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There is a very clear theory of what polygons can be constructed in the plane. One of my professors said that he believed the same ones could be constructed on a sphere through stereographic projection. I'm not sure I buy this and was wondering if anyone had insights into the truth of this statement, or references concerning this topic.

If it is the case that constructability is the same on the plane and sphere, is it true that constructability should be the same on any space homeomorphic to the plane (I realize the plane is only homeomorphic to the sphere minus a point but I can't imagine that affecting the results of this particular question)?

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When you speak of "constructibility", do you mean with straightedge and compass? If so, it seems that you're limited to working in the plane. –  Sammy Black Nov 14 '13 at 20:56
    
@SammyBlack I guess I was hoping for some kind of generalization of a straight edge and compass to instruments that can draw balls and geodesics. Maybe this is too far fetched. –  Jeremy Nov 14 '13 at 21:58

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