Rhombitetraoctagonal tiling

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Short description: Regular tiling of the hyperbolic plane

In geometry, the rhombitetraoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{8,4}. It can be seen as constructed as a rectified tetraoctagonal tiling, r{8,4}, as well as an expanded order-4 octagonal tiling or expanded order-8 square tiling.

Constructions

There are two uniform constructions of this tiling, one from [8,4] or (*842) symmetry, and secondly removing the mirror middle, [8,1+,4], gives a rectangular fundamental domain [∞,4,∞], (*4222).

Two uniform constructions of 4.4.4.8
Name Rhombitetraoctagonal tiling
Image Uniform tiling 84-t02.png Uniform tiling 4.4.4.8.png
Symmetry [8,4]
(*842)
CDel node c1.pngCDel 8.pngCDel node c3.pngCDel 4.pngCDel node c2.png
[8,1+,4] = [∞,4,∞]
(*4222)
CDel node c1.pngCDel 8.pngCDel node h0.pngCDel 4.pngCDel node c2.png = CDel label4.pngCDel branch c1.pngCDel 2a2b-cross.pngCDel nodeab c2.png
Schläfli symbol rr{8,4} t0,1,2,3{∞,4,∞}
Coxeter diagram CDel node 1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 8.pngCDel node h0.pngCDel 4.pngCDel node 1.png = CDel label4.pngCDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 11.png

Symmetry

A lower symmetry construction exists, with (*4222) orbifold symmetry. This symmetry can be seen in the dual tiling, called a deltoidal tetraoctagonal tiling, alternately colored here. Its fundamental domain is a Lambert quadrilateral, with 3 right angles.

Deltoidal tetraoctagonal til.png H2chess 248d.png
The dual tiling, called a deltoidal tetraoctagonal tiling, represents the fundamental domains of the *4222 orbifold.

With edge-colorings there is a half symmetry form (4*4) orbifold notation. The octagons can be considered as truncated squares, t{4} with two types of edges. It has Coxeter diagram CDel node h.pngCDel 4.pngCDel node h.pngCDel 8.pngCDel node 1.png, Schläfli symbol s2{4,8}. The squares can be distorted into isosceles trapezoids. In the limit, where the rectangles degenerate into edges, an order-8 square tiling results, constructed as a snub tetraoctagonal tiling, CDel node h.pngCDel 4.pngCDel node h.pngCDel 8.pngCDel node.png.

Related polyhedra and tiling

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. 

See also

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

External links