Butterfly curve (transcendental) explained

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

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Equation

The curve is given by the following parametric equations:

x = \sin t \left(e ^{\cos} - 2 \cos 4 t - \sin ^{5({t} \over 12 } ) \right) y = \cos t \left(e ^{\cos} - 2 \cos 4 t - \sin ^{5({t} \over 12 } ) \right) 0 \le t \le 12 \pi

or by the following polar equation:

r = e ^{\sin\theta} - 2 \cos 4 \theta + \sin ^{5 \left( (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.

OSCAR'S BUTTERFLY POLAR EQUATION

Developments

In 2006, two mathematicians using Mathematica analyzed the function, and found variants where leaves, flowers or other insects became apparent.^[2]

See also

https://books.google.com/books?id=AsYaCgAAQBAJ&dq=OSCAR+RAMIREZ+POLAR+EQUATION&pg=PA732

• Butterfly curve (algebraic)
• Oscar’s Butterfly Polar Equation

r = (cos 5θ)² + sin 3θ + 0.3 for 0 ≤ θ ≤ 6π(A polar equation discovered by Oscar Ramirez, a UCLA student, in the fall of 1991.)

External links

• Butterfly Curve plotted in WolframAlpha

Notes and References

1. Temple H. . Fay . May 1989 . The Butterfly Curve . Amer. Math. Monthly . 96 . 5 . 442–443 . 10.2307/2325155. 2325155.
2. 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. 10.1080/00207390801923240. Geum. Y.H.. Kim. Y.I.. 122066238.