OK, first off, yes, we’ve done this before. But I’ve yet to discern any concensus, and rather than resurrect the old thread, I’m asking the question again: has the effect been explained? There seem to be three schools of thought at work here:
A) There’s no pattern here at all. A plot of random numbers with the same distribution could easily display the same amount of “structure” because humans are used to seeing structure in images whether said structure is really there or not.
B) The “pattern” is no big suprise. Many of the “lines” visible in the spiral satisfy some of Euler’s prime-generating equations, so that’s nothing new. We know that primes greater than 3 all take the form p = 6n - 1 or p = 6n + 1, which itself forces a certain kind of arrangement, but the fact that the lines in the spiral are broken and chaotic and that those irregularities can’t be explained essentially mean that it’s not a pattern (it’s almost a pattern, but we already know there’s “almost” a pattern to the primes).
C) The reasons for pattern have not yet been found.
Personally, I can’t swallow explanation A. I think you could produce a hundred such comparable plots using random data and see nothing like the structure in the Ulam Spiral. Besides, the pattern becomes absolutely striking the more numbers one plots. To me, explanation A might work for the Bible Code, but not for this.
I realize that the structures in the spiral are largely artefacts of the graphing process, but I don’t think that’s significant. You can see similar effects when the spiral is constructed according to different rules. Which makes me wonder, is there some as-yet-unexplored graphing process, perhaps in three dimensions, that would reveal a truly unexpected and regular pattern?
How could anyone know facts about an as-yet-unexplore graphing process in three, four or more dimensions. Unexplored pretty much means unknown. [sup]IANA geek[/sup]
Unexplored in the context of this application; not unexplored as in Balboa staring at the Pacific. After all, Ulam came up with the thing doodling to pass the time during a boring lecture.
A hundred? Please. Generate 10[sup]10[sup]100[/sup][/sup] and then we’ll have a sample worth talking about. And I’d lay dollars to donuts that a few of them will have much more striking patterns.
The problem here is that humans are extremely poor judges of what’s random and what’s not. In a true random sequence, you will see patterns.
…and we’d be talking about it for quite a while, indeed. Many thousands of years, by my guess. Let’s say we just did, oh, 10[sup]10[/sup], for starters. And lets be wildly optimistic and give ourselves one second to evaluate each one for patterns. Then we’ll come back after the 317 years it would take to do that and compare notes.
The other thing you have to keep in mind is that, no matter how long you’re looking at Ulam’s spiral, you’re only considering an insignificant portion of the sequence of prime numbers. It may well be that any appearance of order in the primes disappears above a certain number that we can’t even reasonably talk about. I’d like as much anyone to be able to predict prime numbers, but until somebody proves that we can, I’m not seeing anything.
I guess one connection I’m not making is that in order for there to be a pattern, the pattern would have to predict all the primes perfectly. I’d love to be able to predict the primes, too, but better minds than mine have tried. That’s a long way off from discerning a pattern. I mean, the twin primes are patterned. The whole p = 6n - 1 | p = 6n + 1 is a pattern. They’re not perfect patterns, but they’re certainly useful.
And why should any pattern have to hold at all? Why can’t it change? We consider fractals to be mathematical patterns, right? But they don’t hold…self-similar but always changing depending on the numbers you’re talking about. I wouldn’t even expect the pattern to hold beyond certain numbers we can talk about, because that’s not what patterns in nature seem to do.
I mean, we already know this one doesn’t hold. Zoomed in close, it looks like a spiral. The 400x400 representation looks like some kind of fuzzy grid to me. The striking image to which I linked in the OP (a better version here ), representing the first 262,000+ primes, displays a completely different, flower-like pattern. Obviously, this image must be scaled, but still…pretty cool. I’ve generated lots of random images and backgrounds, and I’ve never come across anything like that.
As an experiment, and figuring there wasn’t any point drawing a spiral on a table with even numbers in it since none of them (except 2) could be prime, I drew up a table starting at 3 and spiralling outward by twos, i.e. 5, 7, 9, etc. Then I identified the prime numbers, and you know what the pattern revealed?
A guy with waaaaay too much time on his hands on a Friday night.
Also, please to note that that image appears on a website purporting to unseat mathematics in favor of Christianity. I call bullshit.
In fact, the density around a point distance r from the center should be on the order of r/ln(r), and that plot seems to go up faster that that with distance. Again: bullshit.
unexplored means something is undiscovered. Ulam was unknowingly exploring with his doodles and he discovered something new. You have asked for something that hasn’t been discovered. :rolleyes:
What I didn’t suggest in my first post, but will suggest here is that the OP take out a piece of paper, grab a pencil and start doodling, preferably in three dimensions.
Interesting. In your previous post, you say “unexplored” = “unknown.” Here you mention that someone can “unknowingly explore.” Now, you claim it means “undiscovered.” So I guess “explored” means “discovered” then? I’ll be delighted to take credit for having discovered the Ozarks. I think I’ll rename them the Lowzarks, since they’re decidedly unimpressive in terms of height.
Techincally, I wasn’t asking for anything, certainly not facts concerning our as-yet-undiscovered, unknowable, or unexplored plotting method. I was simply wondering about one, if it’s possible that one might exist. I wasn’t asking you to give me one, friend. No, the actual solicitation for cold hard facts came at the beginning of the OP: has this been explained?
What I didn’t suggest in my previous post, but will suggest here, is that you take out a dictionary and look up the word whose meaning you’ve quite nicely explored: nitpick. I guess you can claim to have discovered it, too. :rolleyes:
[sup]IANA geek[/sup]
