How to compute the diameter of 3D torus interconnect?

A 3D torus interconnect is a network topology having p^3 nodes where p > 2. A 3D torus is basically a 3D mesh with links connecting nodes on opposite faces (Am I right?).

The bisection width calculated by me comes out to be 2p^2. However, I am clueless as to how to approach diameter calculation for this. I don't even know where to even begin because I don't understand what purpose the links connecting nodes on opposing faces serve. For those wondering, diameter is the greatest number of hops needed to send a message from one node to another.

For a 1D torus interconnect, the diameter is floor(p/2).

Thus, for a 3D torus interconnect, the diameter is floor(p/2) * 3 since the Manhattan distance should be used for this grid-based interconnect. To visualize that, you can think about a small 4 x 4 x 4 grid and how data packets move through the edges of the grid.