Talk:Ulam spiral

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[edit] OR

If we connect all the points, representing primes when n runs to infinity, we might get a two-dimensional fractal in a square.

This is speculation with no evidence, so it wouldn't belong in an encyclopedia article summarizing what is currently known. Also, it isn't clear what this would mean anyway. As n grows, the sequence of images (all scaled to the same size) doesn't even converge, so it isn't clear what would be meant by the sequence being a "fractal".

To LC and to all the others

As of the same reason the following does not bellong to encyclopedia, sort of speaking: The reasons for this pattern are still not understood.

What kind of patterns should be there? It is true that above thoughts are just speculations for now, but I guess it is worth mention them. The same thing is for every function that covers the square (for example Hilbert curve, Peano curve and reated ones). These curves are in fact sequences of points, aren't they, until n is small enough. If someone has better definition, he can put it on. I would also appeal why original work is thrown away from the wikipedia so easily. Three pictures from the previous version of related article are original works, dedicated to wikipedia, and they were changed with the present ones, which are in fact the same ones. We should respect copyrights and also the effortrights, shouldn't we. -- XJamRastafire 14:01 Aug 25, 2002 (PDT)

The Wikipedia is an encyclopedia. It summarizes what is known on a given subject. If the Mobius spiral is commonly known, then we should create a page for it. If it's only known by its inventor, then it might be more appropriate on everything2. --LC 19:03 Aug 25, 2002 (PDT)

Wheh. -- XJamRastafire 02:06 Aug 26, 2002 (PDT)
In fact as it is said in the article this Ulam's work is just a doodling and it does not tell us much. So why other similar doodling is forbiden here. Don't lean against what belongs to encyclopedia and what doesn't. I do believe that Ulam would agree with this point of view, if he still would be among us. I've tried to direct a reader to resembling subjects and to show some further efforts in this 'doodling'. And this is a pure intuition. We never know what a future may bring, even if we're joking about and all around. I didn't say no lies, so your arguments, LC, are useless. I'll ignore them from now on 'cause in this way we'll get nowhere. Best regards. -- XJamRastafire 02:29 Aug 26, 2002 (PDT)
Original research is not appropriate for an encyclopedia. If you search for "original research" on these pages [1] you'll see many examples of Wikipedians upholding this principle. As LMS said here, "Third, and this is most important: Wikipedia is not the place for original research!" --LC 05:15 Aug 26, 2002 (PDT)
Nice. I'll keep this in mind. Thank you for making things clearer for me. I don't want to be an inventor, based on Buridan's donkey either. With respect. -- XJamRastafire 05:49 Aug 26, 2002 (PDT)

[edit] Random plot

The "random" plot is deceptive, since the trivial fact that there are no even primes above 2 limits the primes to a checkerboard pattern to begin with. It's nowhere near as much of a leap from a random distribution limited to a checkerboard pattern to diagonal lines as it is from a purely random distribution to diagonal lines.

A more believable comparison would be between the Ulam spiral and a random distribution over the odds only. --Fubar Obfusco

That's a good point. Feel free to replace the picture that's there. Of course, there's the question of how far to take this process. We could plot only those numbers coprime to 2 and 3. Or those coprime to 2, 3, and 5. The picture probably raises more questions than it answers, so I've just deleted it. --LC 19:50 Aug 25, 2002 (PDT)

[edit] Are these alignments really totally unexplained ?

Methinks they have an easy explanation as follows (a heuristic explanation, not a proof, alas). Are there more alignments than what my explanation "explains" ?

Suitably restricted half-lines of the Ulam plane (if I may say) are described by parametric equations of the form

   V(N) = 4 N^2 + b N + c

If one looks at this equation modulo 2,3,... any prime p, one observes that for some values of b and c, the proportion of N's such that V(N) is divisible by p is comparatively small. Then, one may assume that the proportion of N's such that V(N) is prime is correlatedly increased.

The proportion of N's such that V(N) is prime will be strongly increased in cases where we are sure a priori that V(N) will never be divisible by 2 nor 3 nor 5 nor 7 (say). This happens for the most "miraculous" half-line,

  4 N^2 - 41

(or similar, sorry, I don't remember the exact equation).

I'm surprised no one has put forth this argument as it seems so obvious. I think it should be mentioned even though it is not a proof. To me it strongly diminishes the alledged "mystery" behind the Ulam alignments.

Note that the same argument also fully proves the existence of white lines: when V(N) can be decomposed into a product of two linear polynomials in N, it is obvious that for N big enough all values of V(N) will be decomposable.

IMO all this, suitably checked and detailed (perhaps by someone with more knowledge than me on the subject), should go to the main page. If no one does it, perhaps I will do it... What do you think, LC (and others) ?

-- FvdP 15:35 Aug 26, 2002 (PDT)

You're right. 4n2-227 is always coprime to the first 9 primes, and it's easy to extend that to any number of primes. The problem here is that most of the published papers are too old to show up in online paper archives like citeseer. When I wrote this page, I found online sources that claimed this was still an open question. I didn't find any online sources giving the answer, but surely an answer this simple must have been found back in the 60s. We really need to go back and look at those old papers, and find exactly what is known about it. In the meantime, I'm removing the statement that the question is still open. --LC 07:46 Aug 27, 2002 (PDT)

What I've told you LC? From the pure intuition :). We should look some years back and I do believe even some years ahead. And this was my intention and nothing else. I can't understand why here we can't mention other types of Ulamlike spirals. They are not my inventions as you've said it. Another point of view is from the outside world of mathematics. They use Ulam prime spiral also in the studies of a hebrew alphabet. We can mention this too. (But I won't :)). Good explanations FvdP. I'll study your remarks further on. A good represantation is also an image, done with a FFT. If a creator has something to do with a creation of prime mysteries, than we have to think at least four dimensional here. -- XJamRastafire 09:10 Aug 27, 2002 (PDT)


Isnt this spiral directly related to the fact that the distance between two prime twins is always of a certain factor (modulo 6 I think), so it is most likely that such structures will occour. how does the spirale look like if you take only prime twins ? how would it look like if you fill a area from te botton left area diagonaly up and down in triangular form? helohe 16:47 23 Mai 2005 (GMT + 1)

[edit] Mysterious alignments in Ulam's spiral

I fell on your page about Ulam's spiral phenomenon and the question as to how mysterious it is, with a conversation with 'FvdP'. Also fell on your remark that maybe some work about this exists but out of the internet circuit.

About 2002 and 2003 I wrote a few webpages attempting to provide an explanation to this phenomenon and I propose you to glance at them at : http://www.geocities.com/dhvanderstraten/ulamtxt.html

Would appreciate any feedback Didier van der Straten vdstrat@attglobal.net

It's not THAT amazing! All odd numbers fall on the diagonals!--SurrealWarrior 21:39, 20 May 2005 (UTC)

[edit] 3D Ulam Spiral

It would be interesting to see if higher dimensionsal Ulam spirals show correspondingly more interesting patterns. Any existing info on this would be appreciated. I personally don't know how exactly the lines in a symmetric 3D Ulam spiral can be drawn, or I would draw it myself and look for patterns. I know that cone shaped spirals have been used, but I don't know if those are entirely symmetric. --Amit 09:02, 12 March 2006 (UTC)

You beat me to it. I was just thinking the same thing today. It would be interesting to see if there are patterns in three or more dimensions. Of course, there's no single obvious way of generating such a "spiral", but multiple heuristics could be tried. -- noosphere 21:37, 8 June 2006 (UTC)
So we need to determine the rules that the heuristic algorithm will follow to create any 3D spiral. We can then construct many such spirals. As far as I know, it's unfortunate though that there isn't a fitness function that can automatically find the most interesting spirals. Assuming this is true, each spiral will have to be displayed using a visualization software, and the interestingness of the structure created by prime numbers in each spiral will have to be manually estimated. My guess though is that the more symmetric the spiral, the better will be the structure. --Amit 22:51, 8 June 2006 (UTC)


[edit] About the Link with the "amazing" ulams rose picture

theres a picture in the link - http://www.abarim-publications.com/artctulam.html that claims to be an embelishment of a prime number spiral. it says it goes from "1 => 262,144" and if that means it only goes to 262,144, then I believe its a prank picture, it really doesnt make sense for a number of reasons. even "embelished". ive seen some really big prime numebr spirals and the only thing that happens is the diagonal lines get bigger and bolder, almost making sub categories out of the smaller ones. i noticed a lot of christian catering, overlays, and assumptions on the theory of the spiral on that website in general, and the book the picture comes from is one of 'those' books. i suggest we remove the link or if someone could explain what "1 => 262,144" and/or what they mean by embelishment, that'd be great. Everything Inane 20:42, 29 September 2006 (UTC)

[edit] Bad redirect here

Ulam's rose redirects here, but the phrase is never used or discussed. 74.128.253.162 02:54, 2 October 2006 (UTC)

[edit] Merge from Number spiral

Please merge any relevant content from Number spiral per Wikipedia:Articles for deletion/Number spiral. Thanks. —Quarl (talk) 2007-02-11 06:20Z

Done, I think. —David Eppstein 07:31, 11 February 2007 (UTC)

[edit] Featured on Digg!

Wow, just saw this article on the Digg mainpage. That means at least 10,000's pageviews. I added the Digg tag at the top of this page. To the authors of this article - Good Work! Jonathan Stokes 07:48, 9 May 2007 (UTC)


[edit] Life

It would be interesting to see what happens when this is run through a few iterations of Conway's Game of Life.

No, it would not. 84.58.176.95 18:58, 10 May 2007 (UTC)
Still, maybe. The final configuration (or configuration cycle) is perhaps obvious but... is it provable ? The question may prove as difficult as the twin primes conjecture. (For what I know = not much, having just stumbled on the question.) --FvdP 18:38, 11 May 2007 (UTC)

[edit] Not so unexpected (bis, see above)

I dislike the following statements:

"What is startling is the tendency of prime numbers to lie on some diagonals more than others, while a random distribution is expected." and "f(n) = 4n2 + bn + c generates an unexpectedly-large number of primes as n counts up {1, 2, 3, ...}" :

expectations depend on the observer, and given my (old) remarks above, only a rather naive observer would "expect random behaviour". So these statements are POV ;-) Not sure how to reword them though, but i'm giving a try. --FvdP 19:10, 11 May 2007 (UTC)

Thanks for cleaning out those silly claims. Lines in the spiral correspond to quadratics, and there is a plausible conjecture about the asymptotic number of primes in a given quadratic, e.g. mentioned in [2]. Experimental data agrees well with it. PrimeHunter 22:53, 11 May 2007 (UTC)

[edit] Explained

I think the text about it not being fully explained as well as the words of awe and wonder should be stricken. For example I think this page does a pretty good job of explaining and debunking the "mystery": What Rose?

--Ericjs 07:53, 2 December 2007 (UTC)