Golden Spiral
The logarithmic spiral that widens by the golden ratio φ every quarter turn, approximated by the Fibonacci arc construction.
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Formulas and key values
- Polar equation
- Growth per quarter turn
- Growth per turn
- Pitch b in r = ae^(bθ)
φ is the golden ratio and a is the radius at θ = 0.
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Start the video01 · Definition
What is the Golden Spiral?
The golden spiral is the logarithmic spiral whose growth factor is the golden ratio φ ≈ 1.618 per quarter turn, i.e. r = a·φ^(2θ/π). It inherits every property of the equiangular family — self-similarity above all — but ties the growth rate to the most famous constant in classical proportion theory.
The version most people meet is the Fibonacci spiral: quarter-circle arcs drawn inside a rectangle tiled by squares of side 1, 1, 2, 3, 5, 8… Because ratios of consecutive Fibonacci numbers converge to φ, the arc construction converges to the true golden spiral, though the two never exactly coincide.
02 · Properties
Key properties
- Growth factor φ per quarter turn; φ⁴ ≈ 6.854 per full turn.
- Equiangular with pitch b = ln φ / (π/2) ≈ 0.3063.
- The Fibonacci quarter-arc spiral is an approximation, not the curve itself.
- Removing the largest square from a golden rectangle leaves a golden rectangle — the geometric engine of the spiral.
- Claims of golden spirals in nautilus shells are mostly overstated; measured shells fit general logarithmic spirals with other pitches.
03 · Questions
Golden Spiral: common questions
- What is the equation of the Golden Spiral?
- Polar equation: r = a φ^(2θ/π), φ = (1+√5)/2 ≈ 1.618. φ is the golden ratio and a is the radius at θ = 0.
04 · Wonders
See it in a math wonder
05 · Keep exploring