Here’s a polyhedron. It has regular faces. It is nonconvex. It is crazy.
This thing can be constructed by taking one of the Johnson solids, the bilunabirotunda (or J91), cutting off two tridiminished icosahedra, and adding on two pentagonal pyramids. You’re left with the two square faces and four of the triangular faces of the bilunabirotunda, along with sixteen new triangular faces. And a headache. The whole thing is kind of Z shaped, which at least partly accounts for Stewart’s designation, “Z4“.
Now, the thing is, if you rotate it 90°, then the square faces still coincide with the square faces of the original bilunabirotunda, but everything else fits inside the bilunabirotunda. And that means you can take a bilunabirotunda and put a Z4-shaped hole through it, making something designated J91/Z4:
The hole is so zigzaggy, there’s no line of sight through it:
So to convince you it really is a hole, I shall now proceed to drop in a penny:
And behold, after possibly a slight detour through several other dimensions, it falls out the bottom:
Try making one of these things. If you do not value your sanity. You will notice I paid pretty much no attention to the colors in building mine: just figuring out the shape of the damn thing and coercing the pieces into place was quite enough.
For even bigger laughs you can notice six bilunabirotundas can be fitted together around a central cube, and that means six of these J91/Z4 things can be put together into one genus 5 object, which in turn can be subtracted from a rhombicosidodecahedron to make a, a, a, oh god make it stop…