Butterfly curve (transcendental)

Last updated August 24, 2024

The butterfly curve is a transcendental plane curve discovered by Temple H. Fay of University of Southern Mississippi in 1989.^[1]

Equation

The curve is given by the following parametric equations:^[2]

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\leq t\leq 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.^[1]

Developments

In 2006, two mathematicians using Mathematica analyzed the function, and found variants where leaves, flowers or other insects became apparent.^[3]

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References

1 2 Fay, Temple H. (May 1989). "The Butterfly Curve". Amer. Math. Monthly. 96 (5): 442–443. doi:10.2307/2325155. JSTOR 2325155.
↑ Weisstein, Eric W. "Butterfly Curve". MathWorld .
↑ 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

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[origin-1] 1 2 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.

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