The superformula is a generalization of the superellipse and was proposed by Johan Gielis around 2000. Gielis suggested that the formula can be used to describe many complex shapes and curves that are found in nature. Gielis has filed a patent application related to the synthesis of patterns generated by the superformula, which expired effective 2020-05-10. In polar coordinates, with the radius and the angle, the superformula is: By choosing different values for the parameters and different shapes can be generated. The formula was obtained by generalizing the superellipse, named and popularized by Piet Hein, a Danish mathematician. 2D plots In the following examples the values shown above each figure should be m, n1, n2 and n3. A GNU Octave program for generating these figures function sf2d(n, a) u = [0:.001:2 * pi]; raux = abs(1 / a(1) .* abs(cos(n(1) * u / 4))) .^ n(3) + abs(1 / a(2) .* abs(sin(n(1) * u / 4))) .^ n(4); r = abs(raux) .^ (- 1 / n(2)); x = r .* cos(u); y = r .* sin(u); plot(x, y); end Extension to higher dimensions It is possible to extend the formula to 3, 4, or n dimensions, by means of the spherical product of superformulas. For example, the 3D parametric surface is obtained by multiplying two superformulas r1 and r2. The coordinates are defined by the relations: where (latitude) varies between −π/2 and π/2 and θ (longitude) between −π and π. 3D plots 3D superformula: a = b = 1; m, n1, n2 and n3 are shown in the pictures. A GNU Octave program for generating these figures: function sf3d(n, a) u = [- pi:.05:pi]; v = [- pi / 2:.05:pi / 2]; nu = length(u); nv = length(v); for i = 1:nu for j = 1:nv raux1 = abs(1 / a(1) * abs(cos(n(1) .* u(i) / 4))) .^ n(3) + abs(1 / a(2) * abs(sin(n(1) * u(i) / 4))) .^ n(4); r1 = abs(raux1) .^ (- 1 / n(2)); raux2 = abs(1 / a(1) * abs(cos(n(1) * v(j) / 4))) .^ n(3) + abs(1 / a(2) * abs(sin(n(1) * v(j) / 4))) .^ n(4); r2 = abs(raux2) .^ (- 1 / n(2)); x(i, j) = r1 * cos(u(i)) * r2 * cos(v(j)); y(i, j) = r1 * sin(u(i)) * r2 * cos(v(j)); z(i, j) = r2 * sin(v(j)); endfor; endfor; mesh(x, y, z); endfunction; Generalization The superformula can be generalized