A Fifty-Faced Polyhedron Featuring Eighteen Regular Octagons

In addition to the eighteen regular octagons, this polyhedron also has eight equiangular hexagons and twenty-four isosceles trapezoids among its fifty faces. I made it using Stella 4d, which you can try, for free, right here.

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.

The Rhombic Octagonoid, a Zonohedron With Ninety Faces

To make this zonohedron with Stella 4d (available as a free trial download here), start with a dodecahedron, and then perform a zonohedrification based on both faces and vertices. It is similar to the rhombic enneacontahedron, with thirty equilateral octagons replacing the thirty narrow rhombic faces of that polyhedron.

I’ve run into this polyhedron from time to time, and have also had students make it. It is the largest zonohedron which can be built using only red and yellow Zome (available here) of a single strut-length (short, medium, or long). I thought it needed a name, so I made one up.