In geometry, the rhombitetraoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr. It can be seen as constructed as a rectified tetraoctagonal tiling, r, as well as an expanded order-4 octagonal tiling or expanded order-8 square tiling.
There are two uniform constructions of this tiling, one from [8,4] or (*842) symmetry, and secondly removing the mirror middle, [8,1<sup>+</sup>,4], gives a rectangular fundamental domain [∞,4,∞], (*4222).
| Name | Rhombitetraoctagonal tiling | ||
|---|---|---|---|
| Image | |||
| Symmetry | [8,4] (
| [8,1<sup>+</sup>,4] = [∞,4,∞] (
= | |
| Schläfli symbol | rr | t0,1,2,3 | |
| Coxeter diagram | = |
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.
With edge-colorings there is a half symmetry form (4*4) orbifold notation. The octagons can be considered as truncated squares, t with two types of edges. It has Coxeter diagram, Schläfli symbol s2. 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, .