Butterfly curve (transcendental)

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:[2]

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

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

${\displaystyle 0\leq t\leq 12\pi }$

or by the following polar equation:

${\displaystyle r=e^{\sin \theta }-2\cos 4\theta +\sin ^{5}\left({\frac {2\theta -\pi }{24}}\right)}$

The sin term has been added for purely aesthetic reasons, to make the butterfly appear fuller and more pleasing to the eye.[1]

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

• 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.)

• Butterfly Curve plotted in WolframAlpha