I'm not sure that this answers the parent's question. The Zariski topology is unusual, and in "some crazy topological shape" I think the parent is looking for a

I'm not sure that this answers the parent's question. The Zariski topology is unusual, and in "some crazy topological shape" I think the parent is looking for a way to map the primes onto a smooth manifold in some finite dimensional space in a way that elucidates some kind of hidden structure.