How can one geometrically characterize the ways of bending/rolling/wiggling (but not folding) an ordinary piece of paper? For example it seems necessary but not sufficient that it be a $C^1$ embedding from an unfolded piece of paper into $\mathbb R^3$. A natural choice seems to be isometric embedding, but then the Nash-Kuiper theorem would imply that I can massage a piece of paper into an arbitrarily small region which doesn't seem possible.
2026-04-04 11:02:57.1775300577
What are the permissible ways to bend but not fold a piece of paper?
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It depends on how you view a piece of paper. We tend to view it as a 2D object, which ignores its thickness. If you do that, it has zero volume. You can then find an embedding that is as continuous as you wish into an arbitrarily small ball in $\Bbb R^3$. If you acknowledge that the paper has thickness, it has a volume greater than zero, and you can't embed it without overlap in a ball any smaller than that volume.