This reminds me of the fixed point theorem. Take a sheet of paper, scrumple it however you choose without tearing it. Place it over an unscrumpled copy of the original, without going outside the edges.
Some point in the scrumpled version is exactly above its original position.
How fascinating. One nitpick: my brief wikipedia research tells me that you probably mean the fixed point property, rather than the fixed point theorem, which appears to be a category rather than a specific idea. But then again maybe I've not fully understood. That seems likely.
That's actually an illustration of the Contraction Mapping Principle.
Another way of illustrating it is that if you're in city X and place a map of city X on the ground, then exactly one point on the map will be above the corresponding physical point.
Contraction mapping is not really required for this phenomenon. Contraction would mean that any two points are closer after the crumpling/mapping than before. In fact contraction will guarantee that there is only one such point which lies over the other.
For the aforesaid property any form of crumpling that does not tear the paper would suffice. Provided no part of the crumpled paper extends beyond the boundary of the pristine sheet (or the city, in your example). The phenomenon relies of Browder's fixed point theorem.
If it helps to reduce one dimension: think of a continuous curve defined over a part of the x-axis [0, 10]. As long as the curve stays inside the box [0,0], [10,10] and every point in the [0,10] part of the x-axis is mapped, it would be impossible to avoid the diagonal. Just try it.
Incidentally, a generalization of the theorem above, called the Kakutani fixed point theorem underlies John Nash's proof (that won him the Nobel) of the existence of equilibria in games.
This reminds me that in Argentina in primary school we would make an improvised football (as in: soccer) by crumpling some paper and then wrapping it in scotch tape. It was of course much smaller than the real ball, but it worked surprisingly well (and broken windows were not an issue). Now I know why :-)
If you, like me, wonder how one can even get funding for research on crumpled paper, Prof. Menon's research area is on condensed matter, "primarily on soft-condensed matter systems".
Understanding the physics of why crumpled paper resists with a specific force can help you design vehicle crumple-zone systems in cars which will save occupants from head on crashes into trees, by having the entire engine crumple with a pre defined resistance.
I understand how guessing at the particular pattern of a crumple is a hard physics problem. What I do not understand is why understanding the macro-properties of crumpled paper is a hard physics problem. For example:
"Do the balls absorb vibrations by trapping pressure waves or by dissipating them? Nobody knows, says Menon, 'but it means there are still plenty of beautiful problems to keep me interested'."
This seems like a rather odd thing to say. What does "trapping pressure waves" even mean? Is Menon asserting that, when I take my shipment out of the container, all of the vibration that the package suffered on its journey is still swirling around in all the crumpled paper?
It's laudable to study everyday physics, but at least when it comes to macro-behavior of crumpled paper, these research questions appear laughable.
"What I do not understand is why [several things]..."
"... these research questions appear laughable."
How about before declaring these questions laughable, you try a little harder to understand why they are interesting? For instance, the fact that we don't have computers powerful enough to simply simulate the question being asked provides one clue of a possible reason why this is more interesting than you think.
Either one of "I don't understand this research" or "this research is laughable" I might accept from a given person, but the combination is unpleasant.
I have a degree in physics, so I guess it was a mistake to leave out the (two) punchlines and leave them implied:
1. "absorb vibrations by trapping pressure waves" is a meaningless concept in physics. Sponges, rubber, styrofoam, springs and crumpled paper all "absorb vibration" in the same way - converting mechanical energy into heat. Apart from a rather pedantic system in which somehow air pressure is used to add angular velocity to a flywheel, there is no sense in which vibrations are "stored". There is also the case of an irreversible deformation, as when a car crumples. And in that case, the object would become monotonically less capable of absorbing energy - a curve that would be easy to measure (place crumpled paper in a vice, repeatedly attempt to squeeze a bit, measure maximum and minimum circumference, repeat, graph)
2. Thermodynamically, any excess energy is going into heat. Period. This could be tested by very carefully measuring the temperature of the crumpled paper before and after being deformed several times, as with the vice above. Eventually (and with paper, sooner rather than later) there's going to be microscopic breakdown at the fiber and even molecular levels, and that increase in entropy must be factored in, too. (Now that is an interesting experimental challenge: how to measure the "looseness" of paper fiber in a crumpled sheet. There are some easy destructive ways but I cannot think of any non-destructive ones.)
So yes, I call shenanigans on this professor and, apart with the (already admittedly) interesting problem of describing the geometry of particular crumples, I think his research is laughable. Although I would enjoy being shown how I am in error, if that be the case.
Some point in the scrumpled version is exactly above its original position.