 7.6 7.8 7.14 8 8 8 8 8.6 8.12 8.16 ∞ ∞ ∞ ∞ ∞.6 ∞.8

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