Construction of parametric equation for equilateral triangle vertices with integer coordinates

Abstract

Abstract In this paper we explore equilateral triangles with integer coordinates vertices lying on the plane a x + b y + c z = 0 where coefficients a, b, c satisfy the Diophantine equation a2 + b2 + c2 = 3d2 for some integer!. If the plane is translated with any integer translation vector, we find planes which also containing equilateral triangles with integer coordinates vertices. Based on that knowledge from some previous studies, another parametric equations for integer coordinates vertices of equilateral triangle in ℝ3 are constructed.