Conditions for crumpling: thin elastic manifold
Metric: preferred distance
between all pairs of points
Elasticity: energy cost for
altering this distance.
Thin manifold: d-dimensional set of points much
smaller in one direction than the others
Energy =
![](slide0005_image018.png)
![](slide0005_image019.png)
Bending B
Stretching S
![](slide0005_image021.png)
![](slide0005_image022.png)
![](slide0005_image023.png)
1
g
R
R
Energy cost simplifies when
local deformation is smooth
![](slide0005_image027.png)
curvature
strain
Smooth: g << 1; R >> h
h