Octagonal tilings with three prototiles

Motivated by theoretically and experimentally observed structural phases with octagonal symmetry, we introduce families of octagonal tilings composed of three prototiles. These tilings are defined by substitution rules with inflation factors δ^(m,n) = m + n(1 + √2), where m, n are nonnegative integers. We consider the cases (m, n) = (1, 1), (2, 2), (1, 2), (3, 2) and describe their basic properties. In addition, we discuss variants of the first three cases, and for (1, 1) we derive the relative frequencies of the vertex stars.