# Number Spirals

April 15, 2004 9:40 AM Subscribe

Number Spirals: Coincidences of order. "In mathematics you don't understand things. You just get used to them."

Fascinating stuff. And pretty irrefutable too, that's a prime attraction of mathematics, I think.

posted by fenriq at 10:58 AM on April 15, 2004

posted by fenriq at 10:58 AM on April 15, 2004

Looks to me like an optical effect that is a simple consequence of two facts. The first is that our eyes and minds have a natural tendency to seek out lines in a field of dots.

The second is that any spiral whose offset is not prime will contain a lot of numbers that are not prime by virtue of being divisible by one of the offset's factors. So there are lots of spirals that are mostly empty, and primes concentrate in the spirals with prime offsets. In particular, every other spiral is empty, which spaces out the spirals that contain primes and makes them easier to see.

(Disclaimer: I wrote a very bad thesis on prime number density several decades ago)

posted by fuzz at 11:28 AM on April 15, 2004

The second is that any spiral whose offset is not prime will contain a lot of numbers that are not prime by virtue of being divisible by one of the offset's factors. So there are lots of spirals that are mostly empty, and primes concentrate in the spirals with prime offsets. In particular, every other spiral is empty, which spaces out the spirals that contain primes and makes them easier to see.

(Disclaimer: I wrote a very bad thesis on prime number density several decades ago)

posted by fuzz at 11:28 AM on April 15, 2004

After reading this I quickly made a similar kind of graph where primes are black dots, composite numbers with only two factors (three factors if you count 1) are red and others are white. It came out quite pretty, I'm almost sure that there's some kind of writing on the right half of the disk...

*smack*

fuzz, I'm not really sure what you mean by an optical effect. From the graph you can see that some curves, which have easily definable mathematical expressions, have a high density of primes. I wouldn't think that has anything to do with human perception as such, even if we found those expressions by just looking at the graph.

posted by ikalliom at 12:21 PM on April 15, 2004

*smack*

fuzz, I'm not really sure what you mean by an optical effect. From the graph you can see that some curves, which have easily definable mathematical expressions, have a high density of primes. I wouldn't think that has anything to do with human perception as such, even if we found those expressions by just looking at the graph.

posted by ikalliom at 12:21 PM on April 15, 2004

*I'm almost sure that there's some kind of writing on the right half of the disk...*

If you can read the language, it says "We apologise for the inconvenience."

posted by DevilsAdvocate at 1:20 PM on April 15, 2004

cool. i like this type of stuff - the very nice info design of the site is an example for us all.

thanks jj - share the rest of your bookmarks with us will ya?

posted by specialk420 at 1:23 PM on April 15, 2004

thanks jj - share the rest of your bookmarks with us will ya?

posted by specialk420 at 1:23 PM on April 15, 2004

For those interested in the mysterious of prime distribution, Prime Obsession is a good read. It's probably the overall best pop-math book about the Riemann Hypothesis.

posted by jcruelty at 2:02 PM on April 15, 2004

posted by jcruelty at 2:02 PM on April 15, 2004

I saw this a few days ago, I think from linkfilter but I'm not sure. Yeah, I see the lines with a high density of primes and they are real. I am not surprised though. Remember for instance that some of those curves might have only even numbers in them. How many primes are even numbers? Only one of them. Obviously curves that go through mostly even numbers are going to be very sparse in the prime department.

There are functions that are denser in primes than others. One of the curves through the spiral happens to be one of those. That doesn't mean there's any significance to it though.

posted by substrate at 2:50 PM on April 15, 2004

There are functions that are denser in primes than others. One of the curves through the spiral happens to be one of those. That doesn't mean there's any significance to it though.

posted by substrate at 2:50 PM on April 15, 2004

Didn't someone purport to have solved the Riemann problem? To have created a function that predicted primes? I remember the proof was undergoing testing or something, and it supposedly had massive ramifications for digital encryption...

posted by lazaruslong at 5:53 PM on April 15, 2004

posted by lazaruslong at 5:53 PM on April 15, 2004

lazaruslong: I don't know that I can speak to the specific information you have about it, but since the Riemann Hypothesis is one of the million-dollar questions at the Clay Mathematics Institute, there would be a lot of coverage (at least, in mathematical circles). However, fifteen minutes of Googling has found nothing. The Riemann Hypothesis is, as with all the Clay problems, very tempting for so-called crackpots to prove while conveniently ignoring a critical step. However, work in attempting to prove the Riemann Hypothesis has often lead to interesting results, so I would not be surprised if you had heard about something like that.

posted by jjray at 7:22 PM on April 15, 2004

posted by jjray at 7:22 PM on April 15, 2004

This person is trying to construct a phase portrait. The reason for the space being wrapped up in a spiral would be the thing that they should be trying to illuminate, which they don't. This is however a valid way of approaching a problem where your equation is self referential. Given that the rule governing whether a particular number is prime or not is sensitive to the number that you're measuring they're at least working in reasonable territory. Good luck to them.

posted by snarfodox at 8:07 PM on April 15, 2004

posted by snarfodox at 8:07 PM on April 15, 2004

What I find more intriguing are the lines and mirrored curves.

Looking at ikalliom's graph you can see the obvious vertical white line. But there is also a pair of white lines emanating left at approximately 30 degrees. What are these?

Along the right centre area are some curves that plainly continue through the horizontal dividing whitespace into the adjacent quadrant, where they more or less dissolve into chaos. What are these? Why do the curves not stop at the strong horizontal line? (Same effect is also visible elsewhere on the graph.)

Adjusting the colours and/or rotating the image leads to other discoveries. For instance, it appears the pattern of mostly-solid curves that swoop from the upper right quadrant across left and toward the centre are mirrored by mostly-white curves on the bottom right quadrant. And now that I've seen that, I also see that the same effect is happening on the left. The strongest curves are mirrored by empty curves.

That 30-degree white line radiating left and down from the centre? It appears to form a boundary for a set of white vertical lines that appear in a pie-shaped wedge in the bottom-left quadrant, between the 30-degree line and centre vertical line. These lines appear to be mirrored on the upper-right, but without a corresponding diagonal of white.

When I look at it with my near-blind eye, the colours nicely fuzz out and I can see a "prime density flume" angling out from the centre, at about 30 degrees, to the upper-right and lower-left, in a sort of tear-drop shape; the point of the tear is at the centre, with the bulb of the tear pointing away. It's subtle, but it's definitely there.

Probably all the answers to everything are in that graph, if one drops acid. :-)

posted by five fresh fish at 8:13 PM on April 15, 2004

Looking at ikalliom's graph you can see the obvious vertical white line. But there is also a pair of white lines emanating left at approximately 30 degrees. What are these?

Along the right centre area are some curves that plainly continue through the horizontal dividing whitespace into the adjacent quadrant, where they more or less dissolve into chaos. What are these? Why do the curves not stop at the strong horizontal line? (Same effect is also visible elsewhere on the graph.)

Adjusting the colours and/or rotating the image leads to other discoveries. For instance, it appears the pattern of mostly-solid curves that swoop from the upper right quadrant across left and toward the centre are mirrored by mostly-white curves on the bottom right quadrant. And now that I've seen that, I also see that the same effect is happening on the left. The strongest curves are mirrored by empty curves.

That 30-degree white line radiating left and down from the centre? It appears to form a boundary for a set of white vertical lines that appear in a pie-shaped wedge in the bottom-left quadrant, between the 30-degree line and centre vertical line. These lines appear to be mirrored on the upper-right, but without a corresponding diagonal of white.

When I look at it with my near-blind eye, the colours nicely fuzz out and I can see a "prime density flume" angling out from the centre, at about 30 degrees, to the upper-right and lower-left, in a sort of tear-drop shape; the point of the tear is at the centre, with the bulb of the tear pointing away. It's subtle, but it's definitely there.

Probably all the answers to everything are in that graph, if one drops acid. :-)

posted by five fresh fish at 8:13 PM on April 15, 2004

By the way, those John von Neumann quotes predate chaos theory by (at least) twenty years (he died in 1957). Times have changed.

posted by snarfodox at 11:54 PM on April 15, 2004

posted by snarfodox at 11:54 PM on April 15, 2004

While searching out von Neumann's exact date of death I came across this by the way:

"At the instigation and sponsorship of Oskar Morganstern, von Neumann and Kurt Gödel became US citizens in time for their clearance for wartime work. There is an anecdote which tells of Morganstern driving them to their immigration interview, after having learned about the US Constitution and the history of the country. On the drive there Morganstern asked them if they had any questions which he could answer. Gödel replied that he had no questions but he had found some logical inconsistencies in the Constitution that he wanted to ask the Immigration officers about. Morganstern strongly recommended that he not ask questions, just answer them!"

posted by snarfodox at 11:58 PM on April 15, 2004

"At the instigation and sponsorship of Oskar Morganstern, von Neumann and Kurt Gödel became US citizens in time for their clearance for wartime work. There is an anecdote which tells of Morganstern driving them to their immigration interview, after having learned about the US Constitution and the history of the country. On the drive there Morganstern asked them if they had any questions which he could answer. Gödel replied that he had no questions but he had found some logical inconsistencies in the Constitution that he wanted to ask the Immigration officers about. Morganstern strongly recommended that he not ask questions, just answer them!"

posted by snarfodox at 11:58 PM on April 15, 2004

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That being said. This hurts my head like math.

posted by Lafe at 9:53 AM on April 15, 2004