## Abstract

We characterize all three point sets in ℝ^{4} with integer coordinates, the pairs of which are the same Euclidean distance apart. In three dimensions, the problem is characterized in terms of solutions of the Diophantine equation *a* ^{2} + *b* ^{2} + *c* ^{2} = 3*d* ^{2}. In ℝ^{4}, our characterization is essentially based on two different solutions of the same equation. The characterization is existential in nature, as opposed to the three dimensional situation where we have precise formulae in terms of *a*, *b*, and *c*. A few examples are discussed, their Ehrhart polynomial is computed and a table of the first minimal triangles of lengths less than \(\sqrt {42}\) is included in the end.

## Keywords

Equilateral triangles System of Diophantine equations Quadratic forms Lagrange’s four square theorem Orthogonal matrices Ehrhart polynomial## Mathematics Subject Classifications (2010)

52C07 05A15 68R05## References

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## Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2014