Prime spirals are a way to visualize the distribution of prime numbers within the set of natural numbers. Accidentally discoverd by Stanislaw Ulam in 1963, the prime spirals show that there is order in the distribution of prime numbers, where one would expect randomness.
Prime numbers are those which are not composed of other numbers. For instance, 3, 5, 7, and 41 cannot be described as the product of two or more smaller numbers. Considering the set of natural numbers, one would expect that, counting up from 1, the frequency of encountering prime numbers would decrease, since the subset being considered, the set of candidate numbers to form a larger, composite number, is increasing. Surprisingly, though, prime numbers can be shown to be prominent among large composite numbers. A program I wrote is calculating that approximately 8% of large (>1,000,000) numbers are primes.
Prime spirals show that, not only do primes appear much more frequently than we would expect, but that they have structure in their positions in the list of natural numbers. During a “boring” presentation, Ulam was doodling on paper. He began writing out natural numbers with 1 in the center of the page, and circling outward. See Figure 1 [1].

Figure 1: Natural Numbers Spiral
Removing composite numbers from the picture shows the beginning of the pattern (Figure 2 [2]):

Figure 2: Prime Spiral
Look at the following sets of numbers: {37, 17, 5}, {41, 19, 5}, {19, 7, 23, 47} and {3, 13, 31}. Each set is arranged as a diagonal line traversing the spiral space. Now consider a grid that is 200×200 numbers (Figure 3 [3]):

200x200 Prime Spiral
Figure 3 obviously shows order in the position of prime numbers (each pixel is a prime number). The image shows the space which the first 400,000 natural numbers occupy. It is clear that many of these numbers are primes and they exhibit the same behavior as those in the set of the first 50 prime numbers.
So, why does this matter? No one really knows. Intuitively, we would expect far fewer prime numbers than there are. And we certainly wouldn’t expect any pattern to their occurrence. The fact that an arbitrary visualization of a expectedly random process yields order is very profound. A simple conclusion is that the very fabric of nature is ordered, that looking even at the most primal systems, i.e., the set of natural numbers, we cannot escape order. Further, we would expect that our universe, the structure of mater, everything we know is a derivation of order from disorder, but prime spirals take us back one more level of abstraction indicating that, in nature, there is no fundamental disorder. The act of counting on your fingers, the number of fish in the seas, the number of cells in your body and the number of particles in the universe all exist in a framework which is fundamentally non-random. There is invisible order within the concept of numbers itself.
[1] http://en.wikipedia.org/wiki/File:Ulam_spiral_howto_all_numbers.svg
[2] http://en.wikipedia.org/wiki/File:Ulam_spiral_howto_primes_only.svg
[3] http://en.wikipedia.org/wiki/File:Ulam_1.png