Golden Ratio Calculator

What Is the Golden Ratio? The golden ratio (φ, phi) is a special mathematical constant approximately equal to 1.6180339887. It is the ratio of two quantities where the ratio of the whole to the larger part equals the ratio of the larger part to the smaller part.

Definition If A > B, then (A + B) / A = A / B = φ ≈ 1.6180339887

Exact Value φ = (1 + √5) / 2 ≈ 1.6180339887… 1/φ = φ − 1 ≈ 0.6180339887…

Remarkably, 1/φ = φ − 1. No other number has this property.

Fibonacci Connection The Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89… As the sequence progresses, the ratio of consecutive terms (8/5, 13/8, 21/13…) converges to φ. The error roughly divides by φ² ≈ 2.618 at each step, so you gain about four decimal places every ten terms: F₁₁/F₁₀ is right to 3 places, F₂₁/F₂₀ to 8, and F₃₁/F₃₀ to 12.

Golden Rectangle A golden rectangle has sides in the ratio 1:φ. Cut a square off it and the remaining piece is another golden rectangle, and you can go on doing that forever.

Width W → Height H = W/φ (landscape) Width W → Height H = W × φ (portrait)

Golden Spiral A logarithmic spiral that grows by a factor of φ for every 90° turn. Found in nautilus shells, galaxy arms, and the arrangement of seeds in sunflowers.

Appearances in Nature

Phyllotaxis is the solid one. Sunflower seeds and pine cone spirals really do come in consecutive Fibonacci counts (8 and 13, or 13 and 21), and there is a mechanism behind it: each new primordium forms in the largest gap left by the previous ones, which drives the divergence angle to 360°/φ². That is a measurable, repeatable result, not a coincidence.

Most of the other nature claims are weaker than they sound, and it is worth separating them:

• Human body proportions. The forearm-to-hand and navel-to-height ratios are quoted at φ constantly. Measure a room full of people and you get a spread from roughly 1.5 to 1.7 with a mean somewhere near 1.6. That mean is close to φ, but it is close to plenty of other numbers too, and the spread is wide enough that the claim is not saying much.
• DNA. The B-DNA helix has a pitch of about 34 Å, which is a real measurement. The “21 Å wide” figure is not: the accepted diameter is about 20 Å. The 21 appears to have been rounded toward the nearest Fibonacci number to make the story work.
• Nautilus shells. A nautilus is a logarithmic spiral, but its growth factor is closer to 1.3 than to 1.618. It is a beautiful spiral that is not the golden spiral.

In Art and Architecture

The art claims deserve more scepticism than they usually get. Mario Livio and George Markowsky both went back to the sources and found very little:

• The Parthenon (447–432 BC) is the standard example, and it does not hold up. The measurements only give φ if you choose where the rectangle starts and stops, and different authors choose differently to get the answer they want. There is no evidence any Greek architect used φ as a design proportion.
• Leonardo’s Vitruvian Man (1490) follows the proportional system Vitruvius wrote down, which is built on simple whole-number fractions, not φ. Leonardo did illustrate Pacioli’s De Divina Proportione about the golden ratio in 1509, so he certainly knew it. He just did not use it there.
• Le Corbusier’s Modulor is the genuine case. He designed the system around φ on purpose in the 1940s and published the reasoning. Deliberate modern use is real; ancient use is mostly retrofitted.

One more property worth knowing. φ has the simplest possible continued fraction expansion, [1; 1, 1, 1, 1, …], where every coefficient is just 1. That makes it, in a precise mathematical sense, the most irrational number: the hardest of all to approximate well with simple fractions. This is the real reason φ shows up in leaf arrangements and seed spirals. A system placing things one after another around a stem wants an angle that never lets the pattern line up into spokes, and the angle that resists that longest is 360° / φ² ≈ 137.5°.

A note on Fibonacci retracement in finance. Stock traders use Fibonacci ratios (23.6%, 38.2%, 50%, 61.8%, 78.6%) as “support and resistance” levels on price charts, and the 61.8% level comes directly from 1/φ. Note that 50% is not a Fibonacci ratio at all, it just got adopted into the set. Whether any of it has predictive value is debated, but the practice is widespread enough to be partly self-fulfilling: traders watch for the levels, place orders there, and the chart sometimes pivots because of the orders rather than because of φ.