Rhombitetraapeirogonal tiling

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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
Wythoff symbol 4 | ∞ 2
Coxeter diagram CDel node 1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node 1.png or CDel node.pngCDel split1-i4.pngCDel 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}.


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 H2 tiling 24i-5.png Uniform tiling i222-t0123.png
Symmetry [∞,4]
CDel node c1.pngCDel infin.pngCDel node c3.pngCDel 4.pngCDel node c2.png
[∞,∞,∞] = [∞,1+,4]
CDel nodeab c1-2.pngCDel ia2b-cross.pngCDel nodeab c1-2.png
Schläfli symbol rr{∞,4} t0,1,2,3{∞,∞,∞}
Coxeter diagram CDel node 1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node 1.png CDel nodes 11.pngCDel ia2b-cross.pngCDel nodes 11.png


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.

H2chess 24id.pngDeltoidal tetraapeirogonal tiling.png

Related polyhedra and tiling[edit]

See also[edit]


  • 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[edit]