A Forest of Eisensteinian Triangles

In 1934, B. Berggren first discovered the surprising result that every Pythagorean triangle is the pre-product of the triangle (3, 4, 5) given as a column vector by a product of three matrices, and that every triangle is obtained in this manner exactly once and in primitive form. In this article, we show a similar result for integer triangles with an angle of 60 and 120 degrees (also known as Eisensteinian triangles). We show that any such triangle is obtained by pre-multiplication of (7, 8, 5) or (13, 15, 7) by a product arising from a set of five matrices.