Its checkerboard coloring can be called a octaoctagonal tiling, and Schläfli symbol of r{8,8}.
= = This tiling represents a hyperbolic kaleidoscope of 8 mirrors meeting as edges of a regular hexagon.
In Coxeter notation can be represented as [8*,4], removing two of three mirrors (passing through the octagon center) in the [8,4] symmetry.
Adding a bisecting mirror through 2 vertices of an octagonal fundamental domain defines a trapezohedral *4422 symmetry.
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.