Quanta Magazine: Mathematicians Identify Threshold at Which Shapes Give Way

But, remarkably, Nash showed there is a rigorous way of maintaining lengths even as you refabricate curves out of twists. First, shrink the sphere evenly, like a deflating balloon. Then add increasingly tight spirals to every curve. By adding infinitely many such twists, you can eventually restore every curve to its original length, even when the original sphere has been crumpled.