The butterfly curve

In polar form, r = e^(sin θ) − 2cos(4θ) + sin⁵((2θ − π)/24). The first term, e^(sin θ), makes r larger above the x-axis than below it, giving the big upper wings and smaller lower ones. The −2cos 4θ term creates four lobes, the basic wing shape.

The last term is the subtle one. Because of the division by 24, it changes very slowly, so each trip around the origin is slightly different from the last. That is what layers the wings, and why the curve takes 12π, six full turns, to draw completely.

The parametric form above is the same curve written with x = r sin t and y = r cos t, which turns it upright. Make sure t runs to 12π; with the default 0 to 2π you get only the first layer. The calculator samples the curve adaptively, so the tight inner loops stay smooth.

Every term is a dial. Change −2cos 4θ to −2cos 6θ and the wings gain extra lobes; change e^(sin θ) to e^(cos θ) and the butterfly turns on its side; replace the fifth power with a cube and the layers spread further apart. Open the polar equation above, replace a number with a letter such as k, and accept the slider to explore.

Curves like this are a good reminder that polar equations are natural for shapes organised around a centre, from flowers to wings, because r directly controls how far each direction reaches.