Leonardo da Vinci, the famous Italian polymath who painted the Mona Lisa, had an understanding of geometry far ahead of his time.
To create the Vitruvian Man—an illustration of the “ideal” human body—in 1490, the Renaissance artist likely relied on mathematical proportions that were 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 could explain why Leonardo da Vinci chose such specific proportions for the arms and legs.
A London dentist thinks he has finally solved the mystery, as a paper published last year revealed.
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Rory Mac Sweeney has discovered an important hidden detail in the crotch of the Vitruvian Man: an equilateral triangle that he believes could explain “one of the most analyzed yet mysterious works in the history of art.”
The Vitruvian Man was partly inspired by the writings of the Roman architect Vitruvius, who believed that the perfect human body should fit into both circles and squares.
Leonardo’s painting uses squares to precisely encompass the “cruciform pose” with arms outstretched and legs inward. Meanwhile, the circle encompasses the pose with arms raised and legs spread.
One popular explanation is that Leonardo da Vinci chose the proportions of the Vitruvian Man based on the theory of the golden ratio, but the measurements don’t exactly match up.
According to McSweeney, “the solution to this geometric riddle has been hiding in plain sight.”
“If you spread your legs…and raise your hands high enough so that your outstretched fingers touch the line above your head…the space between your legs will be an equilateral triangle,” Leonardo wrote in his notes for “The Vitruvian Man.”
When Mac Sweeney did the math on this triangle, he found that the spread of the man’s feet and the height of his belly button created a ratio of approximately 1.64 to 1.65.
This is very close to the tetrahedral ratio of 1.633 – a uniquely balanced geometric form formalized in 1917.
This ratio is used to determine the best way to fill the sphere. For example, if four spheres are connected as closely as possible into a pyramid shape, the height-to-base ratio from their center will be 1.633.
Perhaps Mac Sweeney recognized the importance of this number, as similar trigonometric principles have been used in dentistry since 1864.
Think of a human chin, the Bonville triangle determines the optimal position for its function. Its ratio is also 1.633.
McSweeney doesn’t think this is a coincidence.
Similar to minerals, crystals and other biological accumulation systems found in nature, McSweeney believes the human jaw is naturally organized around a tetrahedral geometry that maximizes mechanical efficiency.
McSweeney believes that if tetrahedral proportions repeat themselves around our bodies, it’s because “human anatomy evolved according to the geometric principles that govern optimal spatial organization throughout the universe.”
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If McSweeney is right, Leonardo may have stumbled upon a universal principle while painting the Vitruvian Man.
Mike Sweeney writes: “The same geometric relationships that appear in optimal crystal structures, biological structures, and the Fuller coordinate system appear to be encoded in human proportions, suggesting that Leonardo intuited fundamental truths about the mathematical nature of reality itself.”
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Whether other scientists agree with McSweeney remains to be seen, but the fact that Leonardo da Vinci mentioned equilateral triangles in his notes suggests that what was between the legs of the Vitruvian Man was important.
The research was published in Journal of Mathematics and Art.
An earlier version of this article was published in July 2025.
