Rhombitetraoctagonal tiling

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.

Rhombitetraoctagonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.4.8.4
Schläfli symbolrr{8,4} or
Wythoff symbol8 2
Coxeter diagram or
Symmetry group[8,4], (*842)
DualDeltoidal tetraoctagonal tiling
PropertiesVertex-transitive

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
Symmetry [8,4]
(*842)
 [8,1+,4] = [∞,4,∞]
(*4222)
=
Schläfli symbol rr{8,4} t0,1,2,3{∞,4,∞}
Coxeter diagram =

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.

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 , 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, .

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.

See also
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