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.
Some point in the scrumpled version is exactly above its original position.