An Algebraic Structure for the Central Mexican Ritual Calendar

TL;DR

This paper introduces an algebraic model of the 260-day Central Mexican ritual calendar, formalizing its structure using cyclic groups and group actions to facilitate analysis and interpretation.

Contribution

It develops a novel algebraic framework representing the calendar as a cyclic group, extending previous interpretations and enabling formal analysis of its structural components.

Findings

Calendar modeled as cyclic group $ ext{Z}_{13} imes ext{Z}_{20}$

Explicit correspondences between day numbers and names derived

Connection established with subgroup generated by permutations

Abstract

This article develops an algebraic model of the 260-day Central Mexican ritual calendar, the \textit{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 \textit{Tonalpohualli}, and they provide a framework for codex analysis.