Constructing 7-Clusters

ACM Computing Classification System (1998): G.2, G.4.

A set of n lattice points in the plane, no three on a line and no four on a circle, such that all pairwise distances and coordinates are integers is called an n-cluster (in R^2). We determine the smallest 7-cluster with respect to its diameter. Additionally we provide a toolbox of algorithms which allowed us to computationally locate over 1000 different 7-clusters, some of them having huge integer edge lengths. Along the way, we have exhaustively determined all Heronian triangles with largest edge length up to 6 · 10^6.