God Plays Dice

If you hover the cursor over today's xkcd, you'll see the following:

Wikipedia trivia: if you take any article, click on the first link in the article text not in parentheses or italics, and then repeat, you will eventually end up at "Philosophy".

I first heard this a few days ago, but with "Philosophy" replaced by "Mathematics". Here's an example:

I clicked on "Random article" which took me to Billy Mercer (footballer born 1896). Following the instructions goes to England (Mercer was English), Country, Geography, Earth, Orbit, Physics, Natural science, Science, Knowledge, Fact, Verification, Formal verification, Mathematical proof, Mathematics.

(A few days ago "fact" went to "information"; the article starts "The word fact can refer to verified information" and someone made "verified" into a link recently. In that case the sequence is fact, information, sequence, mathematics.)

If you keep going you get "quantity", "property (philosophy)", "modern philosophy", "philosophy", "reason", "rationality", "mental exercise", "Alzheimer's disease", "dementia", "cognition", "thought", "consciousness", "mind", "panpsychism", and back to "philosophy".

("rationality" used to go to "philosophy", until someone edited it, leaving the note "Raised the period of the Philosophy article... it was ridiculously low." Of course once someone points out some property of Wikipedia, people will tamper with it.

This doesn't seem to happen if you click on random links, or even second links. The basic reason seems to be a quirk of Wikipedia style -- the article for X often starts out "X is a Y" or "In the field of Y, X is..." or something like that, so there's a tendency for the first link in an article to point to something "more general". Does this mean that "mathematics" necessarily has to be the attractor? Of course not. But it does mean that the attractor, if it exists, will probably be some very broad article.

Edited to add, Thursday, 10:26 am: Try the same thing at the French wikipedia; it doesn't work. This seems to depend on certain conventions that English-language Wikipedians have adopted. However, it seems to work at the Spanish wikipedia, with Filosofía as the target.

The Calkin-Wilf tree now has a Wikipedia page. This is an infinite binary tree with rational numbers at the nodes, such that it contains each rational number exactly once. In the sequence of rational numbers that one gets from breadth-first traversal of the tree,

1/1, 1/2,2/1, 1/3, 3/2, 2/3, 3/1, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, 4/1, ...

the denominator of each number is the numerator of the next; furthermore the sequence of denominators (or of numerators) actually counts something. Plus, there are some interesting pictures that come from plotting these sequences, and some interesting probabilistic properties (see arXiv:0801.0054 for some of the probabilistic stuff, although I actually just found it and haven't read it thoroughly) I've given a talk about this; one day I'll write down some version of it. This is one of my favorite mathematical objects.

It looks like we've got David Eppstein to thank for this. It was introduced in this article by Calkin and Wilf.

One of my favorite bad math jokes ever is now in Wikipedia, and no, I didn't add it.

Namely, exercise 6.24 of Richard Stanley's Enumerative Combinatorics, Volume 2 asks the reader to

"Explain the significance of the following sequence: un, dos, tres, quatre, cinc, sis, set, vuit, nou, deu..."

The answer is that these are the "Catalan numbers", i. e. the numbers in the Catalan language. If this seems random, note that exercise 6.21 is the famous exercise in 66 parts (169 in the extended online version, labelled (a) through (m⁷)), which asks the reader to prove that 66 (or 169) different sets are counted by the Catalan numbers.

I'm telling you about this joke because the Wikipedia article on Catalan numbers begins with a link to the list of numbers in various languages.

An alternative version of this joke (American Mathematical Monthly, vol. 103 (1996), pages 538 and 577) asks you to identify the sequence "una, dues, cinc, catorze, quaranta-dues, cent trenta-dues, quatre-cent vint-i-nou,...", which are the Catalan numbers 1, 2, 5, 14, 42, 132, 429... in the Catalan language. (I'm reporting the spellings as I found them in my sources; the first series is in the masculine and the second is in the feminine, as Juan Miguel pointed out in the comments.)

I heard that the featured articles at Wikipedia were the articles on McCain and Obama, and for the first time ever they had two featured articles at once.

So I went over to check.

But Wikipedia runs on GMT... and so it's Wednesday there. The featured article is Group (mathematics).

The "We" in the subject line is "mathematicians", of course.

Mutilated chessboard problem.

The problem is the folkloric one: given a chessboard with two diagonally opposite corners removed, can you tile it with dominoes? A domino is a rectangle which is the size of two adjacent squares. (If you haven't seen it, think about it; the solution is in the article.)

I've known this problem for as long as I can remember, but I didn't realize it had a name. I didn't realize it needed a name. But apparently it's a standard example of a theorem which is hard for automated theorem-proving programs, so that community needed something to call it, because they can't refer to "that one with the dominoes on the chessboard where you get rid of two squares" in the title of their papers.

Harder question, again if you haven't seen it before: when can you remove two squares and have there be a tiling? (I know the answer. If you want to know it, leave a comment.)

Even harder question: when can you remove four squares and have there be a tiling? This might not be that difficult, but I don't know the answer. (It'll give me something to think about on the walk into school this morning, though.)

The Viquipèdia article "Mathemàtiques" has a bunch of amusing pictures that are meant to be icons of different types of mathematics: a Rubik's cube for abstract algebra, a Koch snowflake for fractal geometry, the Lorenz attractor for chaos theory, dice for probability, an elliptic curve for number theory, and so on.

Some areas don't translate into pictures as well: for category theory they have a commutative diagram, for combinatorics the six permutations of [3], etc.

Also, here are Representacions matemàtiques de diversos camps. (The English version of the article does not have this picture currently; they have a picture of Euclid.)

Much of the article seems to be a straight translation of the English version, but I find myself focusing more on the pictures when reading the Catalan version, because I don't actually read Catalan. But I read French and, to a lesser extent, Spanish, so I can figure things out.

(As to why I'm looking at the Catalan wikipedia -- well, I ended up there because Kowalski said he googled 1.70521 when it appeared in some of his work, and I wanted to see the results.))

Such a search finds Wikipedia articles, usually tables of mathematical constants, in Serbian, English, Esperanto, Catalan, Japanese, Thai, Czech, Turkish, Japanese, Serbo-Croatian, and Bosnian. This is both an illustration of the universality of mathematics and the extent to which Wikipedia is an international enterprise. (My apologies if I misidentified any of these languages!)

But the first hit upon Googling 1.70521 (upon this writing) was Kowalski's post, though it's twenty minutes old. That shows you how fast Google is at indexing. (By the time you read this, who knows? This post might be the first hit.)

Wikipedia has an article entitled 0.999...

The article is quite good, and includes at least sketches of the various proofs that 0.999... = 1 and some interesting historical tidbits; I won't recapitulate those here.

The talk page and the arguments page are interesting, in a different way; one gets to watch people who don't really understand why this should be true, but want to understand it. (Of course there is the occasional crackpot. It might give you a headache, though. It's like sausage being made.