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A London dentist noticed something that historians, mathematicians, and artists hadn’t seen for 500 years — a hidden triangle in the groin of Leonardo da Vinci’s Vitruvian Man that solves one of the greatest enigmas in mathematics and art
For over 500 years, experts have tried to explain why the proportions of the Vitruvian Man do not fit the golden ratio — the number 1.618 that supposedly governs beauty in nature and art.
Furthermore, the measurements of the arms and legs in Leonardo da Vinci’s most famous drawing always varied between 1.64 and 1.65 — close to the golden ratio, but never exact.
Then, a London dentist named Rory Mac Sweeney looked at the drawing from an angle no one had tried. Consequently, he found the answer that had been hidden in plain sight for five centuries.
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What is the Vitruvian Man — and why it matters to modern science
For those unfamiliar with the story, the Vitruvian Man is an ink drawing created by Leonardo da Vinci around 1490. In it, a man with outstretched arms and legs appears simultaneously inscribed in a circle and a square.
Da Vinci was inspired by the Roman architect Vitruvius, who in the 1st century BC described the ideal proportions of the human body in his treatise De Architectura.
Over the centuries, the drawing has become the ultimate symbol of humanity’s search for order and harmony in nature. Moreover, it appears in logos, coins, t-shirts, and book covers around the world.
What few knew is that the proportions governing the drawing remained mathematically unexplained — until now.
The triangle Da Vinci hid in the Vitruvian Man 500 years ago
Mac Sweeney analyzed Leonardo da Vinci’s original notes and found a phrase that had been ignored for centuries.
Da Vinci wrote: “If you open your legs, the space between them will be an equilateral triangle.”
In practice, this means that Leonardo was not using the golden ratio as the basis for the Vitruvian Man. Instead, he was using the tetrahedral ratio of 1.633 — a number that appears in natural structures such as crystals, human skull proportions, and atomic organization.
Thus, the difference between 1.618 (golden ratio) and 1.633 (tetrahedral ratio) may seem insignificant. However, in geometry, this difference completely changes the system of proportions that governs the drawing.
“We were looking for a complicated answer, but Leonardo was already pointing to this triangle,” Mac Sweeney told the British newspaper Daily Mail.
The golden ratio never explained the Vitruvian Man — and now we know why
The golden ratio (1.618) is one of the most popular concepts in mathematics. Indeed, it appears in snail shells, flower petals, and spiral galaxies.
However, when mathematicians precisely measured the Vitruvian Man, the body’s proportions never exactly matched 1.618.
The relationship between the square and the circle in the drawing — the two elements that frame the body — varies between 1.64 and 1.65.
Consequently, for 500 years, scholars assumed that Da Vinci had made a small error or that the concept was merely approximate.
Mac Sweeney proved it was no error. In fact, Da Vinci was using a different geometric system — based on the tetrahedron, nature’s simplest and most efficient three-dimensional shape.
- Golden ratio: 1.618 — DOES NOT fit the Vitruvian Man
- Tetrahedral ratio: 1.633 — fits the actual measurements of the drawing
- Square/circle ratio in the drawing: 1.64 to 1.65
- Equilateral triangle: formed between the feet and the groin when the legs open
- Publication: Journal of Mathematics and the Arts, June 2026
- Researcher: Rory Mac Sweeney, dentist and mathematician from London
The Vitruvian Man mirrors crystals, skulls, and atoms
What makes the discovery even more surprising is where the tetrahedral ratio appears in nature.
For example, it governs the most efficient arrangement of spheres in a three-dimensional space — the so-called optimal packing.
Furthermore, it appears in the proportions of the human skull, in the atomic organization of crystals, and in geometric structures studied by Buckminster Fuller in the 20th century.
Mac Sweeney wrote in the study: “The same geometric relationships that appear in optimal crystalline structures, biological architectures, and Fuller’s coordinate systems appear to be encoded in human proportions.”
That is, Leonardo da Vinci intuited in 1490 — without calculators, without computers — a geometric relationship that formal science only defined in 1917.
This places Da Vinci not only as an artist but as a mathematical thinker four centuries ahead of his time.
Why a dentist solved what mathematicians couldn’t for 500 years
The irony of the discovery lies in the researcher’s profile. Rory Mac Sweeney is neither an art history professor nor a career mathematician. In fact, he is a London dentist with an amateur interest in geometry.
However, it was precisely this external perspective that allowed him to look at the drawing without the assumptions that had guided experts for centuries.
While historians sought complex relationships between the body and the Renaissance cosmos, Mac Sweeney simply measured the triangle that Da Vinci had described in his own notes.
“The solution to this geometric mystery was hidden in plain sight,” he told ScienceAlert.
Just as a planetary alignment observed only once in the history of astronomy revealed hidden patterns in the universe, Da Vinci’s triangle shows that the greatest mysteries are sometimes in the most obvious details.
The debate the discovery reignited — and what still needs to be proven
Despite the enthusiasm, Mac Sweeney’s hypothesis still faces skepticism. After all, the study was recently published in the Journal of Mathematics and the Arts, and the academic community has not yet had time to widely validate or dispute the conclusions.
Furthermore, some experts argue that the triangle may be an aesthetic, not structural, element — and that Da Vinci used multiple geometric systems simultaneously, not just one.
However, the fact that the actual measurements of the drawing align with the tetrahedral ratio (1.633) and NOT with the golden ratio (1.618) is hard to ignore.
The original Vitruvian Man is kept in the Gallerie dell’Accademia in Venice and rarely leaves the vault because it is sensitive to light.
Could the most analyzed drawing in art history still hold other secrets that no one has noticed — simply because everyone was looking in the wrong place?
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