Crumpling paper

States of an inextensible sheet

In the photo above we can see a few of the kinds of states that paper can be folded into. An important thing about paper is that it bends much easier than it stretches, so we call it inextensible. Starting at the lower right corner, we see a orange shape folded in origami paper -- we see that paper can be folded into shapes that are flat almost anywhere and creased along just a few lines. However, working clockwise, we next come to a yellow cone that demonstrates that an inextensible sheet can be folded into what we call a developable surface, which has zero gaussian curvature. Gaussian curvature is the kind of curvature which a sphere has but a cylinder doesn't. That is, at a given point, a sphere curves in two directions, whereas a developable surface curves in just one. A flat inextensible sheet cannot be mathematically (or practically) mapped to a surface that has Gaussian curvature. You can check this experimentally: one can wrap a sheet of paper around a cylinder with little difficulty, but wrapping a sheet around a sphere creases the sheet up. This is why maps of the Earth are invariably distorted when projected onto a sheet.

From another angle we see two crumpled objects on a flat sheet. The purple object was crumpled in my hands, and the white sheet with masking tape was crumpled with a new cylinderical geometry. Next to those is a blue origami dragon, a very complex but ordered folding state of a sheet. Finally, a crumpled ball of the New York Times completes the scene.

An important idea in physics is entropy, a measure of how many possible states of a system have a particular appearance to an observer. To take an example, the origami dragon is a very specific state of folds on a sheet -- a very small misalignment would destroy the appearance of the model. On the other hand, there are many different possible crumpled balls that one could make that would look more or less the same to the eye -- and would look more or less the same if we took more objective observations. Thus the crumpled state has much higher entropy than the flat state or the origami dragon state. A controversal suggestion is that the higher entropy of the crumpled state might cause membranes that are in thermal equilibrium with their environment to crumple up, and people at IBM's Alameden research center have done wonderful simulations of this, as well as writing a basic overview.

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Photograph notes are availible.
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