In some work I've done recently. I have been analyzing the distance matrix of the regular polyhedra in multiple dimensions and have found some curious results.
Starting in 2 dimensions, I have found that if you label the vertices cyclically, where the labeling of the vertices corresponds to the row and column of the distance matrix, the distance matrix will always be a symmetric circulant matrix of size $n\times n$, where n is the number of vertices for the polygon. Of course this is a well-known property of cyclic graphs in general which all the polygons are.
In 3 dimensions, there are only 5 regular polyhedra. However, they all display the same interesting properties. I have found that for all there exists a labeling of their vertices that gives a block bi-symmetric matrix where each block is a symmetric circulant matrix of the same size. In general, this labeling is not as trivial as in the 2 dimensional case and I have found no better way of finding it other than by guess and check. This becomes increasingly difficult for large polyhedra as the number of possible labelings grows as $n!$ where n is the number of vertices of the polyhedra. The following is the structure of the distance matrix for each of the 5 regular polyhedra.
Tetrahedron: $4\times 4$ symetric circulant matrix
Octohedron: $6\times 6$ symetric circulant matrix
Cube: $2\times 2$ block bi-symetric matrix of $4\times 4$ symetric circulant matricies
Icosahedron: $2\times 2$ block bi-symetric matrix of $6\times 6$ symetric circulant matricies
Dodecahedron: $4\times 4$ block bi-symetric matrix of $5\times 5$ symetric circulant matricies
In 4 dimensions the same pattern arises for the polytope I have checked.
Pentachron: $5\times 5$ symetric circulant matrix
Hexadecachron: $8\times 8$ symetric circulant matrix
Tesseract: $4\times 4$ block bi-symmetric matrix (which happens to also be block symmetric circulant) of $4\times 4$ symmetric circulant matrices
This is as many as I have checked. The structure is very interesting because bi-symmetric matrices have many unique properties that can be exploited, and symmetric circulant matrices do as well. I have multiple questions about this phenomenon. Does this pattern persist for all regular polytopes in any dimension? What about the regular polytopes give rise to this distance matrix structure? Is there any way to predict what the structure will be from topological invariants of the graphs? Are there other graphs that satisfy this unique property?
I haven't been able to find any papers written on this phenomenon so any help trying to better understand what is going on would be very helpful.
I had to take out my d20 and stare at it for a bit for this one. I don't have a rigorous proof, but I'm convinced that in any dimension, by definition of what a 'regular' polytope is... you can pick any reference point arbitrarily, since they're all symmetric. You can then define the 'depth' of every other vertex as the minimum number of edges you would have to traverse to move there.
If the distance from the reference point to any two points at the same depth were different, that would have to break the defining symmetry of the object. Any two paths from the reference to a point at a given depth should be identical up to a rotation and or a reflection that preserves distance.
If your data disconfirms this, please let me know! But I suspect that each row in your matrices, no matter how you label the vertices, should have a number of distinct nonzero values equal to the max depth of the vertices as a graph/tree (with arbitrary starting point having value zero) and multiplicities equal to the number of vertices at a given depth. For the cube, $1,3,3,1$, for the icosahedron $1,5,5,1$, for the dodecahedron $1,3,6,6,3,1$, etc.
Hopefully this at least helps explain why the matrices must always be circulant, and why a permutation of the rows and columns might always permit a symmetric or block bisymmetric arrangement.