#### Document Type

Article

#### Publication Date

2007

#### Publication Title

Journal of Integer Sequences

#### 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 p 2(m2 − mn + n2) for some integers m, n. We also show a similar characterization for the sides of a regular tetrahedron in Z 3 : such a tetrahedron exists if and only if the sides are of the form k √ 2, for some k ∈ N. The classification of all the equilateral triangles in 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.

#### Recommended Citation

Ionascu, Eugen J., "A parametrization of equilateral triangles having integer coordinates." (2007). *Faculty Bibliography*. 875.

http://csuepress.columbusstate.edu/bibliography_faculty/875