Butterfly curve (transcendental)

Not to be confused with Butterfly curve (algebraic).

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. ^[1]

Equation

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]

Developments

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.

References

1. Fay, Temple H. (May 1989). "The Butterfly Curve". Amer. Math. Monthly. 96 (5): 442–443. doi:10.2307/2325155. JSTOR   2325155.
2. Weisstein, Eric W. "Butterfly Curve". MathWorld .
3. Geum, Y.H.; Kim, Y.I. (June 2008). "On the analysis and construction of the butterfly curve using Mathematica". International Journal of Mathematical Education in Science and Technology. 39 (5): 670–678. doi:10.1080/00207390801923240. S2CID   122066238.

External links

• Butterfly Curve plotted in WolframAlpha