Unilateral and equitransitive tilings by equilateral triangles

A tiling of the plane by polygons is unilateral if each edge of the tiling is a side of at most one polygon of the tiling. A tiling is equitransitive if for any two congruent tiles in the tiling, there exists a symmetry of the tiling mapping one to the other. It is known that a unilateral and equitransitive (UE) tiling can be made with any finite number of congruence classes of squares. This article addresses the related question, raised in the book Tilings and Patterns by Grünbaum and Shephard, of finding all UE tilings by equilateral triangles. In particular, we show that there are only two classes of UE tilings admitted by a finite number of congruence classes of equilateral triangles: one with two sizes of triangles and one with three sizes of triangles.