Rhombitetraapeirogonal tiling

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

Rhombitetraapeirogonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.4.∞.4
Schläfli symbolrr{∞,4} or
Wythoff symbol∞ 2
Coxeter diagram or
Symmetry group[∞,4], (*∞42)
DualDeltoidal tetraapeirogonal tiling
PropertiesVertex-transitive

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.

See also

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.
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