Take a square sheet of paper. Can you glue opposite sides without ever folding the paper? This is a conundrum that many of the greatest modern mathematicians, like Gauss, Riemann, and Mandelbrot, couldn't figure out. While John Nash did answer yes, he couldn't say how. After 160 years of research, Vincent Borrelli and his collaborators have finally provided a revolutionary and breathtaking example of a bending of a square sheet of paper! And it is spectacularly beautiful!

Although highly appreciated by artists, self-reference has been a nightmare for mathematicians. It took one of the greatest, Kurt Gödel, to provide a better understanding of it. This article deals with paradoxes, recursion, fractals and Gödel's incompleteness theorems.

Introduction I could have started this article by showing you beautiful examples of fractals, found in nature or invented by mathematicians, in two or three dimensions, constructed deterministically or not. But I won’t. In fact, you wouldn’t need to read this article to find such examples. I’d rather start by telling you a story. It […]