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I wonder if the geometry of crinkled aluminum foil has been studied?
           FoilCrinkled
The above is a photo of foil I flattened to reuse. It might be described as a partition into nearly-uncreased polygons, each polygon of not too many sides, and arranged in a rather un-Voronoi like pattern. It superficially resembles a rugged mountain terrain seen from a great height.

I searched a bit for some mathematical analysis of this pattern without luck. Has anyone seen such an analysis? There might be some interesting mathematics here...

Update. Here is Fig.1 from the PNAS article that jc identified, "Three-dimensional structure of a sheet crumpled into a ball," by Anne Dominique Cambou and Narayanan Menon, slices through an equatorial plane of three crumpled spheres:
      CrumpledSpheres

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Have you tried looking up Robert MacPherson to see if his work is relevant? He gave a talk at the 2011 SIAM Conference on Applied Algebraic Geometry at NCSU on the topic of "The geometry of cell structures in foams and metals". He showed pictures which were perhaps not entirely dissimilar from yours, though what I remember more from his talk was the evolution process of foam. He definitely had cells with small numbers of sides. –  Patricia Hersh Nov 5 '12 at 1:15
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You might consider the process as well as the result. The process I imagine produces something not far (in some sense) from a spheroid shape; a first study might be random wrappings of a ball. More advanced studies might involve folding to fill a ball. There are other real life models to consider, such as pocket stuffing. Gerhard "Of Course I'm Doing Laundry" Paseman, 2012.11.04 –  Gerhard Paseman Nov 5 '12 at 1:18

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Crumpled structures are certainly of great interest among some soft matter physicists; you might with this review article of Tom Witten's. He also has a nice webpage with some nice pictures and summaries of papers of his on related topics.

My understanding is that while we have some handle on the behavior of the cone-like and ridge-like singularities that are forced by the crumpling, not much is known about how they end up distributed after crumpling, though see this nice recent PNAS article from UMass on X-ray scans of crumpled metal foil balls.

I might add more later, but these references and their references, etc. should be enough to get you started. There are indeed many beautiful problems in the area of elasticity of thin sheets.

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PNAS article: "Three-dimensional structure of a sheet crumpled into a ball," by Anne Dominique Cambou and Narayanan Menon. "[T]he internal geometry and mechanical properties of the crumpled ball may reflect the history of its preparation." Gerhard's point. Thanks for these great references, jc! –  Joseph O'Rourke Nov 5 '12 at 2:18

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