Constructing 7-Clusters

Authors

  • Sascha Kurz Department of Mathematics, Physics and Informatics University of Bayreuth Bayreuth, Germany
  • Landon Curt Noll Cisco Systems San Jose California, USA
  • Randall Rathbun Green Energy Technologies, LLC Manning, Oregon, USA
  • Chuck Simmons Google, Mountain View California, USA

DOI:

https://doi.org/10.55630/sjc.2014.8.47-70

Keywords:

Erdos Problems, Integral Point Sets, Heron Triangles, Exhaustive Enumeration

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 coordinates are integers is called an n-cluster (in R^2). We determine the smallest 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. Along the way, we have exhaustively determined all Heronian triangles with largest edge length up to 6 · 10^6.

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Published

2015-02-02

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Section

Articles