On the characteristic of integral point sets in $\mathbb{E}^m$

TL;DR

This paper extends the concept of characteristic from integral triangles to higher-dimensional simplices, proving a uniform characteristic property in integral point sets and applying it to develop an efficient construction algorithm that yields new minimal diameter values.

Contribution

It generalizes the characteristic of integral triangles to simplices, proves their uniformity in integral point sets, and introduces an efficient construction algorithm for these sets.

Findings

Proved all simplices in an integral point set share the same characteristic.

Developed an algorithm for constructing integral point sets efficiently.

Determined new exact minimal diameter values for integral point sets.

Abstract

We generalise the definition of the characteristic of an integral triangle to integral simplices and prove that each simplex in an integral point set has the same characteristic. This theorem is used for an efficient construction algorithm for integral point sets. Using this algorithm we are able to provide new exact values for the minimum diameter of integral point sets.