Euclid's 'division of a line in extreme and mean ratio' in Book VI of the Elements (c. 300 BC)
The first surviving definition, used to construct the pentagon and the icosahedron
Science · Mathematics
The golden ratio
c. 300 BC – present
Mathematics — proportion and geometry
The proportion in which a line is divided so that the whole is to the larger part as the larger part is to the smaller, giving the irrational number (1 + √5)/2, approximately 1.618. Euclid called it division "in extreme and mean ratio" in Book VI of the Elements. Luca Pacioli's De divina proportione of 1509, illustrated by Leonardo da Vinci, gave it its mystical reputation, and the American mathematician Mark Barr assigned it the Greek letter phi in the early twentieth century. It is the most irrational of all numbers in the sense that it is hardest to approximate by fractions, which is why it governs the spiral packing of sunflower seeds and pine cones.
How to spot them
- ▸Approximately 1.618, exactly (1 + √5)/2, and denoted by the Greek letter phi
- ▸Euclid's 'division of a line in extreme and mean ratio'
- ▸Its square is itself plus one; its reciprocal is itself minus one
- ▸The limit of the ratio of consecutive Fibonacci numbers
- ▸The continued fraction made entirely of 1s, which makes it the 'most irrational' number
Key facts
Pacioli's 1509 treatise on divine proportion, with Leonardo's drawings of solids (1509)
De divina proportione — the book that made it famous and gave it theological glamour
The rectangle you can cut a square from and be left with a smaller copy of itself
The golden rectangle, whose repeated subdivision traces a logarithmic spiral
The diagonal of a regular pentagon divided by its side
Exactly this proportion, which is why pentagons and pentagrams are full of it
The continued fraction 1 + 1/(1 + 1/(1 + …)), made entirely of ones
The slowest-converging continued fraction, hence the 'most irrational' number
The angle of about 137.5 degrees between successive seeds in a sunflower head
The golden angle — the packing that leaves no gaps and no repeating rows
The ratio approached by dividing each term of Fibonacci's rabbit sequence by the one before
The limit of the ratio of consecutive terms, proved by Kepler
Key ideas
- It is the positive solution of x² = x + 1, so its square is one more than itself and its reciprocal is one less — the only number with both properties.
- Many claims about it in art and architecture — the Parthenon, the Great Pyramid, Leonardo's Mona Lisa — are retrospective and unsupported. Where it genuinely appears is in plant phyllotaxis.
Quick-fire facts
- •Its exact value is (1 + √5)/2 ≈ 1.6180339887, and it satisfies x² = x + 1
- •Euclid treats it in Book VI of the Elements as division 'in extreme and mean ratio'
- •Luca Pacioli's De divina proportione was published in Venice in 1509 with illustrations by Leonardo da Vinci
- •The name 'golden section' (goldener Schnitt) dates only from the 1830s, and the symbol phi was introduced by Mark Barr around 1909, after the sculptor Phidias
- •Kepler proved that the ratio of consecutive Fibonacci numbers tends to it, and called it one of geometry's two great treasures alongside Pythagoras' theorem
- •Its continued fraction is [1; 1, 1, 1, …], making it the number least well approximated by rationals
- •The golden angle, 360° divided by its square, is about 137.5° and governs the arrangement of leaves and seeds
- •Penrose tilings, discovered by Roger Penrose in 1974, have this ratio built into their kite and dart tiles
- •Le Corbusier's Modulor system of proportions, published in 1948, was explicitly based on it
- •The successive powers of the number are themselves related by the same additive rule: each is the sum of the two before
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Sources
- MacTutor History of Mathematics Archive, University of St Andrews — The Golden ratio
- Mario Livio — The Golden Ratio: The Story of Phi, the World's Most Astonishing Number