Butterfly curve (transcendental)

The butterfly curve.

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

Equation

An animated construction gives an idea of the complexity of the curve (Click for enlarged version).

The curve is given by the following parametric equations:

$x=\sin t\!\left(e^{\cos t}-2\cos 4t-\sin ^{5}\!{\Big (}{t \over 12}{\Big )}\right)$

$y=\cos t\!\left(e^{\cos t}-2\cos 4t-\sin ^{5}\!{\Big (}{t \over 12}{\Big )}\right)$

$0\leq t\leq 12\pi$

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.

Oscar's Butterfly

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.

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