How paper folds and algae grow : understanding point defects in thin elastic sheets

If you crumple a sheet of paper, a number of ridges form. They come together at points resembling miniature cones ; remarkably, these point defects tell the paper how to fold. This can be illustrated very beautifully : just depress a flat planar disc of paper with a pencil into your coffee cup and observe how the disc deforms.

Conical defects in crumpled paper are quite different from the familiar icecream cone one obtains by removing a wedge and glueing the sides of the remaining disc together. Surprisingly, if an extra wedge is added to the disc, an infinite number of shapes arises without any need to apply a force. Possible shapes of such an excess cone are illustrated in the figure. In nature certain seawater algae, for example, spontaneously adopt these shapes during their development.

In a recent paper, an ENS team together with a professor of UNAM (Mexico) proposed a simplified model of the e-cone. They construct the shapes and determine the stresses in them. These results compare very well with simple experiments on sheets of paper. They confirm that the configuration with the two-fold symmetry is in fact the ground state. What is maybe more surprising is that all other states with a larger number of folds are stable as well.

references :
- " Morphogenesis of Growing Soft Tissues" Julien Dervaux and Martine Ben Amar Phys. Rev. Lett. 101, 068101 (2008)
- "Conical Defects in Growing Sheets" Martin Michael Müller, Martine Ben
Amar, Jemal Guven, Phys. Rev. Lett. 101, 156104 (2008)

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