One of the things that’s endlessly fascinating to me about math and

science is the way that, no matter how much we know, we’re constantly

discovering more things that we *don’t* know. Even in simple, fundamental

areas, there’s always a surprise waiting just around the corner.

A great example of this is something called the *Ulam spiral*,

named after Stanislaw Ulam, who first noticed it. Take a sheet of graph paper.

Put “1” in some square. Then, spiral out from there, putting one number in

each square. Then circle each of the prime numbers. Like the following:

If you do that for a while – and zoom out, so that you can’t see the numbers,

but just dots for each circled number, what you’ll get will look something like

this:

That’s the Ulam spiral filling a 200×200 grid. Look at how many diagonal

line segments you get! And look how many diagonal line segments occur along

the same lines! Why do the prime numbers tend to occur in clusters

along the diagonals of this spiral? I don’t have a clue. Nor, to my knowledge,

does anyone else!

And it gets even a bit more surprising: you don’t need to start

the spiral with one. You can start it with one hundred, or seventeen thousand. If

you draw the spiral, you’ll find primes along diagonals.

Intuitions about it are almost certainly wrong. For example, when I first

thought about it, I tried to find a numerical pattern around the diagonals.

There are lots of patterns. For example, one of the simplest ones is

that an awful lot of primes occur along the set of lines

f(n) = 4n^{2}+n+c, for a variety of values of b and c. But what does

that tell you? Alas, not much. *Why* do so many primes occur along

those families of lines?

You can make the effect even more prominent by making the spiral

a bit more regular. Instead of graph paper, draw an archimedean spiral – that

is, the classic circular spiral path. Each revolution around the circle, evenly

distribute the numbers up to the next perfect square. So the first spiral will have 2, 3, 4;

the next will have 5, 6, 7, 8, 9. And so on. What you’ll wind up with is

called the *Sack’s spiral*, which looks like this:

This has been cited by some religious folks as being a proof of the

existence of God. Personally, I think that that’s silly; my personal

belief is that even a deity can’t change the way the numbers work: the

nature of the numbers and how they behave in inescapable. Even a deity who

could create the universe couldn’t make 4 a prime number.

Even just working with simple integers, and as simple a concept of

the prime numbers, there are still surprises waiting for us.