An Algebraic Structure for the Central Mexican Ritual Calendar

This article develops an algebraic model of the 260-day Central Mexican ritual calendar, the $\mathit{\text{Tonalpohualli}}$. We represent the calendar as the cyclic group ${\mathbb{Z}}_{13}\oplus {\mathbb{Z}}_{20}$, where each day name is encoded by a numeral-sign pair. From this model, we derive explicit correspondences between day numbers and day names through group actions. We also characterize, in algebraic terms, the twenty 13-day periods, the thirteen 20-day periods, and the partition of days into oriented tetrads. In addition, we describe how these structures relate to a subgroup generated by permutations of the starts of 13-day periods, and we show its connection with a cyclic group of order four and with square rotations. These results formalize and extend previous arithmetic and structural interpretations of the $\mathit{\text{Tonalpohualli}}$, and they provide a framework for codex analysis.