For 0 \le t \le 12\pi , the curve is given by the following
parametric equations: \begin{align} 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{align} or by the following
polar equation: r = e^{\sin\theta} - 2\cos \left(4\theta\right) + \sin^5\left(\frac{2\theta - \pi}{24}\right). The term has been added for purely aesthetic reasons, to make the butterfly appear fuller and more pleasing to the eye. == Developments ==