Ulam spiral
en.wikipedia.org
Ulam spiral
1–10 of 28 posts
Re: Ulam spiral
#2Stanislav Ulam is an incredible man. Mathematician (Ulam Spiral, Monte Carlo Method), Manhattan Project participant, and part-time astronomer who devised a method for nuclear-explosive-powered space travel [0].
Re: Ulam spiral
#3Here's an interactive Ulam spiral with adjustable parameters.
http://bl.ocks.org/syntagmatic/5070320
Open it fullscreen, lower the size and increase the "max" variable to render more numbers. If you've got a Retina display, the size goes down to 0.5 to take advantage of that. You'll see the long diagonal lines mentioned in the article.
Re: Ulam spiral
#4Here's a colorization based on compositeness:
http://blog.morphism.com/2010/05/building-numbers.htmlRe: Ulam spiral
#5Another perhaps surprising aspect, to me at least, is the apparent uniformity of the density of prime numbers in the plane. It's my understanding that the density of the prime numbers decreases as you go higher [1], so why does the plane look so uniformly covered?
Re: Ulam spiral
#6I did a visualization of Ulam spiral a while ago with JS canvas when it was posted here on HN before:
Re: Ulam spiral
#7Stanislav Ulam is an incredible man. Mathematician (Ulam Spiral, Monte Carlo Method), Manhattan Project participant, and part-time astronomer who devised a method for nuclear-explosive-powered space travel [0]. [0]: http://en.wikipedia.org/wiki/Nuclear_pulse_propulsion
He also has a classic result in measure
theory that the French probabilist Le
Cam called 'tightness'. For any probability
measure and for any h > 0, there exists
r > 0 such that the probability mass farther
than r from the origin is less than h.
This result holds in spaces of considerable generality, but I'd have to look up the details now. Details are in P. Billingsley, 'Convergence of Probability Measures', 1999.
I used the result in a paper once. It's nice result and gets used occasionally in advanced work in probability.
Re: Ulam spiral
#8Another perhaps surprising aspect, to me at least, is the apparent uniformity of the density of prime numbers in the plane. It's my understanding that the density of the prime numbers decreases as you go higher [1], so why does the plane look so uniformly covered? [1] http://en.wikipedia.org/wiki/Prime_number_theorem
Perhaps it is because successive rotations of the spiral are proportionally longer, which offsets the decreasing density.
Re: Ulam spiral
#9I wrote a Java application to generate and save variations of the Ulam Spiral a while back. The code is here: https://github.com/PatrickYevsukov/Ulam-Spiral-Explorer An example of the output is here: https://twitter.com/PatrickYevsukov/status/35502312719085977...
Re: Ulam spiral
#10Another perhaps surprising aspect, to me at least, is the apparent uniformity of the density of prime numbers in the plane. It's my understanding that the density of the prime numbers decreases as you go higher [1], so why does the plane look so uniformly covered? [1] http://en.wikipedia.org/wiki/Prime_number_theorem
Perhaps it is because successive rotations of the spiral are proportionally longer, which offsets the decreasing density.
Don't think that would make any difference as long as the size of each dot remains the same. And if you look carefully you can see the density is higher in the center than around the edges (at least to my eye!).
Once you get past 40 or 50 the density declines vary slowly. See chart at bottom of section 2 of this link: