Golden Ratio

Divide a line so that the whole is to the larger part as the larger is to the smaller and you get φ = (1 + √5)/2, the positive root of x² = x + 1. Its genuine mathematical appearances are elegant and numerous: consecutive Fibonacci ratios converge to it, it is the ratio of a regular pentagon's diagonal to its side, and its continued fraction is nothing but 1s, which by a theorem of Hurwitz makes it the hardest number of all to approximate with fractions. That last property is not decorative — it is why sunflower florets and pine cone scales spiral at 137.5°, the golden angle: an irrational turn that resists rational approximation packs seeds without ever lining them up into wasteful rows.

The claims about art and architecture are a different matter. Luca Pacioli's De Divina Proportione of 1509, illustrated by Leonardo, was a book about geometry that made no aesthetic claim; the modern mythology dates mainly to Adolf Zeising in the 1850s and was amplified by twentieth-century design writing. George Markowsky's 1992 paper "Misconceptions about the Golden Ratio" went through the standard examples one by one and found that the Parthenon fits only if you choose the rectangle carefully, the nautilus shell has a spiral ratio nearer 1.33, and there is no evidence the Great Pyramid or the Mona Lisa were designed around it. The mathematics needs no help from the folklore.