The Ulam Spiral of Primes

http://scienceblogs.com/goodmath/2010/06/the_surprises_never_eend_the_u.php?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+scienceblogs%2FCyKN+%28Good+Math%2C+Bad+Math%29

by fogus • 16 years ago

74 36 16 years ago

scienceblogs.com

CapitalistCartr4• 16 years ago

We invented the idea of Prime Numbers. They're an arbitrary creation. As are patterns in them. Random is -NOT- the same as even distribution. Patterns are inevitable. Las Vegas makes a fortune on people who don't understand that. If you flip a coin a hundred times, you'll see patterns in the result. People are good at seeing patterns.

pohl3• 16 years ago

Are you sure we didn't discover them? Numbers and mathematics wouldn't cease to exist were we to disappear.

Your use of the word "random" seems to be a non sequitur. Could you explain that connection?

CapitalistCartr4• 16 years ago

Numbers, yes, but Prime Numbers are dependent on the idea that being divisible by only one and itself are important. There are infinite such ways to group numbers. We do Primes partly because they're useful, and also because they're cool.

I used the word random because the distribution of Primes is random as far as we currently know. http://en.wikipedia.org/wiki/Prime_Numbers#Distribution

vecter• 16 years ago

This is wrong. Humans discovered prime numbers. They are not an invention. They fall naturally out of the integers, and would exist regardless of whether humans did or not. Prime numbers are are some of the most fundamental objects in all of mathematics, and the fact that you can find them on diagonals in these graphs probably indicates a pattern that we can't explain yet.

CapitalistCartr4• 16 years ago

By that reasoning, all ways of grouping, analyzing, and considering numbers are discovered. And those ways are infinite. That primes are important is of our choosing.

zandorg• 16 years ago

It's also such a universal constant that some sci-fi stories or movies (eg, Contact (1997)) use them as a tool to indicate an intelligent civilisation. In other words, transmit 2 blips, 3 blips, 5 blips, 7 blips, and it soon becomes clear it's not random - it's a list of primes.

ascuttlefish• 16 years ago

The question of whether math is invented or discovered is far from settled. I know next to nothing about math, but a quick google search for "math created or invented" brings up so much controversy I find it unlikely that it's been decided. See Mathematical Platonism and its Opposites by Barry Mazur (p. 19 in: http://www.ems-ph.org/journals/newsletter/pdf/2008-06-68.pdf). He says it's The Question that won't go away.

mturmon2• 16 years ago

For more:

http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmet...

I agree that some mathematical objects are "more fundamental than others"; primes are probably among the most basic.

If you're interested in the ontological status of mathematical entities, and the process and meaning of mathematics, check out "The Mathematical Experience" by Davis and Hersh,

http://books.google.com/books?id=lMdz84dWNnAC&printsec=f...

There is a section on primes starting on page 209.

thefool2• 16 years ago

Humans may have invented them, but they are a pretty natural concept.

Once you have a system of numbering, one of the first things one does is to see which numbers can be arrived at by simply adding other numbers to themselves a bunch of times.

I have a pretty limited knowledge of number theory, so I may be completely off mark, but I don't know that people have yet arrived at a nice fundamental theory that can encapsulate the way that primes arrange themselves along the number line.

bitdiddle• 16 years ago

"God made the natural numbers; all else is the work of man." -- Kronecker[1]

[1] http://en.wikipedia.org/wiki/Preintuitionism

rgoddard• 16 years ago

You are going to see a pattern along the diagonal because every other number is even and never prime (excluding 2). So the only way for two primes to be touching is along a diagonal. Given that and our brains tendency to group items which are closer together as being related, anything along the diagonal will stand out more and appear to be a pattern. A more interesting picture would be one where all of the even numbers are omitted.

inferno0069• 16 years ago

All the odds would look like a full checkerboard, not like these diagonals. The fact that some diagonal lines appear to be favored over other nearby diagonal lines of odd numbers is still interesting.

PebblesRox• 16 years ago

Not coloring all the odds, just plotting only odds and still coloring only the primes. The new spiral has a 3 where the old two was and a 5 where the old three was, etc.

Barnabas• 16 years ago

Of course it's interesting. There's no such thing as an uninteresting number. http://en.wikipedia.org/wiki/Interesting_number_paradox

paulgb• 16 years ago

There's also the pattern that every prime (above 3) modulo 6 is congruent to 1 or 5. I suspect that some of the pattern comes from that fact.

nitrogen• 16 years ago

Is there a more generalized version of that pattern, perhaps something like this (I haven't checked):

  let P(n) be the nth prime number: P(1)=2, etc.
  let n be an integer >= 2, and m be an integer > n
  then P(m) ≡ 1 and (P(n) - 1) (mod P(n))

?

bd• 16 years ago

A more interesting picture would be one where all of the even numbers are omitted.

I made canvas renderer to check it:

http://alteredqualia.com/visualization/ulam-spiral.html

It's not checkerboard, but there are horizontal and vertical patterns.

jokermatt999• 16 years ago

I believe this was also shown briefly in the movie Pi. At least, I think that's where I recognize it from. Great movie about finding patterns in numbers and pareidolia.

dunk010• 16 years ago

It really is. One of my favourite movies. Highly recommended.

Dove• 16 years ago

From http://mathworld.wolfram.com/PrimeSpiral.html :

This construction was first made by Polish-American mathematician Stanislaw Ulam (1909-1986) in 1963 while doodling during a boring talk at a scientific meeting.

Awesome. My new excuse for all those doodles in my engineering notebook: valuable, valuable mathematical research.

borism• 16 years ago

if you're like me - more physics than math guy, it's the same Stanislaw Ulam of Teller-Ulam design fame.

mbrubeck• 16 years ago

My friend Peter made a poster of the Ulam Spiral that includes the Python code used to generate it: http://imprompt.us/2010/ulam-spiral-take-2/

crux• 16 years ago

That's my friend Peter, too! He's wicked smart.

shrikant• 16 years ago

[These spirals have] been cited by some religious folks as being a proof of the existence of God. Personally, I think that that's silly; my personal belief is that even a deity can't change the way the numbers work: the nature of the numbers and how they behave in inescapable. Even a deity who could create the universe couldn't make 4 a prime number.

Amen to that.

dagobart• 16 years ago

So, pushing this thought a little further, I'd say such a (or any) deity and us have at least this one thing in common: looking in amazement at those primes.

The Ulam Spiral of Primes

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