Definition: A polytope has **property X** iff there is a function f:N+ → R+ such that for each pair of vertices vi, vj the following holds:


disteuclidean(vi, vj) = f(distcombinatorial(vi, vj))


with distcombinatorial(vi, vj) = shortest path of edges between vi and vj.


That means: for each vi1, vj1, vi2, vj2:


disteuclidean(vi1, vj1) = disteuclidean(vi2, vj2) 


iff


distcombinatorial(vi1, vj1) = distcombinatorial(vi2, vj2) 


Question 1: Is property X already named? What's its common name?


Question 2: Which polytopes have property X? The regular polytopes seem to have it, but are there more?



Combinatorial distance ≡ Euclidean distance
Another way of describing your property X is to say that concentric spheres in the shortest path metric in the graph of the polytope are mapped into concentric Euclidean spehres. I have never heard of these polytopes before, but it is a very natural question. My suspicion is that there are not very many "unsymmetric" ones.


Here is a plan for enumerating them in dimension 3. As mentioned by Martin M. W., in dimension two the only such polytopes are the regular polygons: all edges must be equal, and all angles must also be equal, because of vertices at distance 2 in the graph must have the same Euclidean distance. So in dimension 3, each facet of such a polytope must be a regular polygon. Because a regular polygon fixes the second-smallest distance, all non-triangular facets must be congruent. The 3-polytopes with each face a regular polygon are known: they are the 5 [Platonic solids](http://en.wikipedia.org/wiki/Platonic_solid), the 13 [Archimedean solids](http://en.wikipedia.org/wiki/Archimedean_solid), the infinite family of [prisms](http://en.wikipedia.org/wiki/Prism_%28geometry%29), the infinite family of [antiprisms](http://en.wikipedia.org/wiki/Antiprism), and the 92 [Johnson solids](http://en.wikipedia.org/wiki/Johnson_solid).


Only two prisms satisfy this property, the triangular one and the square one (aka the cube). Most likely, the only antiprism will be the triangular one (aka the octahedron), but this will need some calculations. This leaves finitely many, each of which can be checked (more calculations).


In principle, the same programme could be carried through in dimension 4. Then each facet will be one in the finite list (not) enumerated above, and because of the various distances realized by each possible 3-polytope, not many of them could co-exist. So it could be that the possibilities are even more restricted in dimension 4. Or there could be a combinatorial explosion. It's not clear to me that there would be only finitely many such polytopes (up to similarity) in a fixed dimension $\geq 4$.


Anyone for a computational project? (Note: [Zalgaller's enumeration](http://www.ams.org/mathscinet-getitem?mr=240719) of the Johnson solids takes up almost 100 pages.)


