Maximal integral point sets over $\mathbb{Z}^2$

TL;DR

This paper investigates maximal integral point sets in the plane with integer coordinates, focusing on their size, diameter, and maximality, providing exact values, constructions, and an algorithm for verification.

Contribution

It introduces methods to determine and construct maximal integral point sets with minimal diameter and develops an algorithm to verify their maximality.

Findings

Exact values for certain maximal integral point sets

Construction techniques for arbitrary sizes

An algorithm to verify maximality

Abstract

Geometrical objects with integral side lengths have fascinated mathematicians through the ages. We call a set $P=\{p_1,...,p_n\}\subset\mathbb{Z}^2$ a maximal integral point set over $\mathbb{Z}^2$ 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.