Golden ratio and golden spiral
The golden ratio, φ, about 1.618, is what you get by cutting a line so that the whole is to the longer part as the longer part is to the shorter. The golden spiral is drawn through a rectangle with its sides in that ratio. Luca Pacioli called it the divine proportion in 1509; many modern claims about it, from the Parthenon to the human body, have no evidence behind them.
Meaning
Euclid called it division in “extreme and mean ratio”, and his definition, as the astrophysicist Mario Livio quotes it, is that a line is so cut “when, as the whole line is to the greater segment, so is the greater to the lesser.” The ratio is half the sum of 1 and the square root of 5, or 1.6180339887…. A golden rectangle has its sides in that ratio, and cutting a square from it leaves a smaller golden rectangle; MathWorld notes that the points where the successive squares divide it lie on a logarithmic spiral, sometimes called the golden spiral. The letter phi (φ) for the ratio was apparently first used by Mark Barr at the beginning of the 20th century, after the Greek sculptor Phidias. The number belongs to mathematics. The meanings given to it come from Pacioli, who called it divine, and from modern sacred geometry, which Drunvalo Melchizedek’s publisher describes as “the form beneath our being”.
Origin and history
Euclid defines the division in Book VI of the Elements and uses it to construct the regular pentagon, the icosahedron and the dodecahedron. MacTutor, the history of mathematics archive at the University of St Andrews, reports Thomas Heath’s view that the pentagon construction that needs it was known to the Pythagoreans, which makes them the likeliest first students of it. For centuries it was a property of figures with no number attached. Fibonacci, in his Liber Abaci, gave the parts of a line of length 10 cut in the golden ratio as √125 − 5 and 15 − √125, and Kepler stated in a letter of 1609 that the ratios of neighbouring Fibonacci numbers tend to it.
The divine name is Luca Pacioli’s. His Divina proportione, published in three volumes in 1509, added little to what Euclid and others had proved, MacTutor says, but Pacioli found in the ratio attributes “all befitting God himself”: “just like God cannot be properly defined, nor can be understood through words, likewise this proportion of ours cannot ever be designated through intelligible numbers”. Leonardo da Vinci was a close friend of his. The first known decimal value came in 1597, in a letter from Michael Mästlin at Tübingen to his former student Johannes Kepler, who was drawn to the ratio’s mystical side; MacTutor calls Kepler’s writings on it “a mixture of good mathematics and magic”.
The name “golden section” is modern. Its first known appearance is a footnote in the 1835 second edition of Martin Ohm’s Die reine Elementar-Mathematik, which says that one is “in the habit of calling” the division the golden section, so MacTutor concludes the term was not Ohm’s own. MathWorld gives its first known use in English as an 1875 article on aesthetics in the ninth edition of the Encyclopaedia Britannica.
Is the Parthenon built on the golden ratio?
There is no evidence that it is. Keith Devlin, a mathematician at Stanford, wrote in 2007 that there is “not a shred of evidence” that the Greeks designed the Parthenon on the golden ratio, “and good reason to believe they did not”, and asked of the rectangles drawn over photographs of it: “why put it exactly there and why make the lines so thick?” He found no evidence for the claims about the Egyptian pyramids either, and none for the belief that Leonardo used the ratio in the Vitruvian Man. Euclid himself, Devlin points out, made no claims about beauty, and gave the ratio “the decidedly unromantic name” of extreme and mean ratio. MathWorld’s entry ends by saying that claims of the ratio appearing prominently in art, architecture, sculpture and anatomy “tend to be greatly exaggerated”, and names The Da Vinci Code among the places they are repeated.
Does the golden ratio make things beautiful?
The experiments disagree. In the 1860s Gustav Fechner asked people to pick the most pleasing of ten rectangles, and in Livio’s account 76% of choices fell on those with ratios of 1.75, 1.62 and 1.50, with a peak at the golden rectangle. Michael Godkewitsch concluded in 1974 that the preference was an artefact of where the golden rectangle sat in the range shown. Chris McManus found in 1980 “moderately good evidence” for Fechner’s effect but thought it “very unclear” whether the golden section mattered more than ratios near it, such as 1.5 or 1.6. Livio concludes that “we should abandon its application as some sort of universal standard for ‘beauty’”. Some artists did use it on purpose: Salvador Dalí gave his Sacrament of the Last Supper the ratio in its dimensions and set a dodecahedron over the table, and Le Corbusier built his Modulor system of proportions on it.
The golden spiral and the nautilus
The spiral usually drawn, as here, is a chain of quarter circles, one in each square of the golden rectangle. MathWorld notes that the true golden spiral, a logarithmic spiral, passes through the points where the squares divide the rectangle without being tangent to the sides there. The nautilus shell, often shown as a golden spiral, is not one: Devlin, correcting an earlier column of his own, wrote that the shell does grow in a logarithmic spiral, “But that constant angle is not the golden ratio.” He accepted the ratio’s part in plant growth, and Livio lists the leaf and seed arrangements of plants among its natural appearances.
In sacred geometry
In the sacred geometry taught by Drunvalo Melchizedek, the ratio is part of a design hidden in the body. His publisher lists among the subjects of The Ancient Secret of the Flower of Life (1999) “Leonardo da Vinci’s canon, the mystery of the Mason’s drawing, the golden mean, spirals, the Fibonacci sequence, and the divine phi ratio”. The word divine goes back to Pacioli in 1509. The claim about Leonardo and the body is one Devlin found no evidence for. The ratio also builds the dodecahedron in Euclid.
Where you’ll see it today
whatdoesthissymbolmean.comPhiφ is the usual sign for the golden ratio, first used by Mark Barr after the sculptor Phidias.Read next
- Next in Sacred geometryPlatonic solidsThe five regular solids: the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron, each with faces of one regular shape meeting the same way at every corner. Plato’s Timaeus gave four of them to the elements of fire, air, water and earth, and the fifth to the design of the universe.
- The same shape in Taoist symbolsFu talisman (符)A Daoist talisman: a strip of paper written in a script meant for the gods to read, not people. Daoists use fu (符) to summon deities, command demons and protect a place, and wear them, post them or burn them.
See also
Flower of life, Seed of life, Metatron’s cube, Vesica piscis.
Sources
- The Golden ratio, J J O’Connor and E F Robertson, July 2001, MacTutor History of Mathematics, University of St Andrews, article text
- The golden ratio and aesthetics, Mario Livio, 1 November 2002, Plus Magazine, University of Cambridge, article text
- The Myth That Will Not Go Away (Devlin’s Angle, May 2007), Keith Devlin, Mathematical Association of America (via the Internet Archive), column text
- MathWorld: Golden Ratio, Eric W. Weisstein, MathWorld (Wolfram Research), entry
- MathWorld: Golden Rectangle, Eric W. Weisstein, MathWorld (Wolfram Research), entry
- The Ancient Secret of the Flower of Life, Volume One (publisher’s description), Drunvalo Melchizedek, 1999, Light Technology Publishing, book description
The same shape elsewhere
Browse symbols formed from curves and waves, lines or squares and rectangles.
Researched, edited and approved by Rory Hansen before publishing. Last reviewed October 9, 2026. How we research
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