Dense quasicrystalline tilings by squares and equilateral triangles

Dense square-symmetry tilings of the plane by equilateral triangles and squares are described. Repeated substitution of a vertex of a tiling by groups of vertices leads asymptotically to a limiting density that is independent of the starting pattern and to a family of quasicrystalline patterns with 12-fold symmetry. Diffraction patterns were computed by treating the vertices as point scatterers. As the number of substitutions increases, and as the unit-cell size increases, the diffraction patterns from a single unit cell develop a near-perfect 12-fold symmetry. In addition, the low-intensity background scattering in the diffraction patterns exhibits fractal-like self-similar properties, with motifs of local intensity recursively decorating the more intense features as the number of substitutions progresses.