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

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

2008-04-08

arxiv.org/abs/0804.1280v1

Mathematics

Combinatorics

23 pages, 10 figures, 4 tables

Scientific paper

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.

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