Exploring the geometry of equilibrium
Introduction
I’m a London-based dental surgeon and independent researcher with an interest in geometric proportion and how a single ratio recurs across different structural systems.
My work centres on the ratio √(8/3) ≈ 1.633, which arises exactly in tetrahedral geometry and in hexagonal close-packed crystals.
In a peer-reviewed paper in the Journal of Mathematics and the Arts, I examine how this same ratio offers a candidate account of the proportions in Leonardo da Vinci’s Vitruvian Man, and I explore whether it may also be relevant to aspects of craniofacial and dental architecture.
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“A London dentist may have cracked Leonardo’s geometric code.”
“An elegant mathematical bridge between anatomy and art.”
How Did Leonardo Inscribe the Human Body in a Circle and a Square?
For over a century, scholars have debated how Leonardo settled on the precise relationship between the circle and the square in his Vitruvian Man — and no construction has won consensus. The answer, it turns out, may have been sitting in plain sight, in a line Leonardo wrote on the drawing itself.
Dr Mac Sweeney’s approach is to reframe the question geometrically. Start with what actually needs explaining: on the drawing, the side of the square divided by the radius of the circle measures roughly 1.64 to 1.65. That is the number any proposed construction has to reproduce. So set the human figure aside and ask a purely geometric question — what shape sits most naturally between a circle and a square? Leonardo’s own note answers it: “the space between the legs will be an equilateral triangle.” Taking that triangle as the link between the two and carrying the construction through gives a single, determinate value — √(8/3) ≈ 1.633, within one per cent of the measured proportion
Interestingly, the equilateral triangle Leonardo named in 1490 resembles Bonwill’s triangle, the geometric basis of dental occlusion, which wasn’t formally described until 1864 — a connection Dr Mac Sweeney explores in his work on craniofacial architecture.
my latest book
The Paradox of Lucid Dreaming
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