Is it possible to build a metal wire sphere whose shadow projects the 2d conformal maps of the Riemann Sphere on a flat wall? Is it possible to build a metal wire hollow sphere whose shadow from a nearby point light source projects the 2d conformal maps of the Riemann Sphere on a flat wall? I believe that in theory it should be possible, but I was not sure if there would be any physical impracticabilities that would limit such a shadow project.

This can be done. For example, see

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The technical name for this way of "casting shadows" is called stereographic projection. See

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and so on.