Journal of Integer Sequences, Vol. 10 (2007), Article 07.6.7

A Parametrization of Equilateral Triangles Having Integer Coordinates


Eugen J. Ionascu
Department of Mathematics
Columbus State University
Columbus, GA 31907
USA

Abstract:

We study the existence of equilateral triangles of given side lengths and with integer coordinates in dimension three. We show that such a triangle exists if and only if their side lengths are of the form $\sqrt{2(m^2-mn+n^2)}$ for some integers $m,n$. We also show a similar characterization for the sides of a regular tetrahedron in $\mathbb Z^3$: such a tetrahedron exists if and only if the sides are of the form $k\sqrt{2}$, for some $k\in\mathbb N$. The classification of all the equilateral triangles in $\mathbb Z^3$ contained in a given plane is studied and the beginning analysis for small side lengths is included. A more general parametrization is proven under special assumptions. Some related questions about the exceptional situation are formulated in the end.


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(Concerned with sequence A102698 .)

Received August 28 2006; revised version received June 16 2007. Published in Journal of Integer Sequences, June 18 2007.


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