Packings of Smoothed Polygons (2405.04331v1)

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.

• The paper confirms Mahler's First conjecture by demonstrating that smoothed polygons are the optimal unpackable convex disks in the plane.
• It leverages optimal control theory and hyperbolic geometry to reformulate the classical packing problem using a Lie group framework.
• The study further refines its analysis by addressing chattering solutions and integrating computer algebra techniques for robust geometric insights.

An Academic Overview of "Packings of Smoothed Polygons"

This extensive book by Thomas Hales and Koundinya Vajjha addresses an intriguing problem within discrete geometry: identifying the most unpackable, centrally symmetric convex disk in the plane. Building on previous conjectures, notably by Reinhardt in 1934 and Mahler in 1946, the authors focus on smoothed polygons—polygons with vertices rounded by hyperbolic arcs—as optimal candidates for this property. Specifically, they confirm Mahler's First conjecture, asserting that these smoothed polygons indeed form the class of most unpackable shapes.

Core Contributions

1. Optimal Control Theory Application: The authors leverage optimal control theory to reframe and tackle the unpackability problem. They elegantly translate the geometric problem into a control system on a Lie group with substantial implications for the geometry of packing. This approach brings in tools from hyperbolic geometry and Hamiltonian mechanics, adding depth to the analysis.
2. Solution to Mahler's First Conjecture: Mahler's First conjecture—that the most unpackable disks are smoothed polygons—is successfully resolved. The book entails a rigorous exploration of the structures that constitute the packing problem, including the derivation of dynamical systems representing these structures.
3. Geometrical and Analytical Insights: The complex dynamics of packing are described through structures like Lie groups and hyperbolic geometry. The problem is explored via Pontryagin's Maximum Principle and other control-theoretic instruments, ultimately leading to characterizations of a dense packing's geometry and density.
4. Chattering Solutions Analysis: The authors introduce methods to assess and eliminate chattering behavior (solutions with infinite switching frequency), which is significant in proving the robustness of their geometric configuration.
5. Theoretical and Practical Implications: By establishing that smoothed polygons are optimal for low-density packing, the work provides a refined understanding of packing problems, with apparent implications not just for geometry but also for data compression and material science.

Discussion

The book tackles an unresolved question that dates back to the early 20th century, navigating through various geometric and analytic frameworks. Its utilization of computer algebra systems for assisting proofs indicates a modern methodological blend of analytical reasoning and computational power.

Future Directions

The comprehensive treatment of the Reinhardt problem through these methodologies opens avenues for further exploration of related geometric packing problems in higher dimensions. Theoretical extensions to non-symmetric bodies or non-convex shapes might present a new horizon for this line of inquiry. Additionally, implications for practical optimization problems, like resource distribution and information theory, may warrant exploration, thus bridging mathematical theory with tangible applications.

Overall, "Packings of Smoothed Polygons" stands as a testament to the fruitful intersection of geometry, control theory, and computational methods. It not only provides a resolution to historic conjectures but also sets the stage for contemporary applications and further mathematical exploration.