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We consider a symbolic coding of linear trajectories in the regular octagonwith opposite sides identified (and more generally in regular 2n-gons). Eachinfinite trajectory gives a cutting sequence corresponding to the sequence ofsides hit. We give an explicit characterization of these cutting sequences. Thecutting sequences for the square are the well studied Sturmian sequences whichcan be analyzed in terms of the continued fraction expansion of the slope. Weintroduce an analogous continued fraction algorithm which we use to connect thecutting sequence of a trajectory with its slope. Our continued fractionexpansion of the slope gives an explicit sequence of substitution operationswhich generate the cutting sequences of trajectories with that slope. Ouralgorithm can be understood in terms of renormalization of the octagontranslation surface by elements of the Veech group.

Symbolic coding for linear trajectories in the regular octagon