I am wondering if there exists somewhere an "animation" of one such map (for some lattice / torus), in the style of the kind of $z \mapsto z^2$ maps one encounters in complex analysis classes (one can somewhat convincingly visualize the plane twisting around to cover itself twice, with ramification around $0$). I have seen the plots the absolute values of the function thought of as a meromorphic function on the complex plane, that is not what I am dreaming about...

Is such an animation possible? It is "just" some degree 2 ramified covering of the sphere by a torus, so it is not obvious to me that it would be impossible to animate, but I would have no what program to use, even... does there exist software that can do this kind of thing?

I hope this question makes sense! Looking forward to the answers.

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    $\begingroup$ John Baez talked about it a bit in Week 229 of his This Week's Finds blog. There isn't an animation, but there is a world map that visualizes the function. $\endgroup$ – user856 Jul 7 '15 at 2:32

The following image shows half a torus mapping to a sphere via a scaled Weierstrass $\wp$ function. It was created in Mathematica.

enter image description here

The four branch points lie on the two boundary circles of the half-torus, and map to the endpoints of the two "seams" on the resulting sphere. The other half of the torus maps to the sphere in the same way, with the deck transformation being $180^\circ$ rotation around the horizontal line that goes through the four branch points.

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  • $\begingroup$ Beautiful. Is it possible to extend mapping from a full torus?( two double points on torus settling onto quarter points on the seams.) $\endgroup$ – Narasimham Jul 7 '15 at 6:57
  • $\begingroup$ @Narasimham It's hard to see what's happening with the full torus, because the surface intersects itself during the homotopy. I haven't been able to make a clear animation of it. $\endgroup$ – Jim Belk Jul 7 '15 at 18:45
  • $\begingroup$ This is great! Could you post the code you used? I would love to learn how to make something like this. $\endgroup$ – Lorenzo Najt Jul 7 '15 at 23:35

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