I made this using Stella 4d: Polyhedron Navigator, which you can try for free at this website.
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.
I just found this on Facebook. I made it two years before starting this blog.
Someone once told me that you can’t tile a plane with octagons. I enjoy proving them wrong.