Primitive two-dimensional words and iterated pedal triangles via symbolic coding

The notion of a two-dimensional word arises naturally in the study of combinatorics on words, while the iterative construction of pedal triangles results in a rich dynamical system in the study of geometry. At first, these two classes of objects seem to be unrelated. However, it is known that for all n ≥ 1, the number of primitive two-dimensional words of dimension 2 × n over a binary alphabet agrees with the number of triangles whose first similar pedal triangle is their nth pedal triangle. We construct a finite four-symbol coding of the sorted pedal map and use the resulting branch itineraries to give a bijection between these two classes.