Transcendental plane curve
The butterfly curve. The butterfly curve is a transcendental plane curve discovered by Temple H. Fay of University of Southern Mississippi in 1989.[1]
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 :[2]
x = sin t ( e cos t − 2 cos 4 t − sin 5 ( t 12 ) ) {\displaystyle x=\sin t\!\left(e^{\cos t}-2\cos 4t-\sin ^{5}\!{\Big (}{t \over 12}{\Big )}\right)} y = cos t ( e cos t − 2 cos 4 t − sin 5 ( t 12 ) ) {\displaystyle y=\cos t\!\left(e^{\cos t}-2\cos 4t-\sin ^{5}\!{\Big (}{t \over 12}{\Big )}\right)} 0 ≤ t ≤ 12 π {\displaystyle 0\leq t\leq 12\pi } 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 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]
See also
References ^ a b 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