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Illustrating a Friend's Book on Proportion

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Illustrating a Friend's Book on Proportion

In 1509 the Franciscan friar and mathematician Luca Pacioli published De Divina Proportione in Venice, a treatise on the golden ratio and its role in geometry, architecture and human proportion, and its geometric illustrations, more than sixty woodcuts of regular and semi-regular solids, were supplied by his friend Leonardo da Vinci, whom Pacioli had known since both worked in Milan under Ludovico Sforza's patronage. Leonardo's drawings show the solids as skeletal, transparent frames rather than solid blocks, letting a reader see through each shape to its structure, a genuine innovation in geometric illustration at the time.

Among the solids Leonardo illustrated is a compound of two regular tetrahedra, interpenetrating so their eight outer vertices mark the corners of a cube, which the book presents as an elevated octahedron. The astronomer Johannes Kepler would later name this same compound solid the stella octangula, over a century after Pacioli's book appeared. In the twentieth century, sacred-geometry writers identified this compound-tetrahedron shape with Merkabah, a Hebrew word for chariot properly belonging to an entirely separate and much older tradition of Jewish visionary mysticism concerned with Ezekiel's throne-vision, an identification with no basis in that tradition's own texts and dating only from the 1980s.

Leonardo's documented contribution was narrower and, on its own terms, considerable: precise, structurally legible illustrations of Euclidean solids for a mathematician's treatise, drawn by an artist whose engineering training let him render three-dimensional geometry with a clarity contemporary illustrators had not matched. The stella octangula's spiritual reputation arrived four and a half centuries after Leonardo put ink to the page.

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Sources
De Divina Proportione
Luca Pacioli, Paganino Paganini, 1509
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