Superformula
Johan Gielis’s one formula for stars, flowers, squares and circles: four numbers reshape a single polar curve.
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Formulas and key values
- Polar equation
- Period · in angleIn the graph ≈ 2.513
- Unit circle
m sets the symmetry, n₁ the overall pinch, n₂ and n₃ the shape of the two halves of each lobe; here a = b = 1.
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Start the video01 · Definition
What is the Superformula?
In 2003 the biologist Johan Gielis published a generalisation of the superellipse in the American Journal of Botany: one polar formula with a symmetry number m and three exponents. Varying them produces circles, rounded polygons, stars and many shapes reminiscent of leaves, shells and flowers.
The formula is a superellipse wrapped around the circle m/4 times. With m = 4 and all three exponents equal to 2 it is exactly the unit circle; raising the exponents squares it off; lowering n₁ pinches the sides into a star.
02 · Properties
Key properties
- m = 4 with n₁ = n₂ = n₃ = 2 gives the unit circle.
- With n₂ = n₃ and whole-number m, the shape has m-fold rotational symmetry.
- m = 4, n₁ = n₂ = n₃ = n gives the superellipse |x|ⁿ + |y|ⁿ = 1.
- The radius repeats every 4π/m in angle.
- Published by Johan Gielis in 2003.
03 · Questions
Superformula: common questions
- What is the equation of the Superformula?
- Polar equation: r(φ) = (|cos mφ/4|^(n₂) + |sin mφ/4|^(n₃))^(-1/n₁). m sets the symmetry, n₁ the overall pinch, n₂ and n₃ the shape of the two halves of each lobe; here a = b = 1.
04 · Keep exploring