The butterfly curve is a transcendental plane curve discovered by Temple H. Fay of University of Southern Mississippi in 1989.
Equation For 0 ≤ t ≤ 12 π {\displaystyle 0\leq t\leq 12\pi } , the curve is given by the following parametric equations:
x = sin t ( e cos t − 2 cos 4 t − sin 5 ( t 12 ) ) y = cos t ( e cos t − 2 cos 4 t − sin 5 ( t 12 ) ) {\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:
r = e sin θ − 2 cos ( 4 θ ) + sin 5 ( 2 θ − π 24 ) . {\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.
Developments In 2006, two mathematicians using Mathematica analyzed the function and found variants where leaves, flowers, or other insects became apparent. New developments regarding such curves are still under research by mathematicians.
References
External links Butterfly Curve plotted in WolframAlpha