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Sorry, this is a red herring! Please see my reddit comment. Please go back and down vote this!
After discovering Ulam Spirals I had an idea to draw square spirals with prime edge length, it turned out to have no pattern but after this discussion on reddit.com I tried again with 60 degree turns. This is the result with the first 1000 primes.
This has been tested on Chrome and Firefox so far, please let me know if you have any problems on other browsers.
To draw: take two steps forward, turn right 60 degrees, take three steps forward, turn right 60 degrees, take five steps forward etc...