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	or
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
Symmetry	[∞,4] (*∞42)	[∞,∞,∞] = [∞,1+,4] (*∞222)
Schläfli symbol	rr{∞,4}	t0,1,2,3{∞,∞,∞}
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

Template:Expanded4 table

Template:Order i-4 tiling table

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 Search this book on . (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. Search this book on

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

This article "Rhombitetraapeirogonal tiling" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Rhombitetraapeirogonal tiling. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

---