The Kakeya Conjecture- True Impact of Mathematical Discovery

From coping with the global post-pandemic recovery to the humanitarian crisis of the battalion between Israel and Palestine, we have abandoned the thought of commemorating and honouring recent hypotheses established by the most hard-working and conscientious mathematicians of today. Especially those who dedicate their lives to uncovering results behind theories such as ones that attempt to find the minimum area a room must be at to point a line in every direction. Not only that but to obtain the unnerving end product of this query, there is the commendable and intuitive utilisation of needles. So fellow peers, do not question the significance of mathematics in your daily life, for this article has been orchestrated for the very purpose of realising just that.

The Immediate Solution

Soichi Kakeya, in 1917, had proposed the idea of rotating a needle on a flat surface to point in every direction, to establish the least possible area it can construct, all in good fun of course. Little did he know that his mathematical successors would unveil its vitality in society, in mathematics at least, and rack their brains uncovering its answer. They have discovered its pleasantly surprising connections to concepts like harmonic analysis, number theory and physics.

Intrigued, Russian mathematician Abram Besicovitch, set to work almost immediately in 1919. He suggested that we utilise a deltoid (a fancy triangle) and split it into segments and rotate them such that they overlap, but point themselves in distinguished directions. ambiguously and objectively, the consequence of this theory is that when subdividing these segments further, you end up with a set with no area whilst still accommodating the rotation of the segments in every direction.

Just as one would think that all is well and done, the unending and eminent curiosity of mathematicians erased all such beliefs, as they proposed another aspect to this theory. They suggested the substitution of areas with a different notion of size.

Roy Davies’ input

A new question arose. rather than using an idealised form with no real value, why not utilise a sturdy and solid form with an actual volume, such that a legitimate product can be deducted? questions of Bestovich’s ambiguity arose.

Now before we delve deeper into the topic, let us grasp an understanding of the Minkowski dimension. It is a unit of measurement of the Kakeya sets. It is a comparative degree of the area of the set relative to the thickness of the needle.

This method of thinking was proposed when Roy Davies stated that to accommodate the slightest change in the area of the set, the entire width of each needle must be subject to drastic change. This has been proposed such that theory can fill up twirling needles that can fill up three-dimensional space, and soon enough according to the Minkowski dimension, we were able to achieve the minimum value of 2.7.

Tao and Katz defy the facts!

As time flowed by, the interdisciplinary characteristic of the Kakeya conjecture had been acknowledged, gathering attention amongst numerous mathematicians.

Self-assured, and almost rightfully so, Thomas Wolff established that the Minkowski dimension of a 3d set is no less than 2.5. Terence Tao and Nets Katz prove them wrong! They beat the record and obtained the result of 2.500000001. Although not renowned for the insurmountable distinction in improvement, its significance lies in overcoming an established and technical barrier.

Now, satisfied by their achievement, and greatly motivated and endlessly diligent, they strive for more! they wished to achieve a sort of contradiction. This counterexample is constructed with 3 prerequisites. They wished to establish the factual accuracy of this theory and managed to meet only 2 out of the 3 prerequisites. one, it must be “plany,” which means that whenever line segments intersect at a point, those segments also lie nearly in the same plane. it must also be “grainy,” which requires that the planes of nearby points of intersection be similarly oriented.

In a “sticky” set, line segments that point in nearly the same direction also have to be located close to each other in space. This was one theory they had failed to prove. theoretically, an increase in overlaps provokes the shortening in the size of the set, and there is no better way to enforce these overlaps than to locate them adjacent to one another.

Hong Wang and Joshua Zahl to the rescue!

Following Tao and Katz’s footsteps, Hong Wang and Joshua Zahl aimed to establish this counterexample in semantics, which meant that all three properties (plany, grainy and sticky) ought to co-exist… They analysed the structure of the Kakeya sets. They were in the realisation of the distinction between the idealised version of the needle and its much thicker contradiction, which was what multiple mathematicians sought to comprehend.

Remember in a sticky set, eventually the needles are projected as akin to one another, no matter the size. the smaller they are tattered into, the more” plany” the sets emerge as. Mow they indulged in an area known as projection theory, one not anticipated by their predecessors.

This led them to more inquiry. they attempted to achieve this counterexample by utilising both versions of the Kakeya sets (idealised- which is contained in mere line segments, and its thicker contradiction resembling a real needle), and this was their discovery- they could achieve neither.

Either one would have to accommodate the line segments in a 2d space to make it smaller whilst pointing in various directions , or , in the tangible version, the organisation of the needles in the set would depend upon their various function, leading to other kinds of projections, much too complicated and strenuous that they eventually lose all meaning. In the end, neither seemed to be possible. their efforts seemed fruitless.

This is the nature of the never-ending struggle of these ever-persistent mathematicians, striving despite their crestfallen state. While the Kakeya conjecture may continue to baffle and bewilder, perhaps it serves as a humbling reminder that even in the realm of numbers, there are corners of absurdity waiting to be explored so dear reader, as we bid goodbye to the ever-elusive and whimsical world of the Kakeya conjecture, we realise that the true essence of mathematics lies not only in the precision of its proof but in the utter audacity of these mathematicians to question what seems to be unanswerable.

By: Rokshana Rajendran