See also stellated dodecahedron derivatives and polyhedra addiction.

If you look at the coloring this object has only two sides.

While looking at the first few of these objects I made, I realized they could be thought of as regular polyhedrons that had their vertices duplicated and moved “outward”, and then had all the original edges replaced by new edges that connected each original vertex to the duplicate of each vertex the original edge connected it to. So the result is a sort of “crisscrossing” pattern. Realizing this I wrote a script that automates that procedure, which makes modeling these objects a little faster. A lot of the resulting structures have been extremely complicated and I’m not entirely sure how worth pursuing they are, but last night I tackled the tetrahedron and am fairly pleased with the result. Though I couldn’t decide between making the 4 tips pointy or sharpening the curved edge — I ended up sticking with the latter mostly because the coloring worked that way. I’m also working on a octohedron version now but have some things to figure out still.