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Equilateral Triangles in ℤ4

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Abstract

We characterize all three point sets in ℝ4 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 = 3d 2. In ℝ4, 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.

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Authors and Affiliations

  1. Department of Mathematics, Columbus State University, 4225 University Avenue, Columbus, GA, 31907, USA

    Eugen J. Ionascu

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  1. Eugen J. Ionascu
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Correspondence to Eugen J. Ionascu.

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Honorific Member of the Romanian Institute of Mathematics “Simion Stoilow”.

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Ionascu, E.J. Equilateral Triangles in ℤ4 . Vietnam J. Math. 43, 525–539 (2015). https://doi.org/10.1007/s10013-014-0084-0

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  • DOI: https://doi.org/10.1007/s10013-014-0084-0

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