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An Algebraic Structure for the Central Mexican Ritual Calendar

Ramiro Carrillo-Catalán

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.

An Algebraic Structure for the Central Mexican Ritual Calendar

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 , 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.
Paper Structure (8 sections, 3 theorems, 26 equations, 4 figures)

This paper contains 8 sections, 3 theorems, 26 equations, 4 figures.

Table of Contents

  1. The Central Mexican Ritual Calendar
  2. The Tonalpohualli as a Cyclic Group
  3. The Number and Name of the Days
  4. The Tonalpohualli as a permutations group
  5. The Twenty Trecenas and the Thirteen Veintenas of the Tonalpohualli
  6. The Tetrads of the Tonalpohualli
  7. The Twenty Trecenas of the Tonalpohualli as Permutations and Their Relationship with the Tetrad Decomposition
  8. Conclusions and Perspectives

Key Result

Theorem 1

We have that $\mathbb{Z}_{13}\oplus \mathbb{Z}_{20}\cong\mathbb{Z}_{260}$ via the isomorphisms $\ell$ and $\imath$. Moreover, $\ell$ and $\imath$ are inverse mappings of each other.

Figures (4)

  • Figure :
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (7)

  • Remark 1
  • Theorem 1
  • proof
  • Definition 1
  • Remark 2
  • Theorem 2
  • Theorem 3