Introduction
For years, mathematicians have been intrigued by the idea of finding the worst shape for packing the plane. While hexagons have been proven to be the best way to fill space efficiently, the search for the worst shape has been a challenging endeavor. Recently, a breakthrough came in the form of a new paper by mathematicians Thomas Hales and Koundinya Vajjha, shedding light on this long-standing conjecture.
The Quest for the Worst Shape
The challenge lies in defining a worst shape that is both convex and centrally symmetric. This means ruling out shapes with holes or inward dips, as well as ensuring that the shape maintains symmetry and convexity. Previous attempts pointed to shapes like the rounded octagon as potential candidates for the worst shape, but definitive proof was elusive.
A New Approach
Hales and Vajjha took on the daunting task of proving that the rounded octagon is indeed the worst shape for packing. Employing optimal control theory, they delved into the intricate world of flip-flopping structures to eliminate other possibilities. This innovative approach opened up new avenues in the search for the worst shape, leading to a deeper understanding of the problem.
Challenges and Triumphs
The journey towards a proof was not without its obstacles. The duo encountered candidate shapes that exhibited unusual behaviors, complicating the process of elimination. Despite the complexity of the problem, Hales and Vajjha persisted, unraveling new structures and methods in their pursuit of a solution.
A Step Closer
After years of dedicated work, Hales and Vajjha made significant progress in proving a conjecture put forth by mathematician Kurt Mahler in 1946. Their exploration of smoothed polygons with rounded corners brought them closer to Mahler’s assertions, marking a pivotal moment in their research.
The Road Ahead
While their proof of Mahler’s first conjecture is a substantial achievement, Hales and Vajjha acknowledge that there is still work to be done. The completion of Reinhardt’s conjecture remains a question mark, highlighting the ongoing nature of mathematical inquiry. As they continue to push the boundaries of knowledge, the quest for the worst shape for packing the plane persists, offering new insights and challenges along the way.
