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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2603.18065 |
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| _version_ | 1866910058819878912 |
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| author | Carrillo-Catalán, Ramiro |
| author_facet | Carrillo-Catalán, Ramiro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_18065 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An Algebraic Structure for the Central Mexican Ritual Calendar Carrillo-Catalán, Ramiro General Mathematics 20N99 (Primary), 11A07 (Secondary) 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. |
| title | An Algebraic Structure for the Central Mexican Ritual Calendar |
| topic | General Mathematics 20N99 (Primary), 11A07 (Secondary) |
| url | https://arxiv.org/abs/2603.18065 |