What is the golden ratio? The number behind the myths

The golden ratio is the number (1 + √5) ÷ 2, about 1.618, written with the Greek letter φ (phi). It comes from one way of cutting a line in two: the whole line is to the longer part exactly as the longer part is to the shorter. That simple rule gives a number with unusual properties, which turns up in pentagons and in Fibonacci numbers, and which has attracted more myths than almost any other in mathematics.

In short

• φ is the only positive number that is one more than its reciprocal and one less than its square: 1/φ = 0.618… and φ² = 2.618….
• It shows up in real mathematics: the regular pentagon, the Fibonacci sequence, and the spiral arrangement of seeds in some plants.
• Most famous claims, that the Parthenon, the Mona Lisa or beautiful faces follow it, don't survive careful measurement.

How is the golden ratio defined?

Take a line and cut it into a long part and a short part. There is exactly one place to cut where the ratio of the whole to the long part equals the ratio of the long part to the short part. That shared ratio is the golden ratio. The Greek geometer Euclid, writing around 300 BC, called it cutting a line in "extreme and mean ratio".

Finding the number takes one line of algebra. Call the short part 1 and the long part x. The whole is then x + 1, and the rule says (x + 1) ÷ x = x ÷ 1. Multiply both sides by x and it becomes x² = x + 1, or x² − x − 1 = 0. The quadratic formula gives x = (1 + √5) ÷ 2. Since √5 is 2.2360679…, x = 3.2360679… ÷ 2 = 1.6180339…. The other solution is negative, so it can't be a length.

Worked example: cutting a 100 cm line

• The long part is the whole divided by φ: 100 ÷ 1.618 ≈ 61.8 cm.
• The short part is what's left: 100 − 61.8 = 38.2 cm.
• Check the first ratio: whole ÷ long = 100 ÷ 61.8 ≈ 1.618.
• Check the second: long ÷ short = 61.8 ÷ 38.2 ≈ 1.618.

Both ratios come out the same, and no other cut does that. Cut at 60 cm instead and the two ratios are 1.667 and 1.5, which no longer match.

What makes φ unusual?

The equation x² = x + 1 is the source of everything special about φ. Divide it by x and you get x = 1 + 1/x, so φ is one more than its own reciprocal: 1/φ = 0.618…, the same digits after the decimal point. Read as it stands, the same equation says φ² = φ + 1 = 2.618…. Every power of φ is the sum of the two powers below it, which is the same adding rule that builds the Fibonacci numbers.

That is why the Fibonacci sequence closes in on φ: divide each Fibonacci number by the one before and the answers home in on 1.618, as the Fibonacci note shows ratio by ratio. Geometry gives other routes to it:

• The golden rectangle. A rectangle with sides in the ratio φ to 1 has a strange property: cut a square off one end and the piece left over is a smaller golden rectangle. You can keep cutting forever.
• The pentagon. In a regular pentagon, a diagonal divided by a side is exactly φ, and the diagonals of a five-pointed star cut each other in the golden ratio. This is probably how the Greeks met the number.
• The hardest number to approximate. Written as a continued fraction, φ is 1 + 1/(1 + 1/(1 + 1/…)), with ones all the way down. In a precise sense that makes it the irrational number that fractions approximate worst.

That last property explains the one place φ really does appear in nature. Many plants set each new seed or leaf about 137.5° around from the last, the "golden angle," which is 360° ÷ φ². Because φ is so badly approximated by fractions, the seeds never line up in neat rows and pack the seed head evenly. Like pi, φ is irrational: its decimals never end or repeat.

Where did the name come from?

The idea is ancient; the name is modern. Euclid's Elements defined the ratio and showed how to construct it. In 1509 the Italian mathematician Luca Pacioli, a friend of Leonardo da Vinci, published De Divina Proportione, the "divine proportion," with drawings of solids by Leonardo. The term "golden section," goldener Schnitt, seems to appear first in an 1835 German textbook by Martin Ohm, and the letter φ was first used by the American mathematician Mark Barr at the start of the 20th century, according to MathWorld, in honor of the Greek sculptor Phidias.

The dates matter for the myths. The belief that the golden ratio is the secret of beauty is a 19th-century idea, not an ancient one, and the mathematician Chris Budd notes that the claim that the Parthenon follows it dates only to the 1850s.

Is the golden ratio really in art, buildings and bodies?

Mostly no. Any building, painting or face has dozens of lengths whose ratios fall between 1 and 2. Measure enough of them, choose the end points afterward, and some will come out near 1.618. The table sets the common claims against the evidence, drawing on Budd's survey in Plus magazine.

Claim	What the evidence shows
The Parthenon was designed on the golden ratio	No Greek text says so, and the result depends on which points are measured; the temple's lines also curve on purpose
Leonardo built the Mona Lisa and the Vitruvian Man on it	No direct evidence he used it in any painting; the Vitruvian Man's proportions don't match it, and he wrote of whole-number ratios
Beautiful faces and the navel-to-height ratio follow φ	Not supported; human ratios vary widely, and the "golden face" grids are fitted after the fact
People prefer golden rectangles	The evidence is thin; studies show wide variation, and other rectangles often score as well or better
Le Corbusier used the golden ratio	True: the architect deliberately built his Modulor system of proportions on it in the 1940s

The claims about sunflowers and the nautilus shell are checked in the note on the Fibonacci sequence. The honest cases are the deliberate ones. When a modern architect or designer uses φ on purpose, it's there because they put it there.

Worked example: the Great Pyramid

The Great Pyramid of Giza is the strongest-looking case, so it's worth doing the numbers. Its faces rise at about 51.8°. Egyptian builders set slopes by the seked, the horizontal run for each cubit of height. For the Great Pyramid that was 5½ palms of run for each 7 palms of rise.

• Slant height ÷ half the base: √(7² + 5.5²) ÷ 5.5 = √79.25 ÷ 5.5 = 8.902 ÷ 5.5 ≈ 1.6186. φ is 1.6180.
• Perimeter of the base ÷ height: (4 × 11) ÷ 7 = 44 ÷ 7 ≈ 6.286. Twice pi is 6.283.

The same slope lands close to both φ and π. It can't have been designed to hit both exactly, and one simple whole-number rule produces both near-misses. The numbers alone can't show intent.

How can you check a golden ratio claim?

A short checklist sorts most of them:

1. Is there a document? Ask for a plan, a letter or a treatise from the maker that mentions the ratio. For anything before the 1800s, there almost never is.
2. Who chose the points? If the rectangle or spiral was drawn on top of the picture afterward, it shows only that someone could draw it.
3. How close is close? A ratio of 1.6 or 1.65 is not 1.618. Ask how far off the claim is, and how far off other simple ratios such as 3:2 or 5:3 would be.
4. Were other ratios tested? A building with 40 measurements will have a few near any number you like.

This is ordinary critical thinking applied to geometry, and the same habit of testing a pattern against chance is how science tells a real effect from a coincidence.

Check yourself

• Without looking back, explain why the equation x² = x + 1 follows from the definition of the golden cut.
• Why do ratios of Fibonacci numbers close in on φ?
• Why does the Great Pyramid's slope come close to both φ and π, and what does that say about intent?

Sources

• Euclid's Elements, Book VI, Definition 3 — David E. Joyce, Clark University
• Golden Ratio — Eric W. Weisstein, MathWorld, Wolfram Research
• Myths of maths: The golden ratio — Chris Budd, Plus Magazine, University of Cambridge, 2020
• Great Pyramid of Giza — Wikipedia
• Golden ratio — Wikipedia