Fermat Spiral
The two-armed spiral r² = a²θ whose coils pack tighter as they grow — the pattern behind sunflower seed heads.
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Formulas and key values
- Polar equation
- Area · each ring between consecutive turns of one arm
a is a scale constant (a = 1 in the graph); θ ≥ 0, and the two signs draw the two arms.
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Start the video01 · Definition
What is the Fermat Spiral?
Fermat’s spiral takes the square-root growth law r = a√θ, proposed by Pierre de Fermat in 1636. Because the radius grows ever more slowly, each successive coil sits closer to the previous one — the opposite feel of the logarithmic spiral. Taking both signs of the square root yields two arms that meet smoothly at the origin, dividing the plane into two congruent interlocking regions.
Its claim to fame is equal-area packing: Vogel’s 1979 sunflower model places the n-th seed at r = √n with the golden angle 137.5° between neighbors, filling the disk with uniform density. The same layout now positions mirrors in some concentrated solar power plants.
02 · Properties
Key properties
- Each ring between consecutive turns encloses the same area — the key to uniform seed packing.
- The full curve (both signs) is smooth at the origin, with no cusp.
- It is the special case n = 2 of the parabolic spirals r^n = aⁿθ.
- Coil spacing shrinks like 1/√θ as the spiral grows.
- Vogel’s model: r = c√n, θ = n × 137.5° reproduces sunflower heads.
03 · Questions
Fermat Spiral: common questions
- What is the equation of the Fermat Spiral?
- Polar equation: r^2 = a^2θ, r = ± a√θ. a is a scale constant (a = 1 in the graph); θ ≥ 0, and the two signs draw the two arms.
- What is the area of the Fermat Spiral?
- Area (each ring between consecutive turns of one arm): A = 2π^2 a^2. a is a scale constant (a = 1 in the graph); θ ≥ 0, and the two signs draw the two arms.
04 · Wonders
See it in a math wonder
05 · Keep exploring