The recent paper On the Erdos distinct distance problem in the plane Authors: Larry Guth, Nets Hawk Katz prodded me to get a non-trivial example. Here is what I cannot find: and example of 12 distinct points in the plane with only 5 different distances between points. The regular 12-polygon has 6 different lengths but I cannot do better. http://oeis.org/A186704 implies that there is one>