Time for some Archimedean solids!
A truncated octahedron is basically an octahedron with all six of its corners cut off. The result is a solid that has 6 square faces, and 8 hexagonal faces. If you’re looking at the above photo, you might be thinking, “I don’t see hexagons, I only see triangles!” That’s because the hexagons here are multicolored dark green and light green.
As you might have guessed, I did not actually start with an octahedron and cut the corners off. Instead, I started with another Archimedean solid, the cuboctahedron.
This cuboctahedron is actually identical to another one I made a while ago. Anyway, once I have a cuboctahedron, Tomoko Fuse suggests adding augments, turning it into a truncated octahedron. You can add yet more augments, turning it into a plain old octahedron.
I’m crediting Tomoko Fuse, but to be honest I think I made a test unit and thought, “Why’s it so complicated? What’s with all these extra steps?” And then I made different augments, but they’re functionally equivalent.