Golden Spiral
SpiralsThe logarithmic spiral that widens by the golden ratio φ every quarter turn, approximated by the Fibonacci arc construction.
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Equation
Open the Creator with a ready-made animation brief for this curve — review the scene plan and Manim code before rendering.
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.
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.
Where you'll see it
Golden-rectangle constructions, composition guides in photography and design, and Fibonacci-themed lessons.