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Dentist may have solved 500-year-old mystery in da Vinci's iconic Vitruvian Man
https://phys.org/news/2025-07-dentist-year-mystery-da-vinci.htmlby Paul Arnold, Phys.org
Leonardo's Vitruvian Man and Optimal Geometric Relationships
A London-based dentist may have cracked a centuries-old mathematics puzzle hidden in one of the most famous anatomical drawings in the world—Leonardo da Vinci's Vitruvian Man. This discovery suggests the iconic image reflects the same design blueprint frequently found in nature.
The pen-and-ink drawing of a nude male figure in two superimposed poses, with arms and legs enclosed within a circle and a square, was created by the Renaissance polymath around 1490. It is a study of the ideal human form, partly influenced by the writings of Roman architect Marcus Vitruvius Pollio, who believed the human body has harmonious proportions, just like a well-designed temple.
He proposed that a human figure could fit perfectly inside a circle and a square, but provided no mathematical framework for this geometric relationship. Da Vinci solved it but did not explicitly explain how.
For more than 500 years, how he achieved this perfect fit in one of the world's most analyzed drawings has remained a mystery. It's one that has generated numerous theories and ideas, such as the Golden ratio (1.618...). But none have matched the actual measurements.
However, a new study by London-based dentist Rory Mac Sweeney, published in the Journal of Mathematics and the Arts, is finally providing some answers to da Vinci's geometric method.
Hidden triangle
The paper describes a hidden detail in the Vitruvian Man, namely an equilateral triangle between the man's legs referenced in da Vinci's notes for the drawing. Analysis revealed that this shape corresponds to Bonwill's triangle, an imaginary equilateral triangle in dental anatomy that governs the optimal performance of the human jaw.
. . .
The pen-and-ink drawing of a nude male figure in two superimposed poses, with arms and legs enclosed within a circle and a square, was created by the Renaissance polymath around 1490. It is a study of the ideal human form, partly influenced by the writings of Roman architect Marcus Vitruvius Pollio, who believed the human body has harmonious proportions, just like a well-designed temple.
He proposed that a human figure could fit perfectly inside a circle and a square, but provided no mathematical framework for this geometric relationship. Da Vinci solved it but did not explicitly explain how.
For more than 500 years, how he achieved this perfect fit in one of the world's most analyzed drawings has remained a mystery. It's one that has generated numerous theories and ideas, such as the Golden ratio (1.618...). But none have matched the actual measurements.
However, a new study by London-based dentist Rory Mac Sweeney, published in the Journal of Mathematics and the Arts, is finally providing some answers to da Vinci's geometric method.
Tetrahedral Geometry in Human Craniofacial Architecture. The human skull demonstrates natural organization around tetrahedral and triangular geometries that maximize mechanical efficiency. Bonwill's triangle (green) forms an equilateral triangle connecting the two mandibular condyles and the midpoint of the lower central incisors. The Circle of Spee (purple) and Curve of Spee (blue) represent optimal arrangements for dental occlusion that naturally emerge from these tetrahedral relationships. This geometric organization maximizes force transmission efficiency during mastication, revealing the tetrahedral ratio's presence in human anatomical optimization. Credit: Journal of Mathematics and the Arts (2025). DOI: 10.1080/17513472.2025.2507568
Hidden triangle
The paper describes a hidden detail in the Vitruvian Man, namely an equilateral triangle between the man's legs referenced in da Vinci's notes for the drawing. Analysis revealed that this shape corresponds to Bonwill's triangle, an imaginary equilateral triangle in dental anatomy that governs the optimal performance of the human jaw.
. . .