Equilateral triangles in Z^4

TL;DR

This paper characterizes all sets of three integer-coordinate points in four-dimensional space that are equidistant, extending known three-dimensional results and using solutions to specific Diophantine equations.

Contribution

It provides a complete characterization of equilateral triangles with integer vertices in four-dimensional space based on solutions to a particular Diophantine equation.

Findings

Characterization of equilateral triangles in Z^4

Extension of 3D Diophantine characterization to 4D

Use of two solutions of a Diophantine equation for classification

Abstract

We give a characterization of all three points in $\mathbb R^4$ with integer coordinates which are at the same Euclidean distance apart. In three dimension the problem is characterized in terms of solutions of the Diophantine equations $a^2+b^2+c^2=3d^2$. In $\mathbb R^4$, the characterization is essentially based on two different solutions of the same equation.