Why Georg Friedrich Bernhard Riemann Rocks Your Face Off (Part 1)

I promised more mathematics, and you’re going to get it!

There is a famous mathematician in the mathematics world that has been getting a lot of attention the last century or so. His name is Georg Friedrich Bernhard Riemann, but honestly, you’re not going to find any other Riemann, so I’ll call him just that. Here he is ->

This handsome German gentleman did two things to earn the lofty position this entry so boldly voices:

1) The bulk of his worked helped revolutionize the way that we look at geometry- a subject that hadn’t had too much innovation since Euclid back in 300 B.C. And actually, probably enabled Einstein to fully develop the mathematics behind relativity.

2) In an unsual number theory paper, the only one he had written in the subject, he stated offhandedly that all of the non-trivial zeros of the Zeta function probably have real part 1/2 (I’ll explain this a little better in the next post). He had no clue that a century and a half later this hypothesis would be one of the most sought after proofs in history. It is considered the most critical unsolved problem in mathematics today, and if you were to prove it, Clay Math Institute would give you one million dollars. (See: Millenium Problems)

So let’s talk a little about point one for now, shall we?

We all took Geometry in eighth or ninth grade, so you’re probably thinking either, “I hate math, why am I reading this?” or “What could be so new in geometry? It’s a bunch of lines on a plane, right?”

That’s what held mathematicians back for millenia.

Behold Poincare’s globe:

Let me explain something about Euclid’s geometry. Euclid came up with five “postulates,” which is the mathematical term for “we’re all going to agree that these things are right, because if you don’t you’re being a smart aleck.” They include:

1) Two points can be joined with a straight line.

2) Straight lines can go forever off into space if you have the time to draw it long enough.

3) If you take a short line segment and spin one end of the line around while holding the other end fixed, you get a circle.

4) Right angles are always 90 degrees.

5) If two lines are parallel to each other, they will never intersect. (Technically, if two line segments both intersect another line segment, the two line segments must intersect at one point, or they have to be the same line.)

After reading these, if you’re tracking, you’re probably in agreement with the utter obviousness of these statements. Good. These are all obvious in our perception of the world.

The problem was that some people wanted to prove the fifth postulate. They didn’t think it was obvious, for whatever reason, and for centuries many either labored for years to prove it from the other four postulates (seriously- some people dedicated their lives to proving this postulate and died trying… too bad it’s been proven that you can’t), or they just assumed it was true. Most assumed it was true.

Not anymore in the 19th century. I believe it was Gauss who first mentioned anything about doing geometry without the fifth postulate. He was writing to a friend about it, and he mentioned that weird things happened without it, but without it geometry was still completely consistent!

Poincare developed a model as shown above that demonstrates the hyperbolic space. Imagine a circular swimming pool and you’re swimming in the center. Now imagine that in this swimming pool if you swim you shrink as much as you swim towards the edge. So if you swam half-way towards the edge of the pool, you would be half-sized, and so on. You could never reach the end of the pool, but you could do normal geometry inside this death-trap of a pool, and under the right conditions parallel lines would intersect.

Holy cats.

Think about it.

How this all relates to Riemann:

Riemann developed a whole new way to look at geometry using something called “tensors.” This gave mathematicians a huge tool to create, classify, and study different geometries, whether it had the fifth postulate or not. He also figured out the right way to go higher than three dimensions. Trust me. There are wrong ways.

That was the bulk of his work, but not what he’s most famous for. I’ll post part 2 soon.