Butterfly curve (transcendental)

The butterfly curve with animated construction gives an idea of the complexity of the curve.

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

Equation

For $0\le t\le 12\pi$, the curve is given by the following parametric equations: $$\begin{array}{cc}x& =\sin t(e^{\cos t}-2\cos 4t-{\sin }^5\left(\frac{t}{12}\right))\\ y& =\cos t(e^{\cos t}-2\cos 4t-{\sin }^5\left(\frac{t}{12}\right))\end{array}$$ 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 Template:Math term has been added for purely aesthetic reasons, to make the butterfly appear fuller and more pleasing to the eye.

Developments

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

External links

• Butterfly Curve plotted in WolframAlpha

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