Tetraoctagonal tiling

In geometry, the tetraoctagonal tiling is a uniform tiling of the hyperbolic plane.

Constructions
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+], gives [8,8], (*882). Removing the mirror between the order 2 and 8 points, [1+,8,4], gives [(4,4,4)], (*444). Removing both mirrors, [1+,8,4,1+], leaves a rectangular fundamental domain, [(∞,4,∞,4)], (*4242).

Symmetry
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.