Constructing $7$-clusters

TL;DR

This paper identifies the smallest 7-cluster in the plane with integral distances and coordinates, and develops algorithms to find numerous such clusters, including large-edge Heronian triangles, advancing understanding of integral geometric configurations.

Contribution

It determines the minimal 7-cluster by diameter and provides algorithms to find over 1000 distinct 7-clusters, including large Heronian triangles.

Findings

Smallest 7-cluster with respect to diameter identified

Over 1000 7-clusters computationally located

All Heronian triangles with largest edge up to 6 million enumerated

Abstract

A set of $n$-lattice points in the plane, no three on a line and no four on a circle, such that all pairwise distances and all coordinates are integral is called an $n$-cluster (in $\mathbb{R}^2$). We determine the smallest existent $7$-cluster with respect to its diameter. Additionally we provide a toolbox of algorithms which allowed us to computationally locate over 1000 different $7$-clusters, some of them having huge integer edge lengths. On the way, we exhaustively determined all Heronian triangles with largest edge length up to $6\cdot 10^6$.