Animating the embedding of a flat torus in 3d

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I'm playing around with animations in sage and python, and I'd like to recreate this animation from wikipedia of creating a torus from a flat surface in python:

Torus from rectangle

I'm experimenting with finite fields of the form F_p^2 in sage, and ideally, I want to go from having a flat surface with grid lines representing the integers till p-1, to embedding the torus in 3d. Could any help or guide me in the right direction?

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