Packings of Smoothed Polygons

Abstract

This book uses optimal control theory to prove that the most unpackable centrally symmetric convex disk in the plane is a smoothed polygon. A smoothed polygon is a polygon whose corners have been rounded in a special way by arcs of hyperbolas. To be highly unpackable means that even densest packing of that disk has low density. Motivated by Minkowski's geometry of numbers, researchers began to search for the most unpackable centrally symmetric convex disk (in brief, the most unpackable disk) starting in the early 1920s. In 1934, Reinhardt conjectured that the most unpackable disk is a smoothed octagon. Working independently of Reinhardt, Mahler attempted without success in 1947 to prove that the most unpackable disk must be a smoothed polygon. This book proves what Mahler set out to prove: Mahler's First conjecture on smoothed polygons. His second conjecture is identical to the Reinhardt conjecture, which remains open. This book explores the many remarkable structures of this packing problem, formulated as a problem in optimal control theory on a Lie group, with connections to hyperbolic geometry and Hamiltonian mechanics. Bang-bang Pontryagin extremals to the optimal control problem are smoothed polygons. Extreme difficulties arise in the proof because of chattering behavior in the optimal control problem, corresponding to possible smoothed polygons with infinitely many sides that need to be ruled out. To analyze and eliminate the possibility of chattering solutions, the book introduces a discrete dynamical system (the Poincare first recurrence map) and gives a full description of its fixed points, stable and unstable manifolds, and basin of attraction on a blowup centered at a singular set. Some proofs in this book are computer-assisted using a computer algebra system.