Figure 3.

Robinson substitution rules. These substitution rules provide nonuniform geometric scaling of each tile being substituted for, placing the Robinson tilings in the class of combinatorial substitution tilings (Frank, 2008). By induction, the substitution rules produce the (2m+1)-square from the m-square in one substitution step. For clarity, substitution rules are shown by the arrows on the geometrically constrained Robinson tile set of Figure 1a.