Rhombitetraapeirogonal tiling

Rhombitetraapeirogonal tiling
Rhombitetraapeirogonal tiling Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	4.4.∞.4
Schläfli symbol	rr{∞,4} or $r\left\{\begin{array}{c}\mathrm{\infty }\\ 4\end{array}\right\}$
Wythoff symbol	4 | ∞ 2
Coxeter diagram	File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png or File:CDel node.pngFile:CDel split1-i4.pngFile:CDel nodes 11.png
Symmetry group	[∞,4], (*∞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 [∞,4] or (*∞42) symmetry, and secondly removing the mirror middle, [∞,1⁺,4], gives a rectangular fundamental domain [∞,∞,∞], (*∞222).

Two uniform constructions of 4.4.4.∞

Name	Rhombitetrahexagonal tiling
Image	File:H2 tiling 24i-5.png	File:Uniform tiling i222-t0123.png
Symmetry	[∞,4] (*∞42) File:CDel node c1.pngFile:CDel infin.pngFile:CDel node c3.pngFile:CDel 4.pngFile:CDel node c2.png	[∞,∞,∞] = [∞,1+,4] (*∞222) File:CDel nodeab c1-2.pngFile:CDel ia2b-cross.pngFile:CDel nodeab c1-2.png
Schläfli symbol	rr{∞,4}	t0,1,2,3{∞,∞,∞}
Coxeter diagram	File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png	File:CDel nodes 11.pngFile:CDel ia2b-cross.pngFile:CDel nodes 11.png

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.

File:H2chess 24id.pngFile:Deltoidal tetraapeirogonal tiling.png

Related polyhedra and tiling

*n42 symmetry mutation of expanded tilings: n.4.4.4
Symmetry [n,4], (*n42)	Spherical	Euclidean	Compact hyperbolic				Paracomp.
	*342 [3,4]	*442 [4,4]	*542 [5,4]	*642 [6,4]	*742 [7,4]	*842 [8,4]	*∞42 [∞,4]
Expanded figures	File:Uniform tiling 432-t02.png	File:Uniform tiling 44-t02.svg	File:H2-5-4-cantellated.svg	File:Uniform tiling 64-t02.png	File:Uniform tiling 74-t02.png	File:Uniform tiling 84-t02.png	File:H2 tiling 24i-5.png
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.	File:Spherical deltoidal icositetrahedron.png V3.4.4.4	File:Uniform tiling 44-t0.svg V4.4.4.4	File:H2-5-4-deltoidal.svg V5.4.4.4	File:Deltoidal tetrahexagonal til.png V6.4.4.4	File:Deltoidal tetraheptagonal til.png V7.4.4.4	File:Deltoidal tetraoctagonal til.png V8.4.4.4	File:Deltoidal tetraapeirogonal tiling.png V∞.4.4.4

See also

• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes

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

• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch