Butterfly curve (transcendental)
# Butterfly curve (transcendental)

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

## Equation

The curve is given by the following parametric equations:

    $x=\sin t\!\left(e^{\cos t}-2\cos 4t-\sin ^{5}\!{\Big{t \over 12}{\Big )}\right)$
    $y=\cos t\!\left(e^{\cos t}-2\cos 4t-\sin ^{5}\!{\Big{t \over 12}{\Big )}\right)$
    $0\leq t\leq 12\pi$

or by the following polar equation:

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

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

## Developments

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

