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Leonardo da Vinci, the celebrated Italian polymath who painted the Mona Lisa, had a blase geometric knowing acold up of his time.
To gully the Vitruvian Man successful 1490 – an illustration of the 'ideal' quality body – the Renaissance creator whitethorn person relied connected a mathematical ratio not formally established until the 19th century.
It's 1 of the astir iconic images of each time, and yet for much than 500 years, nary 1 could decipher wherefore da Vinci chose specified circumstantial proportions for the arms and legs.
As revealed successful a paper published this year, a London dentist thinks he's solved the enigma astatine last.
Related: Da Vinci's Genetic Secrets May Soon Be Revealed by Ambitious DNA Project
Rory Mac Sweeney recovered a important hidden detail, tucked successful the Vitruvian Man's crotch: an equilateral triangle that he thinks whitethorn explicate "one of the astir analyzed yet cryptic works successful creation history."
The Vitruvian Man is partially inspired by the writings of Roman designer Vitruvius, who argued that the cleanable quality assemblage should acceptable wrong a ellipse and square.
Da Vinci's drafting uses a quadrate to precisely incorporate a 'cruciform pose', with arms outstretched and legs in. The circle, meanwhile, encompasses a posture wherever the arms are raised and the legs are spread.
The Vitruvian Man by Leonardo da Vinci. (Wikimedia Commons/Public Domain)A fashionable mentation is that da Vinci chose the Vitruvian Man's proportions based connected the Golden Ratio Theory, but the measurements don't rather lucifer up.
According to Mac Sweeney, "the solution to this geometric enigma has been hiding successful plain sight".
"If you unfastened your legs… and rise your hands capable that your extended fingers interaction the enactment of the apical of your head… the abstraction betwixt the legs volition beryllium an equilateral triangle," da Vinci wrote successful his notes for the Vitruvian Man.
When Mac Sweeney did the mathematics connected this triangle, helium recovered that the dispersed of the man's feet and the tallness of his navel created a ratio of astir 1.64 to 1.65.
The greenish equilateral triangle extending from the Vitruvian Man's crotch down to his feet. (Upscaled mentation of representation from Mac Sweeney, J. Math. Arts, 2025)That's precise adjacent to the tetrahedral ratio of 1.633 – a uniquely balanced geometric form, officially established successful 1917.
The ratio is utilized to found the optimum mode to battalion spheres. If 4 spheres are connected arsenic intimately arsenic imaginable into a pyramid shape, for instance, past the height-to-base ratio from their centers volition beryllium 1.633.
A perfectly balanced operation of tetrahedrons. (Mac Sweeney, J. Math. Arts, 2025)Perhaps Mac Sweeney recognized the value of that fig due to the fact that of a akin triangular rule utilized successful dentistry since 1864.
Imagined connected the quality jaw, the Bonwill triangle dictates the optimal presumption for its function. Its ratio is besides 1.633.
Mac Sweeney doesn't deliberation that's a coincidence.
Bonwill's triangle dictates optimal mechanical relation of the quality jaw. (Mac Sweeney, J. Math. Arts, 2025)Similar to minerals, crystals, and different biologic packing systems recovered successful nature, Mac Sweeney thinks the quality jaw people organizes astir tetrahedral geometries, which maximize mechanical efficiency.
Related: Da Vinci Masterpieces Hid a Secret Edible Ingredient, Study Suggests
If the tetrahedral ratio is repeated astir our bodies, Mac Sweeney thinks that is due to the fact that "human anatomy has evolved according to geometric principles that govern optimal spatial enactment passim the universe."
If Mac Sweeney is right, Da Vinci whitethorn person stumbled crossed a cosmopolitan rule portion drafting the Vitruvian Man.
"The aforesaid geometric relationships that look successful optimal crystal structures, biologic architectures, and Fuller's coordinate systems look to beryllium encoded successful quality proportions," writes Mac Sweeney, "suggesting that Leonardo intuited cardinal truths astir the mathematical quality of world itself."
Whether different scientists hold with Mac Sweeney remains to beryllium seen, but the information that da Vinci mentioned the equilateral triangle successful his notes suggests that what lies betwixt the Vitruvian Man's legs is important.
The survey was published successful the Journal of Mathematics and the Arts.
An earlier mentation of this nonfiction was published successful July 2025.
