A peer-reviewed study in the Journal of Mathematics and the Arts reported that an equilateral triangle traced from the navel to the out-turned feet of Leonardo da Vinci’s Vitruvian Man explains how the Renaissance artist balanced the figure between a square and a circle. The research team, led by London restorative dentist Dr. Rory Mac Sweeney, argued that the shape solved a geometric puzzle scholars had debated since the first century BCE.
“That Da Vinci used this shape in a drawing from 1490 indicates his genius insight into the human body. Modern science didn’t even exist then,” said Mac Sweeney, according to a report by the Independent.
Leonardo’s own note above the sketch read, “If you open your legs … the space between the legs will be an equilateral triangle.” Mac Sweeney pointed to that annotation as evidence that the triangle was central to the design rather than an artistic flourish.
By extending the triangle’s lines, the study produced a ratio of about 1.64 between the square’s side and the circle’s radius, virtually matching the “optimal ratio” of 1.633 observed in crystal lattices and close-packed spheres. Mac Sweeney labeled the figure the “tetrahedral ratio,” suggesting it supplies a universal rule connecting atomic structures, stacked fruit, and the mechanics of the human jaw.
The dental link centers on the Bonwill triangle, introduced in 1864 by American dentist William Bonwill to describe the ideal geometry of the mandible. Formed by the two jaw joints and the midpoint between the lower central incisors, each side measures roughly ten centimeters and still guides denture design and bite alignment. Mac Sweeney found that superimposing the Bonwill triangle on the Vitruvian Man reproduced the same 1.64 proportion.
“Leonardo’s equilateral triangle corresponds to Bonwill’s triangle in dental anatomy—the fundamental geometric relationship that governs the optimal function of the human jaw,” said Mac Sweeney in the paper. Vietnam Plus reported that potential applications include craniofacial surgery and prosthetic design.
The triangle also clarified why previous attempts to align the drawing with the Golden Ratio of 1.618 never matched its measurements. Repeating the equilateral triangle around the navel forms a hexagon, echoing the close-packed hexagonal model physicists use to describe efficient sphere packing.
Leonardo created the Vitruvian Man around 1490 to illustrate the Roman architect Vitruvius’s idea that a perfectly proportioned body could fit inside both a circle and a square. For centuries, the exact link between those shapes remained uncertain. “We’ve all been looking for a complicated answer, but the key was in Leonardo’s own words. He was pointing to this triangle all along,” said Mac Sweeney, according to the Independent.
National Geographic Historia noted that Leonardo believed the beauty of mathematics and nature were connected. “This one drawing encapsulates a universal rule of design,” Mac Sweeney told Vietnam Plus, arguing that the hidden triangle reveals a common mathematical language linking anatomy, crystal structure, and classical architecture.
Produced with the assistance of a news-analysis system.