This is the icosahedron, followed by its first stellation.

The first stellation of the icosahedron can be stellated again, and again, and so on. The “final stellation” of the icosahedron is the one right before the stellation-series “wraps around,” back to where it started:

This final stellation of the icosahedron would serve pretty well as a “polyhedral porcupine,” but I was seeking something even better, so I turned my attention to polyhedral compounds. This is the compound of five icosahedra:

The program I use to manipulate these solids is called Stella 4d: Polyhedron Navigator (free trial download available here). My next move, using Stella, was to create the final stellation of this five-icosahedron compound . . . and, when I saw it, I knew I had found my “polyhedral porcupine.”