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My Question

Why is it that the Buddhabrot visually differs so drastically from the Mandelbrot? Based on articles I've read, such as the one on Wikipedia, it seems that the Buddhabrot should simply be a negative of the Mandelbrot and otherwise visually identical.

My Reasoning
Wikipedia states that in order to render a Buddhabrot, you must iterate (complex) points that aren't in the Mandelbrot set through the Mandelbrot function and trace their paths to escape. The points themselves and their paths supposedly visually compose the Buddhabrot.

Mandelbrot Function: $f(z_{n+1})=z_n^2+c$

This is where my confusion arises; why is it that Buddhabrot renderings show activity in regions that are part of the Mandelbrot Set? By definition, any point who's path/trajectory passes through the Mandelbrot Set is part of the Mandelbrot Set because once the path is in the Set, it cannot escape.

Any clarifications on where my comprehension may be wrong or tips on how to render the Buddhabrot are pleasantly welcomed!

Buddhabrot(tilted) (left), Mandelbrot (right)

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By definition, any point whose path/trajectory passes through the Mandelbrot Set is part of the Mandelbrot Set because once its path is in the Set, it cannot escape.

That is true for the Julia sets, where all iterations are for the same $c$, but different start points $z_0$, but not for the Mandelbrot set where $c$ varies and the start point is fixed as $z_0=0$.

If some $z_n$ of the sequence $z_{k+1}=z_k^2+c$, $z_0=0$ is inside the bounded component (note that the M~ set itself, the fractal, is only the boundary curve), this does not mean that the sequence of $\tilde c=z_n$, started again at $\tilde z_0=0$, $\tilde z_1=z_n$, is in any way related to the first sequence. Indeed, $z_{n+1}=z_n^2+c$ is different from $\tilde z_2=z_n^2+z_n$.

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  • $\begingroup$ I see what you mean with regards to the extra information contained in the trajectories described by the Mandelbrot set; trajectories can enter and leave the Set. The broader context of my question is this: I am trying to render a Buddhabrot and currently I am plugging random values into the Mandelbrot Function that are known to diverge. I then plot these values and their escape trajectories. None go throught the Mandelbrot Set, leaving an outline of the Mandelbrot Set. This leads me to believe that none of the trajectories that escape can pass throught the Mandelbrot Set. $\endgroup$
    – sethtadd
    Jul 16, 2017 at 7:50
  • $\begingroup$ I found the flaw in my algorithm, I was plotting the original random $c$ values instead of the more recently calculated $z$ values that describe escape trajectories. Thank you for you great answer though! @LutzL $\endgroup$
    – sethtadd
    Jul 16, 2017 at 8:09

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