Disturbing philosophical fact #1 - You can never cross the same river twice.

Disturbing philosophical fact #2 - You can never crumple the same paper twice. Believe us, we tried!!! Despite the apparent similarity of all uncrumpled sheets, and no matter how accurately the crumpling protocol is repeated, minute differences in the crumpling process grow rapidly, leading to completely different creases patterns for different paper sheets. Once a paper is crumpled and opened up, any subsequent crumpling will tend to happen along the existing creases where the paper is weakened. Once again, however, the paper refuses to crumple in exactly the same way, and new creases appear with successive crumplings. (A fact which greatly upsets our pet ninjas!)

 

We are working to understand what the asymptotic behavior of a sheet crumpled repeatedly many times is. Can the sheet be trained to crumple under a certain protocol without creating new creases, or will creases continue to appear indefinitely? The answer to this question, apart from putting our ninjas' minds at ease, can help shed light on the behavior of other disordered physical systems and their approach to steady state under periodic loading.

The following images show a scanned 3D profile of a crumpled Mylar sheet, and the creases pattern of the sheet after being crumpled once, twice and three times, obtained by calculating the curvature of the scanned surface. Blue and red lines represent ridges and valleys respectively.