OK so I think all the pieces are the same shape? That is counterintiutive to me, and very cool. I guess its a side effect of making the sides have an even number of fingers.
This ought to work for any regular polyhedron I suspect, and maybe for all completely convex polyhedra even at higher complexity. Concave shapes would work but be hard to keep aligned maybe? Have you fooled with any other shapes?
Yes, all the sides are the same shape, an equilateral triangle. I haven’t played with it to create other shapes but it would probably work for those that have a triangular side. A d6 would he made up of squares, a d8 of diamonds etc.