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Butterfly Curve

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ButterflyCurve

There are two curves known as the butterfly curve.

The first is the sextic plane curve given by the implicit equation

y^6=x^2-x^6
(1)

(Cundy and Rollett 1989, p. 72; left figure). The total area of both wings is then given by

A=4int_0^1(x^2-x^6)^(1/6)dx
(2)
=(Gamma(1/6)Gamma(1/3))/(3sqrt(pi))
(3)
=2.8043642106...
(4)

(OEIS A118292). The arc length is

s=9.017346056...
(5)

(OEIS A118811).

The second is the curve with polar equation

r=e^(sintheta)-2cos(4theta)+sin^5[1/(24)(2theta-pi)],
(6)

which has the corresponding parametric equations

x=sint[e^(cost)-2cos(4t)+sin^5(1/(12)t)]
(7)
y=cost[e^(cost)-2cos(4t)+sin^5(1/(12)t)],
(8)

(Bourke, Fay 1989, Fay 1997, Kantel-Chaos-Team, Wassenaar; right figure).

See also

Bean Curve, Butterfly Catastrophe, Butterfly Effect, Butterfly Function, Butterfly Graph, Butterfly Lemma, Butterfly Polyiamond, Butterfly Theorem, Dumbbell Curve, Eight Curve, Piriform Curve

Portions of this entry contributed by Margherita Barile

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References

Bourke, P. "Butterfly Curve." http://astronomy.swin.edu.au/~pbourke/curves/butterfly/.Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 72, 1989.Fay, T. H. "The Butterfly Curve." Amer. Math. Monthly 96, pp. 442-443, 1989.Fay, T. H. "A Study in Step Size." Math. Mag. 70, pp. 116-117, 1997.Kantel-Chaos-Team "Die Butterfly-Kurve." http://www.schockwellenreiter.de/pythonmania/pybutt.html.Sloane, N. J. A. Sequences A118292 and A118811 in "The On-Line Encyclopedia of Integer Sequences."Wassenaar, J. "2D Curves." http://www.2dcurves.com/exponential/exponentialb.html.

Cite this as:

Barile, Margherita and Weisstein, Eric W. "Butterfly Curve." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ButterflyCurve.html

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