# Butterfly curve (transcendental)

The butterfly curve.

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

## Equation

An animated construction gives an idea of the complexity of the curve (Click for enlarged version).

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]