Mesoamerican proportional design and astronomical dualities: rational approximations consistent with $\phi$ and $\pi$ in calendrics and architecture

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 $\pi$ or the golden ratio $\phi$ 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.