Vitruvian Man by Leonardo da Vinci is surely the most iconic image of the Renaissance. Yet this tantalizingly complex drawing still contains plenty of mysteries for today’s scientists to uncover. Last month, a London-based dentist published his new insight into the work, which may reveal the surprising depth of Leonardo’s understanding of human anatomy many centuries before the advent of modern science.

The drawing, inspired by the writings of ancient Roman architect Vitruvius, was produced in around 1490, when Leonardo was in his late 30s. It uses human anatomy to show how a square and circle could correspond by fitting the idealized male body into both overlaid shapes. The elegant sketch combines Leonardo’s knowledge of art, mathematics, anatomy, philosophy, and architecture, exemplifying the multidisciplinary approach that made him the ultimate Renaissance man.

Though Vitruvius had first suggested that a human figure could fit inside both a circle and a square, he had not show how this could be the case. Leonardo took up the challenge, and succeeded, but there has been much speculation about his guiding principles.

Now, Dr. Rory Mac Sweeney has used his professional training as a dentist to analyze the Vitruvian Man from an entirely new perspective. He now believes that the unique drawing “incorporates geometric principles that anticipate modern understanding of optimal biological architecture.”

L: Tetrahedral Geometry in Human Craniofacial Architecture. R: Leonardo’s Vitruvian Man and Optimal Geometric Relationships.. Images courtesy Dr Rob Mac Sweeney.

His findings were recently published in the Journal of Mathematics and Arts. The paper focuses on a third shape within the drawing, that of an equilateral triangle “hiding in plain sight” between the man’s parted legs. This crucial shape is even referred to by Leonardo in his accompanying text.

According to Sweeney, this triangle corresponds to “Bonwill’s triangle,” formed by the contact point of the mandibular central incisors and the right and left mandibular condyles. This principle of modern medicine was established in 1864 by the dentist William Bonwill to explain how the human jaw functions optimally.

In much the same way, Leonardo uses the triangle to relate “the figure’s static positioning to its dynamic capability,” Sweeney explained in his paper.

“The square contains the figure in static cruciform pose—arms horizontal and legs together—while the circle encompasses the figure with arms raised and legs spread to form the equilateral triangle,” he explained. “This geometric relationship between static form and dynamic potential mirrors the structural-functional relationships that govern optimal craniofacial architecture.”

As such, Leonardo’s geometric insights converge with modern dental architecture studies, even as this structural principle wasn’t formally recognized until the 20th century.

An equilateral triangle can be found between the figure’s legs in Leonardo da Vinci’s Vitruvian Man. Image courtesy of Dr Rory Mac Sweeney.

This finding indicates another surprising geometric relationship that Sweeney believes may have been intuited by Leonardo ahead of his time. If the equilateral triangle between the figure’s legs in Vitruvian Man is replicated six times around his navel, it creates the hexagonal pattern behind the ratio of approximately 1.64—a tetrahedral ratio—between the square’s side and the circle’s radius.

This approximation of the tetrahedral ratio is crucial because, as Sweeney explained, the ratio is “a mathematical relationship that defines optimal spatial arrangements in both synthetic and biological systems.”

“Leonardo’s geometric construction successfully encoded fundamental spatial relationships in human form, demonstrating the remarkable precision of his Renaissance vision of mathematical unity between the human figure and natural order,” concluded Sweeney. “Vitruvian Man stands as a testament to Leonardo’s insight that human proportions reflect deeper mathematical principles governing optimal spatial organization.”