Ulam's Spiral

Brendan Moriarty

The Ulam Spiral is a visualization of prime numbers that also reveals that the set of numbers generated by certain quadratic equations have a higher frequency of primes when compared to a random set of integers.

Origin

The spiral is attributed to the mathemetician Stanislaw Ulam, who, when doodling a number spiral during a "boring" presentation noticed the apparent pattern. By erasing non-primes he noticed that lines appeared to form.

To further explore this pattern, Ulam visited Los Alamos National Laboratory to use the MANIAC II computer in order to generate the spiral to greater values.

Purpose

The goal of the interactive examples below is to show how the spiral is constructed, what the spiral looks like for greater values, and to show in real time the spiral being created.

Construction

Identifying Primes

In order to create the spiral, prime numbers must first be identified. One way to do this is to use a sieve. A simple and efficient (for small values) sieve is the classic "Sieve of Eratosthenes", named after the Greek mathemetician Eratosthenes to whom the sieve was attributed.

As shown above (image from Wikimedia Commons), the idea behind the sieve is to incrementally eliminate all multiples less than n -- the upper bound of the number set. The remaining numbers have no factors and are then prime.

Drawing the spiral

Understanding how the spiral goes from seemingly random pixels to forming clear lines can be shown through an interactive example

(Click to start)

This shows the spiral being generated in real time. Each pixel represents a prime and dark space a non-prime.

Explore the Spiral yourself interactively

Pixel maps of the spiral

As the size of the spiral gets bigger, the lines in the spiral become more pronounced.

These images show the spiral generated for different orders of magnitude (click for a to scale image). They were generated and resized using python.

Resources

Intial Proposal
Resume