Jan Bartek – AncientPages.com – The Vitruvian Man is a renowned drawing by Leonardo da Vinci, created around 1490. The writings of Vitruvius, an ancient Roman architect, inspired this artwork. It features a nude male figure in two overlapping positions with arms and legs extended, inscribed within both a circle and a square. Art historian Carmen C. Bambach has lauded it as one of the most iconic images in Western civilization.
Leonardo’s Vitruvian Man and Optimal Geometric Relationships. Leonardo’s construction demonstrates the same tetrahedral principles found in craniofacial architecture. The equilateral triangle (green) that Leonardo explicitly referenced between the figure’s legs, when replicated six times around the navel, creates the hexagonal pattern (yellow) that generates the measured ratio of approximately 1.64 between the square’s side and circle’s radius. This geometric relationship may represent Leonardo’s intuitive recognition. Credit: Journal of Mathematics and the Arts (2025). DOI: 10.1080/17513472.2025.2507568
While not the only drawing influenced by Vitruvius’s writings, this piece uniquely blends artistic and scientific principles, epitomizing the ideals of the High Renaissance. The drawing embodies Leonardo’s vision of ideal human proportions, based on Vitruvius’s concepts but also shaped by his own measurements and insights from contemporaries like Leon Battista Alberti.
Marcus Vitruvius Pollio, an ancient Roman architect, posited that the human body possesses harmonious proportions akin to those of a well-designed temple. He suggested that a human figure could be perfectly inscribed within both a circle and a square, though he did not offer any mathematical basis for this geometric relationship. Leonardo da Vinci later addressed this concept in his renowned drawing, yet he did not provide an explicit explanation of his method. For over five centuries, the precise technique by which da Vinci achieved this ideal fit in one of history’s most scrutinized drawings has remained elusive.
The geometric principles behind Leonardo’s depiction of the relationship between the circle and square have puzzled scholars. However, recent research has begun to reveal new insights into Da Vinci’s intricate illustration techniques.
In a recent study, Rory Mac Sweeney, a dentist based in London, highlights an intriguing detail in Leonardo da Vinci’s Vitruvian Man. He points out an equilateral triangle positioned between the figure’s legs, which is mentioned in da Vinci’s notes accompanying the drawing. Upon analysis, this shape was found to correspond with Bonwill’s triangle—a conceptual equilateral triangle in dental anatomy that plays a crucial role in ensuring the optimal function of the human jaw.
Marcus Vitruvius Pollio, an ancient Roman architect, posited that the human body possesses harmonious proportions akin to those of a well-designed temple. He suggested that a human figure could be perfectly inscribed within both a circle and a square, though he did not offer any mathematical basis for this geometric relationship. Leonardo da Vinci later addressed this concept in his renowned drawing, yet he did not provide an explicit explanation of his method. For over five centuries, the precise technique by which da Vinci achieved this ideal fit in one of history’s most scrutinized drawings has remained elusive.
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
The geometric principles behind Leonardo’s depiction of the relationship between the circle and square have puzzled scholars. However, recent research has begun to reveal new insights into Da Vinci’s intricate illustration techniques.
In a recent study, Rory Mac Sweeney, a dentist based in London, highlights an intriguing detail in Leonardo da Vinci’s Vitruvian Man. He points out an equilateral triangle positioned between the figure’s legs, which is mentioned in da Vinci’s notes accompanying the drawing. Upon analysis, this shape was found to correspond with Bonwill’s triangle—a conceptual equilateral triangle in dental anatomy that plays a crucial role in ensuring the optimal function of the human jaw.
“This analysis reveals that Leonardo’s Vitruvian Man incorporates geometric principles that anticipate modern understanding of optimal biological architecture. The convergence of Leonardo’s measured ratios (1.64–1.65) with the previously termed tetrahedral constant suggests that human anatomy has evolved according to geometric principles that govern optimal spatial organization throughout the universe. From the hexagonal close packing of atoms to the tetrahedral architecture of dental occlusion to the cranial proportions found exclusively in humans, we observe consistent mathematical relationships encoded in biological form.
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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. Vitruvian Man stands as a testament to Leonardo’s insight that human proportions reflect deeper mathematical principles governing optimal spatial organization,” Mac Sweeney wrote in his study.
The study was published in the Journal of Mathematics and the Arts
Written by Jan Bartek – AncientPages.com Staff Writer
