Butterfly curve (algebraic)

Not to be confused with Butterfly curve (transcendental).

The butterfly curve $x^{6}+y^{6}=x^{2}$

The butterfly curve singularity link

In mathematics, the algebraic butterfly curve is a plane algebraic curve of degree six given by the equation

$x^{6}+y^{6}=x^{2}.$

The butterfly curve has a single singularity with delta invariant three, which means it is a curve of genus seven. The only plane curves of genus seven are singular, since seven is not a triangular number, and the minimum degree for such a curve is six.

The butterfly curve has branching number and multiplicity two, and hence the singularity link has two components.

The area of the algebraic butterfly curve is given by

$4\int _{0}^{1}(x^{2}-x^{6})^{\frac {1}{6}}dx={\frac {\Gamma {\big (}{\frac {1}{6}}{\big )}\cdot \Gamma {\big (}{\frac {1}{3}}{\big )}}{3{\sqrt {\pi }}}}\approx 2.804,$

where $\Gamma$ is the gamma function, and the arc length of the curve is $s\approx 9.017$.

References

• Weisstein, Eric W. "Butterfly Curve". MathWorld.

External links

• Sequence A118292 in the OEIS — the area of the algebraic butterfly curve

• Sequence A118811 in the OEIS — the arc length of the algebraic butterfly curve