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Art and culture · Visual arts and geometry

The golden ratio: myth and fact

The golden ratio is said to be in sunflowers, the Parthenon, the Mona Lisa and credit cards. Which is true and which is exaggeration? Learn the maths of φ ≈ 1.618, its link with the Fibonacci numbers and how to test the claims.

  • 18 min
  • 2 simulations
  • Mini quiz

Updated:

Preview

The question

An image doing the rounds online: a spiral drawn over the Mona Lisa's face, a seashell, a Greek temple and a company logo. The same number is written above each one: 1.618. "The golden ratio is the universe's formula for beauty," it says. Can such a bold sentence be true? How do we test whether a claim is true?

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Number line

Where do the Fibonacci ratios go?

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Prediction

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What happens to the ratio of two consecutive numbers in the Fibonacci sequence as the numbers get bigger?

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Number line

Is A4 paper in the golden ratio?

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Prediction

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Which do you think the ratio of the long side to the short side of a sheet of A4 is closest to?

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Frequently asked

5 questions

Is the golden ratio really everywhere?

No. Claims that the Parthenon, the Mona Lisa or the nautilus shell are shaped by the golden ratio are unproven or false. The link in the arrangement of leaves and seeds in plants is real, but it does not hold in every case.

Does any ratio around 1.6 count as the golden ratio?

No. If you accept such a wide range, almost everything turns out "golden". Remembering the cases that fit and ignoring the ones that don't is called confirmation bias. Where to measure and how much deviation to allow must be decided in advance.

So is the golden ratio completely made up?

No. φ = (1 + √5)/2 ≈ 1.618 is a real and rich number: its square is exactly 1 more than itself, and in a regular pentagon the ratio of a diagonal to a side is exactly φ. What is exaggerated is the claim that it is a universal formula for beauty.

What does the Fibonacci sequence have to do with the golden ratio?

In the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34…) each number is the sum of the two before it. If you divide consecutive numbers, the ratios get closer and closer to φ: 8/5 = 1.6; 13/8 = 1.625; 21/13 ≈ 1.615; 34/21 ≈ 1.619.

Is A4 paper in the golden ratio?

No. The sides of A4 are in the ratio √2 ≈ 1.414, and this is a deliberate choice: fold A4 in half and you get A5 with the same ratio. A credit card is about 1.586, and widescreens are 16:9 ≈ 1.78.