The phrase "math is beautiful" is everywhere. Go on instagram and you'll see hundreds of usernames that include some combination of the words "math," "beauty," "love," and "art." Go on Amazon and you'll find dozens, if not hundreds, of books with a similar combination of words in their title.

And I love it. I think it's great. I wish every instagram account name was like this. However, I was recently asked by someone **why** math is beautiful. This person wasn't a math lover, but this person was also not a math hater. This was just a very genuine question — what is it that so many people find beautiful about mathematics?

After a lot of thought I've been able to largely (but not completely, and I'd love to know your thoughts!) summarize it according to the following 4 "S"s: Sensory, Surprise, Simplicity, and Symbiosis. If I had to rank these in terms of importance, I'd say Surprise would win. But they all play their own role. Let's start with the most familiar.

#### Sensory

This is the most familiar type of beauty because it's the same type of beauty we often associate with other art forms like painting and music. Namely, a beautiful painting is beautiful because you can just **look **at it and say "wow, neat." A certain piece of music is beautiful because you can just **hear** it and say "wow, neat."

Similarly, there are some aspects of math that are beautiful to just look at. The Mandelbrot Set immediately comes to mind, as do many other fractals. The golden spiral also looks pretty cool, as do a number of other mathematical objects when depicted on paper.

In some sense, this is the most straightforward and relatable type of mathematical beauty. However, I would venture to guess that this is mostly **not** the type of beauty that we mathematicians have in mind when thinking about mathematical beauty. It is, however, what many who are uninitiated probably think of since it's the most common type of artistic beauty "out there."

So what is the beauty that we (mathematicians) probably have in mind? I'd say it's largely summarized by the next "S".

#### Surprise

As we've seen regularly throughout the Coho Courses, and as anyone who has studied math with any diligence will know, we are almost entirely free to

- make whatever definitions we want, and
- ask whatever questions we want.

What we are most certainly **not** free to do, however, is decide what the answers to those questions — or the consequences of those definitions — are. **That** is up to the gods of logic, and it is up to us to try to figure them out.

Occasionally, those answers are incredibly **surprising**. For example, a circle is a very natural thing for us (humans) to define, and it is then very natural to ask the question: how many diameters does it take to create a single circumference of a circle? The answer to this question, however, is insanely surprising.

Namely, we absolutely could imagine a world where π is, like, 2, or 3, or 4. At worst, maybe π is a fraction like 5/2 or 22/7. Pi is, however, not only irrational, it's transcendental – effectively meaning that there are virtually no "nice" ways to define it. One can't help but be surprised that the answer to such a simple, obvious, and "nice" question can have such a bananas answer.

#### Why Does Surprise = Beauty?

Now we need to ask: why is "surprise" equated with "beauty" in these instances? After all, I get surprised when I stub my toe but I would hardly call that beautiful.

My proposal for an answer to **this **question is largely spiritual. When we encounter these surprising Truths — Truths that were forced upon us by rules of logic that seemingly were themselves forced upon us as well — we feel like we're directly interacting with something "larger" than us, something outside of us.

In other words, uncovering these surprises makes one feel, quite literally, like they're having a discussion with some kind of god*. And **that **experience has been equated with beauty since the beginning of humankind. Discovering a surprise in mathematics is like finding a little secret message, written in stone since the beginning of time, but accessible to our brains through the language of logic. And that's beautiful.

#### Sources of Surprise

We've talked about the transcendentality of π, but what are some of the other sources of surprise in math? There are pure surprises – like seeing π show up in places that seemingly have nothing to do with circles – but there are also some general patterns that we might be able to use to classify some mathematical surprises. The two that come to mind for me are simplicity and symbiosis. Let's take a look at them.

##### Simplicity, and the most popular math tattoo

When things turn out to be simpler than expected, that's surprising, and often beautiful. Consider Euler's identity, which is possibly the most popular math tattoo on Earth (no hate, I have a couple math tattoos and I'll probably eventually get this one too because why not):

Most of the beauty in this equation comes from the "pure" surprise that these 5 fundamental numbers – e, i, π, 1, and 0 – are related to each other at all. What takes the beauty of this formula to the next level is its simplicity.

If someone came to you and said that there was a formula that related these five numbers, there'd be no reason not to think that that formula was huge and complicated. Turns out, though, that it's not. In fact, it's hard to even **imagine** a simpler formula. That adds to the surprise, and therefore to the beauty of this Truth.

##### Symbiosis and The Calculus

Another source of surprise in mathematics is when things secretly have symbiotic relationships with each other. Another word for this would be "unification," but that doesn't start with an "S" so I didn't want to use it.

A classic example of this is the Fundamental Theorem of Calculus, which relates derivatives of functions with integrals of functions. Derivatives were initially studied for their ability to compute the instantaneous rate of change of things, and integrals were initially studied for their ability to compute areas of smooth shapes. The Fundamental Theorem of Calculus, however, shows that these two concepts are actually intimately connected — two different sides of the same coin.

Another example comes from complex analysis. Introduction of the quantity "i", the square root of negative 1, was motivated by studying polynomial equations. However, combining these ideas with those of Calculus leads to surprise after surprise in a field called complex analysis. In other words, the **symbiotic relationship** of these two sets of ideas – complex numbers and calculus – leads to a field that is more beautiful than the sum of its parts.

This happens – and is still happening – often in mathematics, and it's always wonderful when it does. A surprising symbiosis between elliptic curves and modular forms is not only beautiful, but also led to Andrew Wiles' proof of Fermat's Last Theorem. The Langlands Program is a whole research effort studying surprising symbioses between number theory and geometry (and is also relevant for the Fermat's Last Theorem stuff).

#### In Short

There are many ways in which mathematics can be, and is, beautiful. Yes, there are the usual, sensory ways. But I think generally speaking it's the **surprise** of mathematics that makes us feel like we're interacting with something divine, and therefore beautiful. Some of the main sources of surprise are simplicity and symbiosis.

All that said, this is just the opinion of a couple people who like to make t-shirts. I have no doubts that every mathematician has their own, unique reasons for loving this subject and/or thinking that it's beautiful. We'd love to hear your reasons, and we'll incorporate them into the next article we write about such things!

*To be very clear, this is not a religious article, nor are we a religious brand. We don't discriminate against any religion or lack thereof.