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Can we deform a curvy 2-manifold (surface embedded in 3D) so that the resulting homeomorphic surface consists of flat faces only. Like taking a sphere and deforming it to a cube. If that's true, is there a way of knowing when the resulting shape is going to be convex or non-convex?

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Convex shapes can be homeomorphic to non-convex ones (imagine denting a cube), so I'm not sure your second question necessarily makes sense. –  Matt Pressland Feb 27 '13 at 16:41
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This might be close to what you're looking for: en.wikipedia.org/wiki/Triangulation_%28topology%29 –  Jim Feb 27 '13 at 16:45

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I don't know if this is what you're after, but the torus is homeomorphic to this:

enter image description here

(not a perfect representation, but you get the idea)

and you can do the same thing for any closed orientable surface - basically, just add more holes and more pieces as needed.

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Is that shape like a big rectangle with the middle cube removed? –  saadtaame Feb 27 '13 at 17:14
    
@saadtaame Yes. –  Potato Feb 27 '13 at 17:44

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