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.
a sort of stellation of a tetrahedron
I've tried several different things to get the view to align with the center of the object, but nothing seems to work. My last few attempts involved placing a number of cubes or icosahedrons around it, which worked on my computer but not once embedded on the webpage. Now I'm stumped. Update: I've managed to get three.js working and am now learning how to customize it. If I learn enough this could enable me to tackle some projects I've thought about for a really long time, involving customizing objects.)
a sort of stellation of a icosahedron