most recent 30 from http://mathoverflow.net 2013-06-19T08:48:09Z http://mathoverflow.net/feeds/question/111498 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/111498/the-geometry-of-crinkled-aluminum-foil Joseph O'Rourke 2012-11-05T00:44:43Z 2012-11-05T20:19:34Z

<p>I wonder if the geometry of crinkled aluminum foil has been studied? <br /> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img src="http://cs.smith.edu/~orourke/MathOverflow/FoilCrinkled.jpg" alt="FoilCrinkled" /> <br /> 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.</p> <p>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...</p> <p><b>Update</b>. Here is Fig.1 from the <em>PNAS</em> article that <em>jc</em> 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: <br /> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img src="http://cs.smith.edu/~orourke/MathOverflow/SlicesCrumpledSpheres.jpg" alt="CrumpledSpheres" /> <br /></p>

http://mathoverflow.net/questions/111498/the-geometry-of-crinkled-aluminum-foil/111506#111506 jc 2012-11-05T01:48:12Z 2012-11-05T01:48:12Z

<p>Crumpled structures are certainly of great interest among some soft matter physicists; you might with <a href="http://rmp.aps.org/abstract/RMP/v79/i2/p643_1" rel="nofollow">this review article of Tom Witten's</a>. He also has <a href="http://jfi.uchicago.edu/~tten/Crumpling/" rel="nofollow">a nice webpage</a> with some nice pictures and summaries of papers of his on related topics.</p> <p>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 <a href="http://www.pnas.org/content/108/36/14741.short" rel="nofollow">this nice recent PNAS article</a> from UMass on X-ray scans of crumpled metal foil balls.</p> <p>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.</p>