Five reasons
Pacioli gave five reasons for “divine”: the proportion is unique, its three terms recall the Trinity, no ratio of whole numbers expresses it, it never varies, and the dodecahedron, Plato’s figure for the heavens, needs it.
42 / ANTIQUITY TO THE RENAISSANCE
Luca Pacioli
Cut a rod so that the whole is to the longer piece as the longer piece is to the shorter. Pacioli called this the divine proportion, and it shapes every face of the dodecahedron.
Explore the mathematics ↓
01 / THE DIVINE PROPORTION
Where should a rod be cut?
Drag the cut until the whole is to the longer piece as the longer piece is to the shorter.
Check it with Pacioli’s own test, then find the same cut in a face of the dodecahedron.
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THE IDEAOnly one cut works: 6.18 along a rod of 10. There, whole ÷ longer = longer ÷ shorter = φ = (1 + √5) ÷ 2 ≈ 1.618, and the square on the longer piece equals the rectangle of the whole and the shorter piece. A face of the dodecahedron is cut the same way: diagonal ÷ side = φ.
The person behind the ideas
- Published the Summa, a compendium of arithmetic, geometry and bookkeeping, in 1494.
- Leonardo da Vinci drew the solids for Divine Proportion while studying mathematics with him in Milan.
- Published Divine Proportion in Venice in 1509.
More of the mathematics
A cut that repeats
Cut the longer piece by the same rule and its longer part is exactly the old shorter piece. The cut repeats at every size, which is the invariance Pacioli praised.