Discrete segments of Z3Z3 constructed by synchronization of words

We study a natural and naive composition algorithm which takes three input words written on two-letter alphabets and synchronizes them into a word on a three-letter alphabet. We show that in the case where the three input words are compatible Christoffel words, the algorithm provides a synchronization of the letters which allows the geometrical interpretation of the input words to be inherited by the output word forming a 3D discrete line segment. A second approach is considered while applying our composition algorithm to words defined by stripes meeting at a corner of discrete planes. We show that, under certain conditions, the output of the algorithm corresponds to the normal vector of the plane.