Mesoamerican proportional design and astronomical dualities: rational approximations consistent with $φ$ and $π$ in calendrics and architecture

TL;DR

This study shows that ancient Mesoamerican design embedded astronomical and cosmological knowledge within simple integer ratios, notably $5$ and $8$, to realize near-decagonal geometry without invoking $π$ or $φ$. Focusing on the Iguana structure at Guachimontones, it demonstrates that measured proportions $AC/AE$ approximate $8/5$ (and approach $φ$) within construction tolerances, using Monte Carlo uncertainty analysis to quantify robustness. The work argues that proportional modules grounded in calendrical symbolism can reproduce decagonal geometry, suggesting a distinct mathematical tradition where number, measure, and cosmology are integrated in material culture. These findings encourage reevaluation of ancient mathematics as culturally embedded and practically oriented rather than attempting to fit it into Western geometric frameworks.

Abstract

Understanding how ancient Mesoamerican societies integrated mathematical ideas into calendrical design and monumental architecture requires approaches that acknowledge their distinct epistemological frameworks. While explicit textual evidence for concepts such as $π$ or the golden ratio $φ$ is absent, numerical patterns embedded in Mesoamerican calendars, iconography, and ritual architecture reveal a coherent system of proportional reasoning grounded in simple integer ratios. Here we show that the numbers 5 and 8, central to Venus and solar calendrical relations and widely represented in Mesoamerican cosmology, generate rational approximations that reproduce, within known construction tolerances, the geometric relations associated with decagonal layouts. Using high-resolution measurements of the Iguana structure at Guachimontones, we demonstrate that its proportions align with integer ratios consistent with those found in the calendrical system and with the practical geometry of the regular decagon, without requiring knowledge of irrational constants. These findings suggest that Mesoamerican builders employed stable proportional modules that harmonized astronomical cycles, cosmological symbolism, and architectural design. This should not be interpreted as a lack of mathematical sophistication; rather, the material record reveals a distinct mathematical tradition in which number, measure, and cosmology were mutually reinforcing elements of cultural knowledge.

Figures (2)

• Figure 1: Solar and Venusian symbolism involving the numbers $5$ and $8$ in Mesoamerican codices. These numerological motifs occur in depictions of Tonatiuh, Quetzalcoatl, and Xolotl, and frequently combine five-petaled Venus glyphs with eight-rayed solar disks. Their recurrence highlights culturally meaningful dualities that also appear in calendrical design and astronomical relations.
• Figure 2: The Figure shows a digital orthophoto of the Iguana structure at the "Recinto Ceremonial de Guachimontones", in Teuchitlan, Jalisco, México. The orthophoto was taken from weigand, with land isocontours at $1 \text{m}$ intervals. The figure shows a superimposed best fit regular decagon together with the best green circle that bounds the central circular structure. The lengths $AC \approx BD \approx 38m$ and $AE \approx 24m$. In other words: $AC / AE \approx BD / AE \approx 38 / 24 = 19/12 = 1.58333 \approx 1.6 \approx 8/5 \approx \phi$, the golden ratio.