The butterfly curve is a transcendental plane curve discovered by Temple H. Fay of University of Southern Mississippi in 1989.^[1]

Equation

The curve is given by the following parametric equations:^[2]

${\displaystyle x=\sin t\!\left(e^{\cos t}-2\cos 4t-\sin ^{5}\!{\Big (}{t \over 12}{\Big )}\right)}$

${\displaystyle y=\cos t\!\left(e^{\cos t}-2\cos 4t-\sin ^{5}\!{\Big (}{t \over 12}{\Big )}\right)}$

${\displaystyle 0\leq t\leq 12\pi }$

or by the following polar equation:

${\displaystyle r=e^{\sin \theta }-2\cos 4\theta +\sin ^{5}\left({\frac {2\theta -\pi }{24))\right)}$

The sin term has been added for purely aesthetic reasons, to make the butterfly appear fuller and more pleasing to the eye.^[1]

Developments

In 2006, two mathematicians using Mathematica analyzed the function, and found variants where leaves, flowers or other insects became apparent.^[3]