What is the golden ratio?

Where it comes from

From Euclid to “golden”

The first written definition is in Euclid’s Elements, around 300 BCE. Euclid did not call it golden. He called it cutting a line “in extreme and mean ratio”:

“A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.”Euclid, Elements, Book VI, Definition 3 · translated by T. L. Heath

Euclid needed it for geometry, not beauty: his construction of the cut is used later in the Elements to build the pentagon faces of the dodecahedron.

In 1509 the friar Luca Pacioli called it the divine proportion. His book Divina proportione was printed in Venice, and its figures were drawn by his friend Leonardo da Vinci. In 1597 the astronomer Michael Mästlin wrote its value to Johannes Kepler as about 0.6180340.

“Golden” came last. The German phrase goldener Schnitt, golden section, first appears in print in 1835, in a textbook by Martin Ohm, whose footnote suggests the name was already in use. The letter φ was suggested by the American Mark Barr in the early 1900s, reportedly after the Greek sculptor Phidias.

Where it really is

Places φ truly shows up

Every regular pentagon and five-pointed star. In a regular pentagon, each diagonal is exactly φ times the side, and the star’s lines cut each other in the golden ratio. Switch the machine above to Pentagram to see it measured.

Two of the five Platonic solids. The twelve corners of an icosahedron can be placed at the corners of three golden rectangles crossing at right angles. The dodecahedron is built from pentagons, so φ runs through it too.

Plant spirals. Through the golden angle, sunflower heads, pineapples and pine cones often show spirals in Fibonacci numbers. A 2016 study of 657 sunflowers grown by the public found Fibonacci structure in 82% of spiral counts.

Art made on purpose. Le Corbusier built his Modulor measuring system between 1943 and 1950 on the golden section and the Fibonacci numbers. Salvador Dalí’s 1955 The Sacrament of the Last Supper, at the National Gallery of Art in Washington, measures 166.7 × 267 cm, a ratio of 1.60, and has a huge dodecahedron over the table.

The details

What the measurements show

The Great Pyramid. Using a base of 755.79 feet and a height of 481.4 feet, the slant height divided by half the base comes to about 1.62, within 0.1% of φ. But four times the base divided by the height is 6.28, within 2% of 2π, so the same pyramid “proves” π as well. Markowsky found no sign the Egyptians knew of φ at all.

The Parthenon. With its measured width of about 101 feet and height of about 45 feet, the front comes out at 9 to 4, a ratio of 2.25. Drawings that show a golden rectangle on it leave parts of the building outside the box.

The nautilus. Clement Falbo measured shells at the California Academy of Sciences. They fit rectangles of about 1.33 to 1, ranging from 1.24 to 1.43. A golden spiral grows about 6.85 times per turn; the nautilus about 3.

The face. A 2008 review found the popular “phi mask” describes the faces of fashion models and fits most people’s preferences poorly. A 2010 study ran four experiments and found faces look best when the eyes-to-mouth distance is about 36% of the face’s length and the eye spacing about 46% of its width, the proportions of an average face.