# Butterfly curve (transcendental)

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

The butterfly curve is a transcendental plane curve discovered by Temple H. Fay. The curve is given by the following parametric equations:

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

or by the following polar equation:

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