100% Prime
April 22, 2013 2:06 PM Subscribe
"Each prime number is represented by a bright, white square, whereas a non-prime ("composite") is grey. Visitors can select difference spatial arrangements of these numbers, ranging from several variants of the well-known Ulam Spiral, over the Archimedian spiral, to the more sophisticated 3D Hilbert curves."
Prime Explorer, via the generally-quite-nifty Information Aesthetics.
Prime Explorer, via the generally-quite-nifty Information Aesthetics.
OK. I'm getting every square painted in a red, yellow and green gradient. No grey squares or white squares...
posted by mr_roboto at 2:33 PM on April 22, 2013
posted by mr_roboto at 2:33 PM on April 22, 2013
OK, the Ulam spiral shows the curious and well-known result of primes appearing to cluster along certain lines. What is the point of arranging the primes into space-filling curves?
posted by Nomyte at 2:47 PM on April 22, 2013 [1 favorite]
posted by Nomyte at 2:47 PM on April 22, 2013 [1 favorite]
I remember that there was a ton of press last year when Shinichi Mochizuki came forth with a proof of the abc conjecture, which would be an unbelievably profound achievement if it can be verified.
Unfortunately I'm not yet far enough along in my math major to really understand why.
posted by Whitall Tatum at 2:56 PM on April 22, 2013
Unfortunately I'm not yet far enough along in my math major to really understand why.
posted by Whitall Tatum at 2:56 PM on April 22, 2013
Um, they're all yellow? It says the primes are white. Where are the primes.
posted by brenton at 3:29 PM on April 22, 2013 [1 favorite]
posted by brenton at 3:29 PM on April 22, 2013 [1 favorite]
What fractional dimension is occupied by primes? Is there an approximation?
posted by Eideteker at 3:31 PM on April 22, 2013
posted by Eideteker at 3:31 PM on April 22, 2013
Um, they're all yellow? It says the primes are white. Where are the primes.
Turns out there aren't any. The whole thing was a prank that got out of hand. Weird sense of humor, that Euclid guy.
posted by Now there are two. There are two _______. at 3:33 PM on April 22, 2013
Turns out there aren't any. The whole thing was a prank that got out of hand. Weird sense of humor, that Euclid guy.
posted by Now there are two. There are two _______. at 3:33 PM on April 22, 2013
I have often had the dream of finding some arrangement of a natural number line (possibly wrapping it around a complex 3D object) in which all the primes line up in a simple way, allowing me to generate huge series of prime numbers. I pledge to only use this knowledge for evil.
Fortunately for the rest of the world, I am unable to find such an arrangement and lack the knowledge to take advantage it even if one existed.
In summary, this tool is useless. Pretty though.
posted by AndrewStephens at 3:41 PM on April 22, 2013 [2 favorites]
Fortunately for the rest of the world, I am unable to find such an arrangement and lack the knowledge to take advantage it even if one existed.
In summary, this tool is useless. Pretty though.
posted by AndrewStephens at 3:41 PM on April 22, 2013 [2 favorites]
> (possibly wrapping it around a complex 3D object)
Oh, I'm sure that's been done, somewhere.
You know how the universe is foamy, on an astronomical scale? Walls of galaxies surrounding nearly empty bubbles. So, my hunch is, every time some new intelligent species discovers something fundamental about the structure of the universe -- >pop< we see a new bubble. No telling what they see, they're gone.
Obviously this must be less than once a century or so or we'd have observational proof by now.
----
(Shorter: I couldn't figure it out either, and my math major wife won't be home for hours ...)
posted by hank at 4:04 PM on April 22, 2013
Oh, I'm sure that's been done, somewhere.
You know how the universe is foamy, on an astronomical scale? Walls of galaxies surrounding nearly empty bubbles. So, my hunch is, every time some new intelligent species discovers something fundamental about the structure of the universe -- >pop< we see a new bubble. No telling what they see, they're gone.
Obviously this must be less than once a century or so or we'd have observational proof by now.
----
(Shorter: I couldn't figure it out either, and my math major wife won't be home for hours ...)
posted by hank at 4:04 PM on April 22, 2013
Nice video of the Ulam spiral.
And - as Nomyte observed: what's the point of wrapping around the 2D and 3D Hilbert curves? There doesn't seem to be any sort of emergent pattern in these cases (unless I'm missing something).
posted by crazy_yeti at 4:21 PM on April 22, 2013 [1 favorite]
And - as Nomyte observed: what's the point of wrapping around the 2D and 3D Hilbert curves? There doesn't seem to be any sort of emergent pattern in these cases (unless I'm missing something).
posted by crazy_yeti at 4:21 PM on April 22, 2013 [1 favorite]
I have often had the dream of finding some arrangement of a natural number line (possibly wrapping it around a complex 3D object) in which all the primes line up in a simple way, allowing me to generate huge series of prime numbers.
John Conway once described a fairly small sequence of rational numbers (like, twelve or so), and a process by which one can, using those 12 or so fractions, sequentially generate every prime number, without bound. It's a great trick, since it seems to compress the infinity of the primes into those magical fractions. It is a trick, though, and produces a wonderful "Ah ha!" moment when it's explained.
posted by lex mercatoria at 5:20 PM on April 22, 2013 [1 favorite]
John Conway once described a fairly small sequence of rational numbers (like, twelve or so), and a process by which one can, using those 12 or so fractions, sequentially generate every prime number, without bound. It's a great trick, since it seems to compress the infinity of the primes into those magical fractions. It is a trick, though, and produces a wonderful "Ah ha!" moment when it's explained.
posted by lex mercatoria at 5:20 PM on April 22, 2013 [1 favorite]
Is this all in base 10? Do these patterns appear if you do this in base 7 or base 9 number systems?
posted by Brodiggitty at 5:31 PM on April 22, 2013
posted by Brodiggitty at 5:31 PM on April 22, 2013
It's possible that this thing only shows two things:
1) People are good at picking patterns out of noise (I don't know if the distribution of the primes is random or not in a mathematical sense, but whenever I see something like this I instinctively don't believe it's significant until I've seen a control version)
2) That sometimes a line on one simple geometric shape will map onto another line on another object when transformed by some topological process. For example, most of the shapes here start at the middle and work outwards. It seems reasonable that a radial line on one of these will also appear on another, as the amount the perimeters of the shapes increase by is similar.
Is there a mathematician around to shed any light on what's going on or if there's anything cool to look out for?
posted by Ned G at 5:48 PM on April 22, 2013
1) People are good at picking patterns out of noise (I don't know if the distribution of the primes is random or not in a mathematical sense, but whenever I see something like this I instinctively don't believe it's significant until I've seen a control version)
2) That sometimes a line on one simple geometric shape will map onto another line on another object when transformed by some topological process. For example, most of the shapes here start at the middle and work outwards. It seems reasonable that a radial line on one of these will also appear on another, as the amount the perimeters of the shapes increase by is similar.
Is there a mathematician around to shed any light on what's going on or if there's anything cool to look out for?
posted by Ned G at 5:48 PM on April 22, 2013
Pardon me if I'm wrong, but I think this fits along with a lot of the things on Information Aesthetics: cool to look at, but not really providing any additional insight into the data presented.
posted by zsazsa at 6:00 PM on April 22, 2013 [3 favorites]
posted by zsazsa at 6:00 PM on April 22, 2013 [3 favorites]
Is this all in base 10? Do these patterns appear if you do this in base 7 or base 9 number systems?
Buh? No, the identity of the number doesn't change based on its character string representation. The number referred to as 710 is just as prime as 1112. Also, escabeche works on number theory, let's send up a flare.
posted by Nomyte at 6:35 PM on April 22, 2013 [1 favorite]
Buh? No, the identity of the number doesn't change based on its character string representation. The number referred to as 710 is just as prime as 1112. Also, escabeche works on number theory, let's send up a flare.
posted by Nomyte at 6:35 PM on April 22, 2013 [1 favorite]
Oh hi there.
I don't know if there's so much to see here. The implicit question in "I don't know if the distribution of the primes is random or not in a mathematical sense" is a very good one, though! In many ways, primes behave like random numbers, but in other ways they don't (e.g. only one of them is even and all the rest are odd, which is pretty far from random!) Exactly what it means to say that numbers produced by a completely deterministic process are "random" is a philosophically sticky point, and trying to understand the various things one might mean by "random" has generated a huge amount of interesting number theory over the years. The biggest recent development in the area is certainly the proof by Ben Green and Terry Tao that the primes contain infinitely long arithmetic progressions of any length (which would certainly be true for random sequences of numbers with the same density as primes) -- and nowadays, by work of Green and Tao together with Tammy Ziegler, we know that much more patterns appear among the primes just as often as chance would suggest, were the primes truly random. So the primes "act random" in many interesting senses.
None of which has anything to do with this applet, I'm afraid.
posted by escabeche at 7:08 PM on April 22, 2013 [5 favorites]
I don't know if there's so much to see here. The implicit question in "I don't know if the distribution of the primes is random or not in a mathematical sense" is a very good one, though! In many ways, primes behave like random numbers, but in other ways they don't (e.g. only one of them is even and all the rest are odd, which is pretty far from random!) Exactly what it means to say that numbers produced by a completely deterministic process are "random" is a philosophically sticky point, and trying to understand the various things one might mean by "random" has generated a huge amount of interesting number theory over the years. The biggest recent development in the area is certainly the proof by Ben Green and Terry Tao that the primes contain infinitely long arithmetic progressions of any length (which would certainly be true for random sequences of numbers with the same density as primes) -- and nowadays, by work of Green and Tao together with Tammy Ziegler, we know that much more patterns appear among the primes just as often as chance would suggest, were the primes truly random. So the primes "act random" in many interesting senses.
None of which has anything to do with this applet, I'm afraid.
posted by escabeche at 7:08 PM on April 22, 2013 [5 favorites]
Brodiggitty: "Is this all in base 10? Do these patterns appear if you do this in base 7 or base 9 number systems?"
Prime numbers do not rely on the base they're written in to be prime. 7 in base ten is 111 in base two is 13 in base four, and prime every time. The wrapping in these graphics is sequential, or based on squares, n-sided figures, etc - also not dependent upon a particular base.
posted by notsnot at 7:27 PM on April 22, 2013 [1 favorite]
Prime numbers do not rely on the base they're written in to be prime. 7 in base ten is 111 in base two is 13 in base four, and prime every time. The wrapping in these graphics is sequential, or based on squares, n-sided figures, etc - also not dependent upon a particular base.
posted by notsnot at 7:27 PM on April 22, 2013 [1 favorite]
Expected this to be the same guy from the Teller-Ulam configuration. Isn't. I am dissapoint!
posted by sourcequench at 7:35 PM on April 22, 2013
posted by sourcequench at 7:35 PM on April 22, 2013
AndrewStephens: A simple, 6 place, residue table will line up all the primes under either 1 or 5. I posted about this here.
Lots of cool things with prime numbers have been posted to mefi which might give you some other ideas:
Factorization diagrams
Fractran a programming language based on ratios of primes
El Patron de los Numeros Primos another nifty prime visualization.
Also, did you know there's a different set of primes for complex numbers? This breaks it down: Gaussian Primes. And that's only the beginning! Look up the Heegner numbers. Fun stuff!
posted by wobh at 8:26 PM on April 22, 2013
Lots of cool things with prime numbers have been posted to mefi which might give you some other ideas:
Factorization diagrams
Fractran a programming language based on ratios of primes
El Patron de los Numeros Primos another nifty prime visualization.
Also, did you know there's a different set of primes for complex numbers? This breaks it down: Gaussian Primes. And that's only the beginning! Look up the Heegner numbers. Fun stuff!
posted by wobh at 8:26 PM on April 22, 2013
That's my first exposure to the Ulam spiral, and it's frightening in the way that good math is often frightening for me. It reveals a hidden, unexpected order in the world of number. I'm not sure why my brain parses that hidden order as frightening. Perhaps I've been reading too much Lovecraft and not enough Plato.
Don't read e^(i*pi) + 1 = 0 after midnight.
posted by justsomebodythatyouusedtoknow at 8:38 PM on April 22, 2013 [2 favorites]
Don't read e^(i*pi) + 1 = 0 after midnight.
posted by justsomebodythatyouusedtoknow at 8:38 PM on April 22, 2013 [2 favorites]
The thing's hollow - it goes on forever - and - oh my God - it's full of stars!
Seriously though, it's utterly fascinating, aesthetically speaking. I'm using it to zone out during lunch while randomly thinking of numbers and infinity.
posted by the cydonian at 10:22 PM on April 22, 2013
Seriously though, it's utterly fascinating, aesthetically speaking. I'm using it to zone out during lunch while randomly thinking of numbers and infinity.
posted by the cydonian at 10:22 PM on April 22, 2013
Thanks notsnot et al for the answers on base-systems. The last teacher I had who even touched on the topic of base 10 was Mrs.Dill in grade 2. In hindsight it seems like a complex topic to discuss at that age but it stuck with me.
posted by Brodiggitty at 2:17 AM on April 23, 2013
posted by Brodiggitty at 2:17 AM on April 23, 2013
In college a math professor of mine once referred to a particular statement as "having no right to be true." It's a saying for when you find something to be true and profound, and it reveals a wide-sweeping relationship between things that, when considered in the orthodox framework, are completely unrelated.
Prime numbers come from algebra. They shouldn't necessarily make pretty curves when graphed in 2 dimensions. The fact that they do has no right to be true. That's why they are so beautiful.
posted by cotterpin at 3:51 AM on April 23, 2013
Prime numbers come from algebra. They shouldn't necessarily make pretty curves when graphed in 2 dimensions. The fact that they do has no right to be true. That's why they are so beautiful.
posted by cotterpin at 3:51 AM on April 23, 2013
Switching between different Ulam Spirals and watching the numbers realign themselves into different patterns - I feel like I'm looking into the Matrix.
posted by EndsOfInvention at 4:04 AM on April 23, 2013
posted by EndsOfInvention at 4:04 AM on April 23, 2013
Primorial number system (primorial base/radix)
posted by Eideteker at 4:32 AM on April 23, 2013 [1 favorite]
posted by Eideteker at 4:32 AM on April 23, 2013 [1 favorite]
So, my hunch is, every time some new intelligent species discovers something fundamental about the structure of the universe -- >pop< we see a new bubble. No telling what they see, they're gone.
You are *happy campers*, but already you know.
I will not talking about *silly Androsynth*, now is stop asking.
If you are say the question another time it is *frumple* too much and Orz are *dancing* for *dissolving* the *campers*.
posted by FatherDagon at 12:03 PM on April 23, 2013 [2 favorites]
You are *happy campers*, but already you know.
I will not talking about *silly Androsynth*, now is stop asking.
If you are say the question another time it is *frumple* too much and Orz are *dancing* for *dissolving* the *campers*.
posted by FatherDagon at 12:03 PM on April 23, 2013 [2 favorites]
Expected this to be the same guy from the Teller-Ulam configuration. Isn't. I am dissapoint!
But sourcequench, it's absolutely the same Stan Ulam! If you can get your hands on it, his autobiography Adventures of a Mathematician is a very lively read, also the book From Cardinals To Chaos is a great account of his mathematical work, if you have the background to follow it.
posted by crazy_yeti at 1:43 PM on April 23, 2013
But sourcequench, it's absolutely the same Stan Ulam! If you can get your hands on it, his autobiography Adventures of a Mathematician is a very lively read, also the book From Cardinals To Chaos is a great account of his mathematical work, if you have the background to follow it.
posted by crazy_yeti at 1:43 PM on April 23, 2013
Look up the Heegner numbers. Fun stuff!
Go ahead! Jump right in the deep end of the pool! :-)
posted by crazy_yeti at 1:47 PM on April 23, 2013
Go ahead! Jump right in the deep end of the pool! :-)
posted by crazy_yeti at 1:47 PM on April 23, 2013
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I still don't understand it, but it's pretty.
posted by HuronBob at 2:16 PM on April 22, 2013