Maximal integral point sets over ℤ²

Abstract

Geometrical objects with integral side lengths have fascinated mathematicians through the ages. We call a set P = {p 1 , . . . , p n } ⊂ Z 2 a maximal integral point set over Z 2 if all pairwise distances are integral and every additional point p n+1 destroys this property. Here we consider such sets for a given cardinality and with minimum possible diameter. We determine some exact values via exhaustive search and give several constructions for arbitrary cardinalities. Since we cannot guarantee the maximality in these cases, we describe an algorithm to prove or disprove the maximality of a given integral point set. We additionally consider restrictions as no three points on a line and no four points on a circle.

On particular diameter bounds for integral point sets in higher dimensions

“…2 Classification of planar integral point sets Facher sets are very dominating examples of planar integral pont sets. In [7], facher sets of characteristic 1 are called semi-crabs. For 9 ≤ n ≤ 122, the diameter d(2, n) is reached on a facher point set [8].…”

“…Every cross set has characteristic 1; in [7], cross sets with only 2 points out of one of the lines are called crabs. Definition 8.…”

On diameter bounds for planar integral point sets in semi-general position

“…It turned out to be very easy to construct a planar integral point set of n points with n − 1 collinear ones and one point out of the line (so-called facher sets); the same holds for 2 points out of the line (we refer the reader to [6], where some of such sets are called crabs) and even for 4 points out of the line [7]. For 9 ≤ n ≤ 122, the minimal possible diameter is achieved at a facher set [4].…”

On particular examples of planar integral point sets and their classification

“…Facher sets are very dominating examples of planar integral pont sets. In [14], facher sets of characteristic 1 are called semi-crabs.…”