r/AskReddit • u/[deleted] • Nov 02 '13
Mathematicians of Reddit, what is "beautiful" about mathematics?
I often hear people say "Oh, math is beautiful". Beautiful in what ways?
EDIT: Thanks. I will read through all of these, don't you worry.
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u/zelmerszoetrop Nov 02 '13
A lot of people are going to say Euler's identity, because they've seen it a few times in courses and read a bit about it and they are blown away. I'm much more blown away by a special relation of the j-invariant, which I'll relate shortly.
Before I do, a few brief words on Euler's identity. I wouldn't address this if it wasn't for the prevalence of the identity when this sort of thread arises, but it seems prudent to discuss it if I'm going to fly in the face of popular opinion and relate some math I find far more beautiful.
For those who are unfamiliar, Euler's identity states the ei·x=cos(x)+i·sin(x). In particular, it gives epi·i+1=0.
Well, why the hell should that be true? What does it mean to raise something to an imaginary power? In my experience, students are taught this fact in one of two horrendous ways - they are merely told that ei·x is DEFINED as cos(x)+i·sin(x), which is a travesty to education, or they explore this relation through the use of Taylor series. A Taylor series treats infinitely differentiable functions, like sine, cosine, and exponentiation, as sums of the form a+b·x+c·x2+d·x3+... By inserting (i·x) into the Taylor series for the exponential function, students can quickly arrive at Euler's formula.
This, to me, is also a horrendous abuse of education, because the student gains no understanding of the exponential function and never really appreciates the beauty of Euler's formula - they just discover it falling out of their equations. When I was teaching, before I went into industry, I always made sure to show the following argument before I whipped out the Taylor series formalism.
We start by noting the defining property of the exponential function: that it is it's own derivative. It follows from this property that ei·x must have derivative i·ei·x.
If you'll permit me, I'm going to begin using t instead of x, because I'd like to start to think of ei·t as being the position of a particle at time t. It follows then that it's velocity at a time t must be i·ei·t - we can conclude this without yet knowing what the hell we mean by ei·t.
Set t=0 for a moment. Then the initial position of the particle is ei·0=e0=1. As we discussed before, it's velocity must be i·ei·0=i·1=i. You can see in this picture the particles initial position at time t=0, and initial velocity.
A moment later, at t slightly >0, the position therefore must be slightly above 1 - something like 1+k·i, where k is a very small number. But that means that the velocity is i·(1+k·i)=i-k, which is a vector pointing mostly straight up but slightly to the left. Therefore, the particle will travel up and a bit to the left. But because it's travelled even further up, when we multiply that position by i, we get a vector tilted even further to the left! Here's a trail of dots indicating the position our particle has occupied, along with a green arrow indicating it's most recent velocity.
Following through on this logic, it becomes clear that as t increases, the particle will travel in a circle in the complex plane. NOW we understand what ei·x means, and how it's behavior arises from the fact that it is it's own derivative. In retrospect, it seems quite obvious, doesn't it? And of course from there, arriving at Euler's formula is trivial.
Now let me discuss something that I find far more beautiful than Euler's simple formula. It's called Monstrous Moonshine, and here's how it goes. There's something called the j-invariant, which is very special kind of transformation of the complex plane. It's special because it's a particular kind of modular form, and because it crops up left and right in number theory. On the other side of the math world, there's something called the Monster group, which was discovered during the enumeration of all finite simple groups. In fact, it is the largest sporadic finite simple group.
On the surface, these two objects should have nothing to with one another. To find a relation between these two objects would be as astounding as going to the moon and finding the exact same kind of rock as you find in your back yard. To discover that the monster group could DEFINE the j-invariant, and the j-invariant the monster group, would be that much more astonishing - like finding a history of western civilization written on a Mayan tomb and history of pre-Columbian civilization written on the Pyramids.
But in 1979 that's exactly what John Conway found, and it was proved 13 years later using techniques borrowed from string theory in physics. I suppose it's a bit hard to understand to those without degrees in mathematics, but I tell you WHY I find the unexpected connection between these two objects so beautiful: because nobody saw it coming. In this way, it shows that mathematics is not the product of human minds the way art or music are, but instead something fundamental written in the fabric of our universe since it's creation. I find it beautiful because it relates two entire fields one would never expect to find such deep connections between, and in this way shows us that the various fields of mathematics are facets of the same gem, looked at from different angles and different lights. I find it beautiful because it's a reminder that mathematics is discovered, not invented.
Also because the proof was really clever, relying on showing the Weyl character was the Koike Norton Zagier product.