Mandelbrot Set on the Riemann Sphere

Mandelbrot on the projection of the bottom of the Rienman sphere straight down. The mandelbrot st on the projection of the top half of the Riemman sphere straight down.

The mappings are B_1(0) \to \hat{\mathbb{C}}:

\phi^- (z) = \frac{z}{2-|z|}

\phi^+(z) = \frac{1}{z(2-|z|)}

For most practical purposes you’ll want to (as seen in above picture) restrict the domain of \phi^+ to not include some neighborhood of 0 to make it a bounded function on \mathbb{C}.

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4 Responses to “Mandelbrot Set on the Riemann Sphere”

  1. happyseaurchin Says:

    is there an image i can see of this on the riemann sphere…
    i am having trouble visualising it….

  2. happyseaurchin Says:

    put in another search on google and this page pops up again… i remember asking you if it was possible and i thought this was your answer for a moment… any progress on seeing this in ’3d’?

  3. happyseaurchin Says:

    have to say, it still doesn’t look how i imagined it… i guess there are lots of ways of doing it…

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