The bizarre mathematical conundrum of Ulam's Spiral

Illustration for article titled The bizarre mathematical conundrum of Ulams Spiral

If there's anything we learn from math teachers and the Da Vinci Code, it's that prime numbers are magic. They can do anything, and be anywhere. Including a doodle on a math paper.

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In the 1960s, a gentleman known as Stanislaw Ulam was making his way through a miserable meeting by doodling on a piece of paper. Unlike most of us, who only manage to do 3D cubes and obscene drawings of people we don't like, Ulam tried filling his paper with math. And he discovered something very strange. Ulam drew a '1' at the center of his paper. Directly to the right of the one he drew a '2.' Above the two he drew '3', and continued spiralling the numbers outwards toward from the one. When he was done filling up the page, he decided to circle all the prime numbers - the numbers divisible only by one and themselves.

Illustration for article titled The bizarre mathematical conundrum of Ulams Spiral
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What he found was a lot of diagonal lines. They crisscrossed the paper, sometimes in short bursts and other times in long strings. While there are plenty of singularities and outliers, a large plot of the primes on Ulam's Spiral shows a remarkable density of diagonals. Further plotting with computers show that these diagonals appear even when the numbers get high, and even when the spiral doesn't originate with the number one. Change the spiral from one that's plotted on a grid to one that's plotted on a circular spiral, and the lines will change direction, but they'll still be there. Plot it on the hexagon - more lines.

It's things like this that make prime numbers so eery. They keep showing up in nature, in important functions, and in pure mathematical play. (I think they're the ghosts of ancient Greek numerals.)

Via Good Math.

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DISCUSSION

Corpore Metal

This subject and these diagrams and images remind me of the movie, Pi.

There were some comments below trying to tie math together with religion or the lack of religion. As an atheist, I think this is pointless. Throughout the entire history of math, people have look at its beauty and hidden orders and wondered if, as Galileo put it, we are reading the mind of God or not.

I personally think it's entirely possible to have an ordered universe on the mathematical level and still have no Grand Maker of Mathematics. Why can't these eternal, universal patterns just be? Why do they have to have any meaning at all? Why can't math just be and be nothing beyond that?

People see and discover the beautiful, and vastly clever patterns in math and some of them think, "Wow, what a clever hack!" But that doesn't mean there is a hacker.

Or on the other hand, maybe god is math and nothing more—as some scientists have suggested. If so, that is a very strange, alien and remote god to think about, not the god hoped for by some religion.

These hidden patterns in math aren't really good to build a decent religion out of because they aren't easily accessible by the public, not without a lot of training and education. These patterns might strike some as beautiful and elegant but they have almost no relevance to the daily struggles of people's personal lives.

I could show you pictures of the Julia set and still, your son is dead or you husband has left you and you still have to get up and go to work in the morning. Not a very helpful or comforting god if you ask me. Better to look for god elsewhere.

We could, as the Pythagorians did, apply mysticism to this stuff but, if we are being honest (And the Pythagorians weren't.), it's a very sterile and alien mysticism to build a cult around.