On Tuesday the winners of the most prestigious award in mathematics — the Fields Medal — were announced. The award is given to 2-4 mathematicians under the age of 40 every four years, “to recognise outstanding mathematical achievement for existing work and for the promise of future achievement.”
This award is typically given for mathematical research that solves or extends complicated problems that mathematicians have struggled with for decades, or even centuries.
And, believe it or not, one of the winners, Manjul Bhargava, reformulated a 200-year old number theory problem in a rather unconventional way: by using a Rubik’s cube and Sanskrit texts.
But before we get to his idea, let’s go over some background.
Here’s an interesting mathematical idea: if two numbers that are each the sum of two perfect squares are multiplied together, the resulting number will also be the sum of two perfect squares.
Let’s take the numbers 25 (a perfect square of 5) and 36 (a perfect square of 6) and just add them to each other for simplicity’s sake. The resulting equation would read:
(25 + 25)(36 + 36) = 3600, where 3600 is the perfect square of 60.
You can keep trying this with all different number combinations — as long as the 4 numbers in the brackets are perfect squares — and the output will always also be a perfect square. (Yes, maths is awesome.)
But let’s get back to Bhargava. His grandfather, Purushottam Lal Bhargava, was the head of the Sanskrit department of the University of Rajasthan — and so “Bhargava grew up reading ancient mathematics and Sanskrit poetry texts”, according to Quanta Magazine.
In one of these manuscripts, he discovered a “generalization [similar to the above maths were just covered] developed in the year 628 by the great Indian mathematician Brahmagupta” that stated:
“If two numbers that are each the sum of a perfect square and a given whole number times a perfect square are multiplied together, the product will again be the sum of a perfect square and that whole number times another perfect square”.
Later on, when he was a graduate student, Bhargava learned that Carl Gauss — a leading mathematical figure of the 18th and 19th century “came up with a complete description of these kinds of relationships” — provided that the expressions only had two variables, and only quadratic forms (meaning x2,but not x3).
Essentially, Gauss came up with a ‘composition law’ that would tell you which quadratic form you’d get if you multiplied two of these kinds of expressions together.
But here’s the downside: Gauss’ law took him approximately 20 pages to write out. Ouch. No one wants to read through all of that.
Bhargava was interested in uncovering an easier way to describe Gauss’ law — and perhaps if he could tackle higher exponents, according to Quanta Magazine.
Then one day, Bhargava thought of a solution when he was sitting in his room, looking at a Rubik’s Cube. He took a mini-cube (a 2×2 Rubik’s cube) and “realised that if he were to place numbers on each corner of the mini-cube and then cut the cube in half, the eight corner numbers could be combined in a natural way to produce a binary quadratic form”, according to Quanta Magazine.
If you think about a Rubik’s cube, you’ll note that there’s 3 ways to cut the cube in half. Using that information, “Bhargava discovered [that these three forms] add up to zero” with respect to Gauss’ method. And ta-da, by simply dividing a Rubik’s cube, he “gave a new and elegant reformulation of Gauss’ 20-age law“.
Afterwards, Bhargava worked with a Rubik’s Domino (a 2x3x3 shape) — and he realised that he could go beyond the limited quadratic forms of Gauss’ law.
Eventually, he went on to discover 12 more compositional laws, which later became his Ph.D. thesis.
According to Quanta Magazine, Barghava said that he’s always been interested in ideas like this — the “problems that are easy to state, and when you hear them, you think they’re somehow so fundamental that we have to know the answer.”
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