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:


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


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

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.



OSCAR'S BUTTERFLY POLAR EQUATION

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