In geometry, the tetraoctagonal tiling is a uniform tiling of the hyperbolic plane.
There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,4] or (*842) orbifold symmetry. Removing the mirror between the order 2 and 4 points, [8,4,1<sup>+</sup>], gives [8,8], (*882). Removing the mirror between the order 2 and 8 points, [1<sup>+</sup>,8,4], gives [(4,4,4)], (*444). Removing both mirrors, [1<sup>+</sup>,8,4,1<sup>+</sup>], leaves a rectangular fundamental domain, [(∞,4,∞,4)], (*4242).
| Name | Tetra-octagonal tiling | Rhombi-octaoctagonal tiling | |||
|---|---|---|---|---|---|
| Image | |||||
| Symmetry | [8,4] (*842) | [8,8] = [8,4,1<sup>+</sup>] (*882) = | [(4,4,4)] = [1<sup>+</sup>,8,4] (*444) = | [(∞,4,∞,4)] = [1<sup>+</sup>,8,4,1<sup>+</sup>] (*4242) = or | |
| Schläfli | r | rr =r1/2 | r(4,4,4) =r1/2 | t0,1,2,3(∞,4,∞,4) =r1/4 | |
| Coxeter | = | = | = or |
The dual tiling has face configuration V4.8.4.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4242), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*42) orbifold.