[This is a sequel to my previous post Magforming the Johnson Solids. Please see that for a disclaimer and (if you want to understand the words in this post) the geometrical background. If you only want to see some pretty pictures, then ignore all this stuff and just scroll down…]
Last time, we investigated the shapes that can be built out of regular polygons, focusing on convex shapes – roughly speaking those with no holes, dents, or spikes. There are exactly 98 theoretically magformable convex polyhedra: all 5 of the Platonic solids, 11 Archimedean solids (out of 13 in total), 4 prisms (of an infinite family), 4 antiprisms (ditto), and 74 Johnson solids (out of 92).
Once you drop the requirement for convexity, the only limits are your imagination and the size of your magformer collection. In principle there are inifinitely many magformable polyhedra, because you can always stick more bits on. See robopenguin here, for example.
So what to do? Let’s return to the starting point of every discussion of polyhedra: the Platonic Solids.