In his 2025 paper in the Journal of Mathematics and the Arts, Mac Sweeney proposes that Leonardo appears to provide the clue in his own writing. In the annotation below the drawing, Leonardo states that the space between the spread legs forms an equilateral triangle. The equilateral triangle thus becomes the candidate mediating geometry.

The construction follows from elementary Euclidean geometry – a compass-and-straightedge derivation available in Leonardo’s time. Two equal circles intersect, each passing through the other’s centre, forming the figure known as the vesica piscis. Within their intersection, an equilateral triangle and a square share a common baseline. The diagonal of the square divided by the altitude of the triangle yields precisely √(8/3), approximately 1.633

That is within one per cent of the proportion measured on Leonardo’s drawing.

This ratio expresses the proportional relationship between the equilateral triangle and the square. It appears in whichever pair of dimensions the construction uses. In the vesica, it appears between the square’s diagonal and the triangle’s altitude. In Leonardo’s drawing, the triangle’s side equals the circle’s radius – so the ratio appears between the square’s side and the circle’s radius, the very proportion measured on the drawing.

Different pairings, the same proportional content. The invariant is the equilateral geometry that links radial and orthogonal organisation.

The same geometry recurs when radial systems organise into stable arrangements in space. Spheres are the purest radial objects. When equal spheres are arranged as densely as possible, equilateral triangles and regular tetrahedra emerge in the local geometry of the packing, and the same √(8/3) ratio appears in the close-packed structure. That this is the densest possible arrangement of equal spheres was conjectured by Johannes Kepler in 1611 and proved by Hales and Ferguson in 1998.

This is how Dr Rory Mac Sweeney proposes Leonardo da Vinci inscribed the human body in both a circle and a square: not by squaring the circle, but by mediating between radial and orthogonal geometry through the equilateral triangle.

The proportional signature of that mediation is √(8/3)