Mandelbrot Set on the Riemann Sphere

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:

hmm
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…