3 renamed: topological -> combinatorial, geometric -> euclidean

TopologicalCombinatorial distance ≡ geometricalEuclidean distance

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:

disteuclideuclidean(vi, vj) = f(disttopologicalcombinatorial(vi, vj))

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

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

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

iff

disttopologicalcombinatorial(vi1, vj1) = disttopologicalcombinatorial(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?

2 edited tags
1

Topological distance ≡ geometrical distance

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:

disteuclid(vi, vj) = f(disttopological(vi, vj))

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

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

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

iff

disttopological(vi1, vj1) = disttopological(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?