{"id":862,"date":"2010-06-22T11:58:52","date_gmt":"2010-06-22T11:58:52","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2010\/06\/22\/the-surprises-never-eend-the-ulam-spiral-of-primes\/"},"modified":"2010-06-22T11:58:52","modified_gmt":"2010-06-22T11:58:52","slug":"the-surprises-never-eend-the-ulam-spiral-of-primes","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2010\/06\/22\/the-surprises-never-eend-the-ulam-spiral-of-primes\/","title":{"rendered":"The Surprises Never Eend: The Ulam Spiral of Primes"},"content":{"rendered":"

One of the things that’s endlessly fascinating to me about math and
\nscience is the way that, no matter how much we know, we’re constantly
\ndiscovering more things that we don’t<\/em> know. Even in simple, fundamental
\nareas, there’s always a surprise waiting just around the corner.<\/p>\n

A great example of this is something called the Ulam spiral<\/em>,
\nnamed after Stanislaw Ulam, who first noticed it. Take a sheet of graph paper.
\nPut “1” in some square. Then, spiral out from there, putting one number in
\neach square. Then circle each of the prime numbers. Like the following:<\/p>\n

\"ulam.png\"<\/p>\n

If you do that for a while – and zoom out, so that you can’t see the numbers,
\nbut just dots for each circled number, what you’ll get will look something like
\nthis:<\/p>\n

\"ulam200.png\"<\/p>\n

That’s the Ulam spiral filling a 200×200 grid. Look at how many diagonal
\nline segments you get! And look how many diagonal line segments occur along
\nthe same lines! Why do the prime numbers tend to occur in clusters
\nalong the diagonals of this spiral? I don’t have a clue. Nor, to my knowledge,
\ndoes anyone else! <\/p>\n

And it gets even a bit more surprising: you don’t need to start
\nthe spiral with one. You can start it with one hundred, or seventeen thousand. If
\nyou draw the spiral, you’ll find primes along diagonals.<\/p>\n

Intuitions about it are almost certainly wrong. For example, when I first
\nthought about it, I tried to find a numerical pattern around the diagonals.
\nThere are lots of patterns. For example, one of the simplest ones is
\nthat an awful lot of primes occur along the set of lines
\nf(n) = 4n2<\/sup>+n+c, for a variety of values of b and c. But what does
\nthat tell you? Alas, not much. Why<\/em> do so many primes occur along
\nthose families of lines?<\/p>\n

You can make the effect even more prominent by making the spiral
\na bit more regular. Instead of graph paper, draw an archimedean spiral – that
\nis, the classic circular spiral path. Each revolution around the circle, evenly
\ndistribute the numbers up to the next perfect square. So the first spiral will have 2, 3, 4;
\nthe next will have 5, 6, 7, 8, 9. And so on. What you’ll wind up with is
\ncalled the Sack’s spiral<\/em>, which looks like this:<\/p>\n

\"Sacks<\/p>\n

This has been cited by some religious folks as being a proof of the
\nexistence of God. Personally, I think that that’s silly; my personal
\nbelief is that even a deity can’t change the way the numbers work: the
\nnature of the numbers and how they behave in inescapable. Even a deity who
\ncould create the universe couldn’t make 4 a prime number.<\/p>\n

Even just working with simple integers, and as simple a concept of
\nthe prime numbers, there are still surprises waiting for us.<\/p>\n","protected":false},"excerpt":{"rendered":"

One of the things that’s endlessly fascinating to me about math and science is the way that, no matter how much we know, we’re constantly discovering more things that we don’t know. Even in simple, fundamental areas, there’s always a surprise waiting just around the corner. A great example of this is something called the […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[24,43],"tags":[],"class_list":["post-862","post","type-post","status-publish","format-standard","hentry","category-goodmath","category-numbers"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-dU","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/862","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=862"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/862\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=862"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=862"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=862"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}