What is the equation of the Logarithmic Spiral?

Polar equation: r = a e^(bθ). a is the radius at θ = 0 (a = 1 in the graph) and b sets how fast it widens.

What is the arc length of the Logarithmic Spiral?

Arc length (from the pole to θ): L = (√(1 + b^2))/b r(θ). a is the radius at θ = 0 (a = 1 in the graph) and b sets how fast it widens.

Spirals

Logarithmic Spiral

The self-similar spiral r = a·e^{^(bθ)} that crosses every radius at the same angle — nature’s growth curve.

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b (growth rate)0.17

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Formulas and key values

Polar equation: $r = a e^{b\theta}$
Growth per turnIn the graph ≈ 2.91: $e^{2\pi b}$
Tangent–radius angle: $\cot\alpha = b$
Arc length · from the pole to θ: $L = \frac{\sqrt{1 + b^2}}{b}\,r(\theta)$

a is the radius at θ = 0 (a = 1 in the graph) and b sets how fast it widens.

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01 · Definition

What is the Logarithmic Spiral?

The logarithmic (or equiangular) spiral r = a·e^{^(bθ)} multiplies its radius by a fixed factor with every turn, instead of adding a fixed amount. The result is perfect self-similarity: zooming in or out gives back the same spiral, only rotated. Jacob Bernoulli was so taken with the property that he asked for the curve on his tombstone with the motto “eadem mutata resurgo” — though the stonemason mistakenly carved an Archimedean spiral.

Its defining feature is the constant angle between the curve and every ray from the center, with cot of that angle equal to b. This is why hawks approach prey along one and why the nautilus shell, which grows without changing shape, follows it.

02 · Properties

Key properties

Equiangular: the tangent makes a constant angle α with every radius, cot α = b.
Self-similar: scaling by e^{^(2πb)} equals rotating by one turn.
Each turn multiplies the radius by e^{^(2πb)}.
Total arc length from the pole to angle θ is finite: r(θ)·√(1+b²)/b.
Special case b → 0 degenerates to a circle.

03 · Questions