Butterfly curve (transcendental)

For ${\displaystyle 0\leq t\leq 12\pi }$, the curve is given by the following parametric equations:^[2] $${\displaystyle {\begin{aligned}x&=\sin t\!\left(e^{\cos t}-2\cos 4t-\sin ^{5}\left({\frac {t}{12}}\right)\right)\\y&=\cos t\!\left(e^{\cos t}-2\cos 4t-\sin ^{5}\left({\frac {t}{12}}\right)\right)\end{aligned}}}$$ or by the following polar equation: $${\displaystyle r=e^{\sin \theta }-2\cos \left(4\theta \right)+\sin ^{5}\left({\frac {2\theta -\pi }{24}}\right).}$$

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

In 2006, two mathematicians using Mathematica analyzed the function and found variants where leaves, flowers, or other insects became apparent.^[3] New developments regarding such curves are still under research by mathematicians.