When the Japanese mathematician Sōichi Kakeya posed a seemingly simple question in 1917 – what is the smallest area a needle can sweep out when rotated in all directions – little did he know that it would fuel decades of mathematical exploration and discovery. This question, known as the Kakeya conjecture, has since led mathematicians to uncover surprising connections between harmonic analysis, number theory, and even physics.

While variations of the Kakeya conjecture have been proven in easier settings, it still remains unsolved in normal, three-dimensional space. However, two mathematicians have recently made significant progress by providing a new proof that addresses a major obstacle that has hindered solutions for decades.

Kakeya was primarily interested in sets in the plane that contain a line segment of length 1 in every direction. Traditionally, the smallest such set was thought to be a deltoid – a triangle with slightly caved-in sides – which has half the area of a disk with a diameter of 1. However, in 1919, the Russian mathematician Abram Besicovitch demonstrated that it was possible to construct a thorny-looking set with an arbitrarily small area by arranging needles in a specific way.

This set, called a Besicovitch set, can be formed by splitting a triangle into thinner triangular pieces and carefully rearranging them to overlap as much as possible while protruding in slightly different directions. By repeating this process infinitely, a set with no area can be obtained while still accommodating a needle pointing in any direction. This counterintuitive result can be extended to higher dimensions.

A key development came when mathematicians began reframing the Kakeya problem by introducing the concept of fattened sets. These are sets where each line segment is fattened up, resulting in extremely thin rectangles or tubes. The size of these sets, as measured by their area or volume, changes as the width of the needle is varied.

The mathematician Roy Davies showed in the 1970s that as the total area changes, the width of each needle must change significantly. For example, to achieve a total area of 1/10 of a square inch, the needles would need to be approximately 0.000045 inch thick. As the area decreases further, the needles become exponentially thinner.

The “size” of the Kakeya set is measured using a quantity called the Minkowski dimension, which indicates the rate at which the number of balls needed to cover the set grows as the diameter of each ball decreases. While familiar dimensions, such as 1 for a line segment and 2 for a square, exist, Minkowski’s definition allows for sets with fractional dimensions, such as 2.7.

Through the exploration of the Kakeya conjecture and the concept of Minkowski dimension, mathematicians continue to delve into the intricate world of geometric puzzles and their far-reaching implications.

Sources:

– Source 1: https://www.quantamagazine.org/mathematicians-finally-move-closer-to-solving-an-intractable-geometry-problem-20220223/

– Source 2: [Add source if available]