Golden ratio
Also called phi, extreme and mean ratio, divine proportion, golden section
Euclid’s Elements; the golden name is modern
Geometry symbolsGolden ratio meaning
The golden ratio is a cut where the whole stands to the longer part as that part stands to the shorter. Euclid constructed it and did not call it golden. The Parthenon and nautilus stories are mostly measurement after the fact.
A cut where the whole stands to the larger part as that part stands to the smaller.
The name. Euclid says extreme and mean ratio: the whole to the greater part as the greater to the less. Phi, the letter, is an early-twentieth-century nickname, usually credited to Mark Barr, after the sculptor Phidias.
Where it comes from
Euclid constructs it (Elements II.11, VI.30) and uses it to build the pentagon. He does not call it golden, divine, or beautiful. Luca Pacioli’s De divina proportione (1509), with drawings by Leonardo, is the Renaissance sermon on its properties. The German goldener Schnitt is nineteenth-century; Martin Ohm’s textbook of 1835 is the citation usually given. Consecutive Fibonacci numbers approach the ratio. Kepler noticed that kinship. Fibonacci’s rabbit problem is not a theory of art.
Geometry
A proportion, not a mood
Take a length. Cut it so the long piece is to the short as the whole was to the long. Do it again and the same shape returns. The number is (1 + √5) / 2, about 1.618, and it is irrational. In the pentagram the diagonals and the sides are in this ratio, which is why the star will not sit on a rational grid. That is the content. Awe is optional and should be aimed at the cut, not at a catalog of buildings.
Modern
The ratio is real. The treasure hunt is not.
The Parthenon, the nautilus, the Mona Lisa, and the human body are regularly measured until they yield 1.618. The measurements move. A logarithmic spiral is not a golden spiral unless its growth is this one. Phidias did not leave a note. Pacioli’s “divine” is a theologian’s adjective for a mathematical property he found marvelous, not a survey of Greek temples. The pentagram in this cabinet is a place the ratio actually lives.
Near this
- PentagramFive lines in one stroke. The doctrines came later, and they disagree.
- Vesica piscisTwo equal circles, each through the other’s center. A third shape appears.
- PiHow many diameters fit around a circle. The letter came later, from a printer’s book.
- Radical signThe check that asks what number, times itself, makes this one.
Sources
- Euclid, Elements II.11 and VI.30
- Luca Pacioli, De divina proportione (1509)
- Martin Ohm, 1835, for “goldener Schnitt”
Named when the entry depends on a text, a date, or a standard. Catalog numbers are not invented.
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symbol.wiki, edited as Sigilla. Last reviewed 2 October 2026. A reading of public tradition, not a ruling.