# What shape does a piece of paper make when it is pushed from the edges?

When I push a piece of (A4) paper oriented landscape to me from the shorter edges, it makes a pretty shape, resembling a bell-curve. I seem to remember these sort of situations being a motivation for or concrete instance of some theorems in differential geometry, but apart from that I have no idea how to determine what the true shape of the paper in this situation.

Not a great example (as I'm pushing with one hand to take the photo) but similar to what I'm after. (Hey, if you can generalize to a one-sided push I might just award the checkmark!)

• All I'd ask is an illustration of the specific sort of shape you're thinking about - the details of the pushing can make a huge difference in the answer... – Steven Stadnicki Aug 19 '13 at 5:00
• Done (hopefully) – Hugh Aug 19 '13 at 5:09
• This doesn't have a whole lot to do with differential geometry since the problem is basically one-dimensional. Maybe try physics SE instead? – Scaramouche Aug 19 '13 at 8:11
• The solution is the curve minimizing the bending energy $\int \kappa(s)^2\,\mathrm ds$ for prescribed length, endpoints, and tangents, a.k.a. the elastica. See e.g. Sec. 9 onwards (and the bottom of Fig. 11) of Raph Levien's "The elastica: a mathematical history". – Rahul Aug 19 '13 at 8:19
• Buckling of sheets - the shape is similar to a standing wave and depends a lot on how the ends are constrained while compression. – Macavity Sep 1 '13 at 18:13

## 1 Answer

It is the Elastica. For small wave heights, the equation is like $\sin^2 k x$ , where $k$ depends on bending rigidity $EI$, applied force $P$. More accurately described in terms of Elliptic functions.

The differential equation is simply: $\text{curvature} = -k\, y$

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