Equilateral Triangles and the Fano Plane

Abstract

We formulate a definition of equilateral triangles in the complex line that makes sense over the field with seven elements. Adjacency of these abstract triangles gives rise to the Heawood graph, which is a way to encode the Fano plane. Through some reformulation, this gives a geometric construction of the Steiner systems S(2, 3, 7) and S(3, 4, 8). As a consequence, we embed the Heawood graph in a torus, and we derive the exceptional isomorphism PSL2(픽7) ≃ GL3(픽2). The study of equilateral triangles over other finite fields shows that seven is very specific.