Tessellation in Blue, Consisting of Rhombi and Concave, Equilateral Octagons

To make this tessellation, I started with a single regular dodecagon, then reflected it over one of its own diagonals, and then repeated such reflections numerous times. Before coloring the image, on aesthetic grounds, I hid the decagon-diagonals from view.

Tessellating the Plane With Squares, Regular Octagons, and 45/135 Degree Rhombi

The tessellation of the plane using only squares and regular octagons is well-known, and not difficult to construct. What I did here was to add another polygon to the mix: rhombi containing 45 and 135 degree angles, “seeded” in the initial part of the tessellation, at this figure’s center. This addition greatly increases the rigor of this construction.

This is a radial tessellation with eight-fold symmetry. I couldn’t continue it outwards any further without crashing Geometer’s Sketchpad, the software I use to make tessellations, but this is taken far enough to see the pattern.