Orbits of edge-to-edge tilings by equilateral triangles, squares and regular hexagons

Abstract

An edge-to-edge tiling of the Euclidian plane by equilateral triangles, squares and regular hexagons is said to be of type (t,s,h) if there are exactly t orbits of triangles, s orbits of squares and h orbits of hexagons under the symmetry group of the tiling. We prove that there exist tilings of type (t,s,h) for every t ⩾ 92, s ⩾ 2, h ⩾ 43. We completely determine the values of t and h for which tilings of type (t,1,h) exist.