Archive for March, 2008

Ode to the Cadbury Egg

March 27, 2008

egg

Cadbury Egg, you satisfy my craving for sugar

I look but ne’er find fellow fans of your delicious nougat

That sumptuous shell of chocolate

May you forever be manufactured

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Why Georg Friedrich Bernhard Riemann Rocks Your Face Off (Part 2)

March 26, 2008

Let me tell you a little about prime numbers.

Prime numbers are surprising, slippery, beautiful numbers. The fact that no other number divides into a prime besides itself and one allows for a lot of unique and interesting results, but they are the most difficult to attack for a mathematician.

For example, there is a famous unsolved problem called the Golbach conjecture: every even number greater than two can be written as the sum of two prime numbers. So 4 is 2+2, 6 is 3+3, 8 is 5+3, and so on. The challenge has been around since 1742. People have tested every even number up to 1,000,000,000,000,000,000 and have found that it’s true. But how do you prove that no matter how far you go up the number line this will always work out? What if there’s a number x that is fifty digits long that can’t be written as the sum of only two primes?

Another example of the challenge of prime numbers is the fact that it is very time-consuming to check whether a number is prime or not. Unless you have a few mathematical tricks up your sleeve, you have to try dividing a number by all the prime numbers below it and if it never gets divided evenly you can add that number to the list of primes. That’s not too bad, right? Not for large numbers. When we look at numbers that are hundreds of digits long it can take hundreds of computers months with specialized algorithms to figure out if that number is prime or not. There has to be a formula or something, right?

Let’s talk about Riemann.

Younger Riemann

In 1859, after Riemann had made his name a little well-known in the Mathematical circuit from his geometry, he wrote a paper titled “On the Number of Primes Less Than a Given Quantity.” The paper basically said, “give me a number, and I can tell you how many prime numbers are below it.”

That’s what we need, right? Here’s what he gave us

pcf

Don’t worry about understanding this completely. I add this visual to feel smart. The pi(x) is the function that counts the number of prime numbers under x. On the right is a bunch of complicated technical limits, integrals, and big-O notations. But the point is that Riemann shows us a prime number formula! This is all provided that “the non-trivial zeros of the zeta function all have real part 1/2.” This is the Riemann Hypothesis.

And this little hypothesis has neither been proven or disproven since. We can’t really say much until it’s been settled, and when Riemann tried to prove it while writing his paper he wrote, “Certainly someone would wish for a stricter proof here; I have meanwhile temporarily put aside the search for this after some fleeting futile attempts, as it appears unnecessary for next objective in my investigation.” It has since turned out to be the most crucial part to his paper and a major hinge to most number theory research today.

Let me show you the zeta function:

Riemann Zeta Function

Put in a complex (imaginary) number n, and out pops a number on the other side. If you put in a correct n, you get zero. And that’s what Riemann looked at- the zeros. He said whenever this function was zero, the n looked like “1/2 + (number) * i”.

Why do prime numbers matter, especially when you’re dealing with numbers larger than the number of atoms in the universe? (This is not an exaggeration.)

Whenever you make a purchase online and there’s a little padlock in the corner of your browser, that encryption is based on the fact that huge prime numbers are hard to pick out, and large non-primes are hard to divide up into primes. This encryption, the encryption that keeps the majority of internet information safe, is called RSA encryption.

It’s quite possible that once the Riemann Hypothesis is worked out we’ll have to find some new type of encryption.

 

Oof.

This has been a long trek. I’ma step away from math for a bit.

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

March 19, 2008

Riemann’s picture that you’ll find at any website about Riemann… probably his best.

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:

HYPERBOLIC TESSELLATION!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.

3.14…

March 14, 2008

HAPPY PI DAY!

I only have four minutes to post this, so I’m going to talk about one thing- this formula:

BEAUTIFUL!

That’s right. If you take the irrational constant e (2.718…) a raise it to the power of the complex number i ( sqrt(-1) ) times the irrational constant pi (3.14159265…), it equals negative one.

Think about it.

It’s been dubbed the most pleasing equation in mathematics. It fuses together the most basic constants into one elegant phrase.

I will probably post more math stuff later in honor of this great day.

Walking Tacos!

March 12, 2008

WALKING TACO!!!

I love walking tacos.

I walked into work today to find that the management had decided to treat us lowly workers to some walking tacos. After we had received them they ordered us not to sit down until we had finished the meal. We laughed nervously but gratefully at the joke and they handed us a bag of doritos with taco meat.

Surprisingly, source W has nothing to say about walking tacos. It merely brings up an article about “Frito Pie.” Someone should get on that.