There are many visualizations of the primes in 2D and many are fascinating. I've always wondered what it would look like in 3d or 4d or higher. what if the primes were mapped on to some crazy topological shape? Is there some shape and dimension out there that produces a "perfect" pattern?
> I've always wondered what it would look like [...] if the primes were mapped on to some crazy topological shape?
This is easy: consider the spectrum of the ring Z, Spec Z (by definition, the spectrum of a commutative ring consists of its prime ideals; all prime ideals of Z (with the exception of the zero ideal) are generated by the prime numbers; in this sense, we can identify all elements of Spec Z (with the exception of the zero ideal, which is "just another point" of Spec Z) with the prime numbers if we desire).
On Spec Z, we can define a topology, the Zariski topology, by defining
V(I) := {P \in Spec Z | I \subseteq P},
where I is an ideal of Z, as the closed sets of the Zariski topology.
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.
