Gielis Supershape
r = (abs(cos((m · θ)/4))^(n₂) + abs(sin((m · θ)/4))^(n₃))^(−1/n₁)
Open in the app The dials and keys named below are the app's.
What it draws
Inside the brackets sit |cos(m·θ/4)| and |sin(m·θ/4)|, each raised to an exponent of its own and added. That pair runs through its pattern m/2 times in a turn of θ, with two tips each time, so the figure carries exactly m lobes. Setting m = 0 leaves a plain circle.
The outer power −1/n₁ turns that sum into a radius. At the opening values the bracket is 1 wherever one of the two terms is ±1 and the other 0, and it falls to 2·2^(−3.5) where they are equal in size. So the radius runs from 1 at the first set of angles out to ≈ 2.38 at the second, giving a five pointed star with rounded tips.
One formula, many shapes
Set m = 4 and all three exponents to the same p, and the radius becomes (|cos θ|^p + |sin θ|^p)^(−1/p). That is the superellipse |x|^p + |y|^p = 1: a circle at p = 2, a square standing on its corner at p = 1, and a rounded square as p grows. The other three letters are the freedom added on top. m makes the symmetry count anything at all, while n₂ and n₃ sharpen the two halves of each lobe separately.
n₁ is the letter with a sign. Negative values give radii that are the reciprocals of the positive ones, so at −2 the figure runs from 0.42 out to 1 and each tip has become a notch: the star is inside out. At n₁ = 0 the exponent is undefined and nothing is drawn. One more detail sets Turns of θ: the curve closes in a single turn while n₂ and n₃ agree, and needs two once they differ with an odd m.
History
Johan Gielis published this in 2003, in the American Journal of Botany, as one transformation covering a wide range of natural and abstract shapes. It generalises the superellipse, and it is usually called the superformula.
Try
- Drag m to 6: six lobes instead of five. The count follows the letter exactly, all the way to 10.
- Drag n₁ to −2 and the figure turns inside out, the five points becoming five notches on a body of radius 1.
- Drag n₃ to 10: the lobes stretch and lose their mirror symmetry, the tips reaching ≈ 3.0 where they reached 2.38.
- Replace the formula with superformula(θ, m, n₁, n₂, n₃). The outline is the same to the last digit: Wavelace has the formula as a function, which the three Supershape solids use.
- Drag n₁ to 7, so that all three exponents agree. The star deflates into a gently lobed blob, its radius between 1 and 1.28.