[math-fun] Bettsian but not vertex-transitive

Given some positive integer n, we consider a finite set S of points in R^n. We are interested in sets which are Bettsian (the set of distances from a point X in S to the other points in S is independent of the choice of X). I have shown that in R^2, all Bettsian sets are vertex-transitive. For R^4 and higher dimensions, I have counter-examples (Bettsian sets which are not vertex-transitive). At the time of writing, the problem is still open for R^3. http://cp4space.wordpress.com/2012/11/21/betts-revisited/ Sincerely, Adam P. Goucher