The Beauty of Math I: Simplicity & Complexity

There are many reasons why people call math beautiful. To most, this adjective would be the last they would use to describe math: how can something so cold and logical contain beauty? Obviously this is a very subjective topic so all I can do is try and convey why I think it is beautiful.

“Life is not complex. We are complex. Life is simple, and the simple thing is the right thing.”
Oscar Wilde

In many ways, both philosophically and literally, math is a mirror of life. Some of my favourite mathematics comes from seemingly simple problems that hide a lot of complexity. In life, we tend to ignore ‘simple’ questions and move on to ‘harder’ or ‘more practical’ problems. At some stage in your life, you have probably wondered ‘who am I?’ and ‘why am I here?’. Do you spend much time trying to answer these now? Or do you just get on with your life? These questions are extremely simple ones to ask, yet to answer them is extremely difficult. We are often made to feel like these questions are not worth spending time on. Thankfully the situation in mathematics is very different: a lot of beautiful mathematics has come from actually taking the time to answer some ‘simple questions’.

Let’s start with some simple mathematics and see where it leads us. One of the first real bits of mathematics that people learn is Pythagoras’ theorem. This theorem tells us how the sides of a right-angled triangle are related: the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Most people remember the formula $a^2+b^2=c^2$ . If I give you two of the side lengths, this formula lets you work out the third. Most of the time, picking values for $a$ and $b$ give you ‘terrible’ numbers for $c$ . For example, if $a=b=1$ , then $c=\sqrt{2}\approx 1.41421356\dots$ and $c$ is irrational (yuck!). But sometimes things work out much nicer. For example, if $a=3$ and $b=4$ , then $c=5$ ! There are other whole number solutions as well. For example, $a=5$ , $b=12$ and $c=13$ . These nice solutions are called Pythagorean triples. A natural question to ask is: how many Pythagorean triples are there? This is a simple question to ask. It turns out that there are infinitely many whole number solutions to $a^2+b^2=c^2$ . In fact, with a bit of math, we can show that all Pythagorean triples can be written as $a=n^2-m^2$ , $b=2mn$ , and $c=n^2+m^2$ (up to similarity of triangles) for whole numbers $m$ and $n$ . For example, if we pick $m=1$ and $n=2$ we get the Pythagorean triple $(3,4,5)$ . Picking other values for $m$ and $n$ gives us other Pythagorean triples and all Pythagorean triples can be found this way! This gives us a complete answer to our first ‘simple’ question, but there is a fair bit of math behind the solution to this problem. Having said that, you learn all the tools to answer this question in high-school but it would take some creativity to come up with this yourself!

“Simplicity is the greatest adornment of art.”
Albrecht Durer

We are now going to tweak this question slightly and see what happens. Notice that the two smallest numbers of the Pythagorean triple $(3,4,5)$ differ by 1. Our question is: are there other Pythagorean triples with this property and if so, how many? The answer to the first part of this question is yes: the triple $(20,21,29)$ is another example. In order to systematically generate more examples, we will use our answer to the first question. We know that the two smallest numbers of a Pythagorean triple have to be of the form $m^2-n^2$ and $2mn$ for some numbers $m$ and $n$ . For these to differ by one means that

$m^2-n^2-2mn=\pm 1$

Using some high-school algebra, we can write this as $(m-n)^2-2n^2=\pm 1$ . If we let $x=m-n$, then we can write this as $x^2-2n^2=\pm 1$ . This is a very famous equation called Pell’s equation. It takes a lot more sophistication to solve this equation in whole numbers and it is intimately related to algebraic number theory (something you probably won’t see in high-school). This is a big step up from our previous problem, but it turns out that the solutions to this Pell equation can be found by looking at $\sqrt{2}$ . The $\sqrt{2}$ can be written as a continued fraction in the following way:

$\sqrt{2}=1+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\dots}}}}$

To get exactly $\sqrt{2}$ , you must continue the fraction indefinitely. However, chopping off the continued fraction early gives us a fraction which is a good approximation of $\sqrt{2}$ . Surprisingly, the numerator and denominator of these fractions are solutions to Pell’s equation! For example, if we chop the fraction off after the second 2 we get

$1+\frac{1}{2+\frac{1}{2}}=\frac{7}{5}$

and $7^2-2\cdot 5^2=49-50=-1$ . Using this solution to Pell’s equation, we get the solution $m=12$ , $n=5$ to our original problem. This gives us the Pythagorean triple $m^2-n^2=119$ , $2mn=120$ , $m^2+n^2=169$ and we can see that the smallest two numbers differ by 1! Using this process, we can generate infinitely many Pythagorean triples with this property.

It’s amazing that such a slight change to the question can produce such a huge change in how we answer it. To fully understand Pell’s equation requires a lot of advanced mathematics and it has surprising links to other areas of math. Despite this, the actual problem itself can be understood by a high-school student. This is our first glimpse at simple problems leading to complex, underlying mathematics.

We are going to make one final tweak to our problem. This last tweak will take us to the cutting edge of mathematics! The solution to our first problem showed us that there were infinitely many whole number solutions to the equation $a^2+b^2=c^2$ . Our last simple question is: what happens if we change the power from 2 to some other number? For example, are there any whole number solutions to $a^3+b^3=c^3$ or $a^{101}+b^{101}=c^{101}$ ? To put it more precisely: are there any whole number solutions to the equation $a^n+b^n=c^n$ for $n\geq 3$ ? This is one of the most famous questions in mathematics and was originally asked by Pierre de Fermat in 1637. Fermat believed that the answer to this question was that there are no solutions to this equation. He also believed that he could prove this and left a message in the margin of one his textbooks claiming he had a proof. The problem has since been known as Fermat’s last theorem.

“Always try the problem that matters most to you.”
Andrew Wiles

It is unlikely that Fermat actually had a valid proof of this theorem: it took 358 years for mathematicians to confirm that Fermat was right in 1995. The proof requires a huge amount of mathematical machinery from many different areas: algebraic number theory, representation theory, group theory, geometry, and analysis. The beauty of this problem however is that it draws on all of these fields. In fact, this problem was the driving force behind the development of algebraic number theory and led to many other interesting discoveries and ideas. One of the main ingredients in the proof of Fermat’s last theorem were elliptic curves. These amazing objects appear in a number of areas of mathematics and are connected to some extremely surprising math (see monster moonshine, for example). It’s mind-boggling that such a simple tweak to Pythagoras’ theorem could lead to some of the most amazing objects in mathematics! It certainly shows the power of pursuing ‘simple’ questions.

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