Butterfly curve (transcendental)

The butterfly curve is a transcendental plane curve discovered by Temple H. Fay. The curve is given by the following parametric equations:

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

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

or by the following polar equation:

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

Butterfly curve (transcendental) with Mathematica : ParametricPlot[{Sin[t] (Exp[Cos[t]] - 2 Cos[4 t] - Sin[t/12]^5), Cos[t] (Exp[Cos[t]] - 2 Cos[4 t] - Sin[t/12]^5) }, {t, 0, 20 Pi}]

See also

Butterfly curve (algebraic)

References

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" from MathWorld.

Mathematics Encyclopedia