Alternative Prime Spirals

The Ulam spiral is based on the idea of arranging integers in a rectilinear 2D spiral.

And noticing that certain diagonal patterns fall out that aren't explainable by simple equations that describe some of the "holes".

What other orderings might reveal interesting patterns? How about a 2D Hilbert curve?

Or maybe a 3D one? How might you continue a spiral in a cubic 3D space? Would you get diagonal planes describing the primes? How about a higher dimensional space — maybe higher dimensional planes?