Leonardo's Vitruvian Man: modern craniofacial anatomical analysis reveals a possible solution to the 500-year-old mystery

Abstract

For over 500 years, Leonardo da Vinci’s geometric system for establishing the precise relationship between the circle and square in his Vitruvian Man drawing has remained a mystery. This paper demonstrates that Leonardo’s explicit textual reference to ‘an equilateral triangle’ between the figure’s legs provides his construction method and reveals the anatomical foundation for his proportional choices. The analysis shows that Leonardo’s equilateral triangle corresponds to Bonwill’s triangle in dental anatomy—the foundational geometric relationship governing optimal human jaw function. Leonardo's systematic construction yields a ratio of 1.64–1.65 between the square’s side and circle’s radius, matching both published measurements of the original drawing and the tetrahedral ratio of 1.633 found in optimal spatial arrangements. This ratio corresponds to modern calculations of optimal human craniofacial proportions (cranial architecture: 1.64 ± 0.04), suggesting Leonardo identified geometric principles of optimal spatial organization that contemporary science recognizes as important to human anatomical optimization. The findings position Vitruvian Man as both artistic masterpiece and prescient scientific hypothesis about the mathematical relationships governing ideal human proportional design. Highlights Leonardo's textual reference to ‘an equilateral triangle’ between the figure's legs provides insight into his geometric approach, offering a potential solution to the long-standing question of his circle-square relationship. Leonardo's equilateral triangle corresponds to Bonwill's triangle in dental anatomy–the geometric relationship governing optimal human jaw function established in 1864, suggesting Leonardo may have identified similar principles centuries earlier. The measured ratio of 1.64–1.65 in Leonardo's drawing approximates the tetrahedral ratio of 1.633–a mathematical relationship that appears in optimal sphere packing, tensegrity structures, and Fuller's Vector Equilibrium. Modern anatomical measurements, including cranial architecture ratios of 1.64 ± 0.04 found exclusively in humans, suggest convergence around similar mathematical relationships across biological systems. The analysis shows how combining art historical scholarship with contemporary anatomical knowledge can illuminate sophisticated mathematical thinking underlying Renaissance artistic achievement.