Leonardo da Vinci, the famous Italian polymath who painted the Mona Lisa, had a sophisticated geometric understanding far ahead of his time.
1490 in drawing the Vitruvian Man—an illustration of the “ideal” human body—the Renaissance artist could rely on a mathematical relationship that was not formally established until the 19th century.
It’s one of the most iconic images of all time, but for more than 500 years no one has been able to decipher why da Vinci chose such specific proportions of the hands and feet.
As revealed in a paper published earlier this year, a London dentist believes he has finally cracked the riddle.
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Rory Mac Sweeney has found an important hidden detail tucked into the crotch of the Vitruvian Man: an equilateral triangle that he believes may explain “one of the most scrutinized yet mysterious works in the history of art”.
The Vitruvian Man is partly inspired by the writings of the Roman architect Vitruvius, who argued that the perfect human body must fit inside a circle and a square.
Da Vinci’s drawing uses a square to accurately position the “cross pose” with arms outstretched and legs tucked in. Meanwhile, the circle includes a posture where the arms are raised and the legs are extended.
A popular explanation is that da Vinci chose the proportions of the Vitruvian Man based on the theory of the golden ratio, but the measurements do not exactly match.
According to Mac Sweeney, “the solution to this geometric mystery is hiding in plain sight.”
“If you open your legs … and raise your hands so that the outstretched fingers touch the line of the top of the head … the space between the legs will be an equilateral triangle,” da Vinci wrote in his notes to the Vitruvian Man.
When Mac Sweeney calculated this triangle, he found that the spread of a man’s feet and the height of his navel create a ratio of approximately 1.64 and 1.65.
This is very close to the tetrahedral ratio of 1.633, a uniquely balanced geometric form officially established in 1917.
The ratio is used to determine the optimal method of packing the balls. For example, if the four spheres are joined as closely as possible in the shape of a pyramid, then the ratio of the height to the base from their centers will be 1.633.
Perhaps Mac Sweeney recognized the significance of this number due to a similar triangle principle used in dentistry since 1864.
Imagining the human jaw, Bonwill’s triangle dictates its optimal function. It also has a ratio of 1.633.
Mac Sweeney doesn’t think that’s a coincidence.
Similar to minerals, crystals and other biological packing systems found in nature, Mac Sweeney believes that the human jaw naturally organizes itself around tetrahedral geometries that maximize mechanical efficiency.
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If the tetrahedral relationship repeats itself around our bodies, Mac Sweeney believes it’s because “human anatomy evolved according to geometric principles that govern optimal spatial organization throughout the universe.”
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If Mac Sweeney is right, Da Vinci may have stumbled upon a universal principle when he painted the Vitruvian Man.
“The same geometric relationships that appear in optimal crystal structures, biological architectures, and Fuller’s coordinate systems appear to be encoded in human proportions,” writes Mac Sweeney, “suggesting that Leonardo intuited fundamental truths about the mathematical nature of reality itself.”
Whether other scientists agree with Mac Sweeney remains to be seen, but the fact that da Vinci mentioned an equilateral triangle in his notes suggests that what is between the Vitruvian Man’s legs is important.
The study was published Journal of Mathematics and the Arts.
An earlier version of this article was published in 2025. in July