Imagine a long corridor with many doors that open and close in a strict rhythm. If you send sound down it, the echo from each door will distort the melody, and a crooked wall at the end will throw off the entire rhythm. It would seem that understanding what's happening deep inside is now impossible.

But there's a trick. Measure how much sound went left and how much went right, and subtract one from the other: you get a number. And this number, oddly enough, does not depend on how crooked the wall at the end is. It is determined only by how often the doors open and how the corridor itself is built. In physics, such systems are called open lattices with periodic driving. Boundaries indeed strongly alter the signal shape, but the total imbalance of left and right flows remains unchanged.

The authors propose to catch this effect using cold atoms — pass them through a lattice of light and watch where they deflect. For electronic devices, for instance, those based on surface acoustic waves, the same imbalance can be calculated differently depending on the contact model. One method gives a signal close to twice the electron charge times frequency, another gives something quite different. But at the core is always the same topological number, the winding number, hidden deep within the material.

So even if the edge of the system looks like crumpled paper, inside it sits an invisible counter that cannot be thrown off.