Mandelbrot集的边界 作为多项式纽线的极限

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The Mandelbrot Curves - A better technique for calculation desmos.com/calculator/coamqcajzq

These are the first 50 in a series of implicit polynomial curves that converge to the boundary of the Mandelbrot set.  The curves are computed using iteration of the function f(z) = z² + c, where z and c are complex numbers.  Each curve is the locus of points c where the magnitude of z is equal to 2 after a set number of iterations.  Credit to user "The Fractalistic" on YouTube for the technique, with a modification to slightly speed up the calculations.  Be patient if you decide to zoom in to this plot, as it will still take some time for the calculations to run to completion.

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给定一个$n$次复系数多项式$p(z)$, 集合$\{z∈\Bbb C:|p(z)|=c\}$称为多项式纽线(polynomial lemniscate)
在笛卡尔坐标系可以写成$f(x,y)=c^2$, 其中$f(x,y)=p(z)\overline{p(z)}$是一个$2n$次多项式.
当 $p$ 是 1 次多项式时，所得曲线是一个圆，其圆心是 $p$ 的零点。
当 $p$ 是 2 次多项式时，曲线是卡西尼卵形线，其焦点是 $p$ 的零点。

$\lvert z^6+z^5+z^4+z^3+z^2+z+1\rvert=1$

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\begin{aligned}
z&=x+iy\\
f(z)&=z^2+z\\
Q(z)&=|z|^2\\
\text{多项式纽线}&\\
Q(z)&=4\\
Q(f(z))&=4\\
Q(f(f(z)))&=4\\
&\vdots
\end{aligned}youtube.com/shorts/775og3l0NIc