The Poincar\'e\--Birkhoff Theorem , also called Poincar\'e Last Geometric Theorem, is a landmark result in area preserving dynamics. It was formulated by Poincar\'e based on his investigations in celestial mechanics. The theorem may be easily stated: a periodic twist map of an infinite closed strip, or closed annulus, has at least two geometrically distinct fixed points. V.I. Arnold realized...