8 ms·

Ulam spiral

90 points by conductor 13y ago

prezjordan 13y ago
Stanislav 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 http://en.wikipedia.org/wiki/Nuclear_pulse_propulsion
- graycat 13y ago
  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.
- stiff 13y ago
  It's Stanislaw, since he was Polish, Stanislav would be a Russian name. He is most importantly the co-inventor of the hydrogen bomb: http://en.wikipedia.org/wiki/Thermonuclear_weapon http://en.wikipedia.org/wiki/Thermonuclear_weapon There is a great autobiography by him called "Adventures of Mathematician".
  - downer99 13y ago
    Agreed, linking through Stanislaw Ulam turned out to be evermore interesting, even though the Ulam Spriral itself is a good starting point, given its relevance to the field of programming. http://en.wikipedia.org/wiki/Stanislaw_Ulam http://en.wikipedia.org/wiki/Stanislaw_Ulam
  - lutusp 13y ago
    > There is a great autobiography by him called "Adventures of Mathematician". A terrific book, still available but pricey (ca. $30 for a paperback). I have a dog-eared copy acquired 40 years ago in my library. The closest Ulam comes to revealing atomic secrets in his book is to say the method he and Teller chose to produce a thermonuclear reaction requires the "repetition of certain arrangements." I always thought that was the perfection of obscurity.
- RBerenguel 13y ago
  Wasn't he also involved in an early atmosphere on Earth simulation experiment? Edit: sure not, it's Urey. Four letters, starts with U. Mental index hash fail :)
oscilloscope 13y ago
Here's an interactive Ulam spiral with adjustable parameters. http://bl.ocks.org/syntagmatic/5070320 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.
jsmorph 13y ago
Here's a colorization based on compositeness: http://blog.morphism.com/2010/05/building-numbers.html
rcthompson 13y ago
Another 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 http://en.wikipedia.org/wiki/Prime_number_theorem
- oakwhiz 13y ago
  Perhaps it is because successive rotations of the spiral are proportionally longer, which offsets the decreasing density.
  - spikels 13y ago
    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: http://primes.utm.edu/howmany.shtml http://primes.utm.edu/howmany.shtml
- yetanotherphd 13y ago
  The density of primes is 1/log(n), so it does drop off, but not so fast that it would be obvious for small numbers. I tried creating giant Ulam spirals one time, and I did correct for this, which made it look a lot more uniform. I had initially hoped to find new patterns this way, but nothing turned up. While numerical experiments are fun, there are a huge number of potential avenues that could be explored. Finding the interesting ones is basically what mathematics is.
  - darkmighty 13y ago
    How did you correct for it? Did you just nonlinearly scale the final picture or did you scale the axis on the spiral itself and then sampled the result?
bd 13y ago
I did a visualization of Ulam spiral a while ago with JS canvas when it was posted here on HN before: http://alteredqualia.com/visualization/ulam-spiral.html http://alteredqualia.com/visualization/ulam-spiral.html https://news.ycombinator.com/item?id=1452301 https://news.ycombinator.com/item?id=1452301
patrickyevsukov 13y ago
I 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 https://github.com/PatrickYevsukov/Ulam-Spiral-Explorer An example of the output is here: https://twitter.com/PatrickYevsukov/status/355023127190859776 https://twitter.com/PatrickYevsukov/status/35502312719085977...
cjlm 13y ago
If you like this then you'll probably enjoy the visualizations on this site[0]. [0] http://www.numberspiral.com http://www.numberspiral.com
- alxndr 13y ago
  Wow, those are pretty cool. Page 8 with the factor curves look really interesting. http://www.numberspiral.com/art/19.gif http://www.numberspiral.com/art/19.gif
eknkc 13y ago
Basic explanataion: http://www.youtube.com/watch?v=iFuR97YcSLM http://www.youtube.com/watch?v=iFuR97YcSLM BTW, these guys really love numbers and math, it's fun to watch their enjoyment while explaining things like an Ulam spiral. And I think he's on crack or something.
grinich 13y ago
Here's a visualizer for the Ulam Spiral, Triangular, Pentagonal, Hilbert Curve, and many more. You can also visualize {2-5}-factor primes, superperfect numbers, smooth numbers... http://www.bigblueboo.com/prime/ http://www.bigblueboo.com/prime/
arketyp 13y ago
I wonder how this generalizes to higher dimensions.
- mrcactu5 13y ago
  http://math.stackexchange.com/questions/59369/prime-spirals-on-surfaces-of-revolution http://math.stackexchange.com/questions/59369/prime-spirals-...
mrcactu5 13y ago
Patterns related to Primes are difficult - outside present technology. On MathOverflow, Joseph O'Rourke and Terence Tao offer more pretty pictures and discussions on patterns related to the spiral http://mathoverflow.net/questions/102075/prime-spiral-distribution-into-quadrants http://mathoverflow.net/questions/102075/prime-spiral-distri...
coldcode 13y ago
I tried 9.3 / 1 and it looks like a photograph record.
csense 13y ago
vihart talks about Ulam spiral, Pascal's Triangle, cellular automata: http://www.youtube.com/watch?v=Yhlv5Aeuo_k http://www.youtube.com/watch?v=Yhlv5Aeuo_k

90 points by conductor 13y ago

Stanislav 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 http://en.wikipedia.org/wiki/Nuclear_pulse_propulsion

graycat 13y ago
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.
stiff 13y ago
It's Stanislaw, since he was Polish, Stanislav would be a Russian name. He is most importantly the co-inventor of the hydrogen bomb: http://en.wikipedia.org/wiki/Thermonuclear_weapon http://en.wikipedia.org/wiki/Thermonuclear_weapon There is a great autobiography by him called "Adventures of Mathematician".
- downer99 13y ago
  Agreed, linking through Stanislaw Ulam turned out to be evermore interesting, even though the Ulam Spriral itself is a good starting point, given its relevance to the field of programming. http://en.wikipedia.org/wiki/Stanislaw_Ulam http://en.wikipedia.org/wiki/Stanislaw_Ulam
- lutusp 13y ago
  > There is a great autobiography by him called "Adventures of Mathematician". A terrific book, still available but pricey (ca. $30 for a paperback). I have a dog-eared copy acquired 40 years ago in my library. The closest Ulam comes to revealing atomic secrets in his book is to say the method he and Teller chose to produce a thermonuclear reaction requires the "repetition of certain arrangements." I always thought that was the perfection of obscurity.
RBerenguel 13y ago
Wasn't he also involved in an early atmosphere on Earth simulation experiment? Edit: sure not, it's Urey. Four letters, starts with U. Mental index hash fail :)

oscilloscope 13y ago

Here's an interactive Ulam spiral with adjustable parameters. http://bl.ocks.org/syntagmatic/5070320 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.

jsmorph 13y ago

Here's a colorization based on compositeness: http://blog.morphism.com/2010/05/building-numbers.html

rcthompson 13y ago

Another 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 http://en.wikipedia.org/wiki/Prime_number_theorem

oakwhiz 13y ago
Perhaps it is because successive rotations of the spiral are proportionally longer, which offsets the decreasing density.
- spikels 13y ago
  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: http://primes.utm.edu/howmany.shtml http://primes.utm.edu/howmany.shtml
yetanotherphd 13y ago
The density of primes is 1/log(n), so it does drop off, but not so fast that it would be obvious for small numbers. I tried creating giant Ulam spirals one time, and I did correct for this, which made it look a lot more uniform. I had initially hoped to find new patterns this way, but nothing turned up. While numerical experiments are fun, there are a huge number of potential avenues that could be explored. Finding the interesting ones is basically what mathematics is.
- darkmighty 13y ago
  How did you correct for it? Did you just nonlinearly scale the final picture or did you scale the axis on the spiral itself and then sampled the result?

bd 13y ago

I did a visualization of Ulam spiral a while ago with JS canvas when it was posted here on HN before: http://alteredqualia.com/visualization/ulam-spiral.html http://alteredqualia.com/visualization/ulam-spiral.html https://news.ycombinator.com/item?id=1452301 https://news.ycombinator.com/item?id=1452301

patrickyevsukov 13y ago

I 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 https://github.com/PatrickYevsukov/Ulam-Spiral-Explorer An example of the output is here: https://twitter.com/PatrickYevsukov/status/355023127190859776 https://twitter.com/PatrickYevsukov/status/35502312719085977...

cjlm 13y ago

If you like this then you'll probably enjoy the visualizations on this site[0]. [0] http://www.numberspiral.com http://www.numberspiral.com

alxndr 13y ago
Wow, those are pretty cool. Page 8 with the factor curves look really interesting. http://www.numberspiral.com/art/19.gif http://www.numberspiral.com/art/19.gif

eknkc 13y ago

Basic explanataion: http://www.youtube.com/watch?v=iFuR97YcSLM http://www.youtube.com/watch?v=iFuR97YcSLM BTW, these guys really love numbers and math, it's fun to watch their enjoyment while explaining things like an Ulam spiral. And I think he's on crack or something.

grinich 13y ago

Here's a visualizer for the Ulam Spiral, Triangular, Pentagonal, Hilbert Curve, and many more. You can also visualize {2-5}-factor primes, superperfect numbers, smooth numbers... http://www.bigblueboo.com/prime/ http://www.bigblueboo.com/prime/

arketyp 13y ago

I wonder how this generalizes to higher dimensions.

mrcactu5 13y ago
http://math.stackexchange.com/questions/59369/prime-spirals-on-surfaces-of-revolution http://math.stackexchange.com/questions/59369/prime-spirals-...

mrcactu5 13y ago

Patterns related to Primes are difficult - outside present technology. On MathOverflow, Joseph O'Rourke and Terence Tao offer more pretty pictures and discussions on patterns related to the spiral http://mathoverflow.net/questions/102075/prime-spiral-distribution-into-quadrants http://mathoverflow.net/questions/102075/prime-spiral-distri...

coldcode 13y ago

I tried 9.3 / 1 and it looks like a photograph record.

csense 13y ago

vihart talks about Ulam spiral, Pascal's Triangle, cellular automata: http://www.youtube.com/watch?v=Yhlv5Aeuo_k http://www.youtube.com/watch?v=Yhlv5Aeuo_k