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 }⊂ℤ² a maximal integral point set over ℤ² 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.

Keywords:

• integral distances
• diameter
• exhaustive search
• maximality

2000 AMS Subject Classifications :

• 52C10
• 52C45
• 05D99
• 11D99
• 52-04
• G.2.1

Acknowledgements

We would like to thank the anonymous referees for many helpful comments to improve the presentation of the article.