# Rhombitetraoctagonal tiling

Rhombitetraoctagonal tiling :
| Poincaré disk model of the hyperbolic plane ||
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 4.4.8.4 |
| Schläfli symbol | rr{8,4} or $r{\begin{Bmatrix}8\\4\end{Bmatrix}}$ |
| Wythoff symbol | 4 | 8 2 |
| Coxeter diagram | or |
| Symmetry group | [8,4], (*842) |
| Dual | Deltoidal tetraoctagonal tiling |
| Properties | Vertex-transitive |
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 | | |
| Symmetry | [8,4] (*842) | [8,1$^+$,4] = [∞,4,∞] (*4222) = |
| Schläfli symbol | rr{8,4} | t$_{0,1,2,3}${∞,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 s$_2${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, .

## Related polyhedra and tiling

*n42 symmetry mutation of expanded tilings: n.4.4.4 vte :
| Symmetry [n,4], (*n42) | Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracomp. |
| Symmetry [n,4], (*n42) | *342 [3,4] | *442 [4,4] | *542 [5,4] | *642 [6,4] | *742 [7,4] | *842 [8,4] | *∞42 [∞,4] |
| Expanded figures | | | | | | | |
| Config. | 3.4.4.4 | 4.4.4.4 | 5.4.4.4 | 6.4.4.4 | 7.4.4.4 | 8.4.4.4 | ∞.4.4.4 |
| Rhombic figures config. | V3.4.4.4 | V4.4.4.4 | V5.4.4.4 | V6.4.4.4 | V7.4.4.4 | V8.4.4.4 | V∞.4.4.4 |
Uniform octagonal/square tilings vte :
| [8,4], (*842) (with [8,8] (*882), [(4,4,4)] (*444) , [∞,4,∞] (*4222) index 2 subsymmetries) (And [(∞,4,∞,4)] (*4242) index 4 subsymmetry) |||||||
| {8,4} | t{8,4} | r{8,4} | 2t{8,4}=t{4,8} | 2r{8,4}={4,8} | rr{8,4} | tr{8,4} |
| Uniform duals |||||||
| V8$^4$ | V4.16.16 | V(4.8)$^2$ | V8.8.8 | V4$^8$ | V4.4.4.8 | V4.8.16 |
| Alternations |||||||
| [1$^+$,8,4] (*444) | [8$^+$,4] (8*2) | [8,1$^+$,4] (*4222) | [8,4$^+$] (4*4) | [8,4,1$^+$] (*882) | [(8,4,2$^+$)] (2*42) | [8,4]$^+$ (842) |
| h{8,4} | s{8,4} | hr{8,4} | s{4,8} | h{4,8} | hrr{8,4} | sr{8,4} |
| Alternation duals |||||||
| V(4.4)$^4$ | V3.(3.8)$^2$ | V(4.4.4)$^2$ | V(3.4)$^3$ | V8$^8$ | V4.4$^4$ | V3.3.4.3.8 |