xkcd, philosophy, and Wikipedia

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 hidden mathematics of bathrooms

From the xkcd blog: urinal protocol vulnerability.

The basic premise here is the following: there's a long row of urinals (n of them), and a line of men who want to use them. The first man picks a urinal at the end. Each man after that picks one of the urinals which is the furthest from any of the occupied urinals. Nobody ever leaves. How many men have to show up before one of them will be forced into using a urinal adjacent to one that's already occupied? Call this number f(n).

You might think that f(n)/n approaches some limit, but it doesn't; it oscillates between 1/3 and 1/2 based on the fractional part of log₂ n. If n = 2^k + 1 then this "greedy" algorithm for filling the urinals works and every other urinal gets filled: f(2^k + 1) = 2^k-1 + 1. If n = 3 x 2^k-1 + 1 then the worst possible thing happens and only every third urinal gets filled, and f(3 x 2^k-1 + 1) = 2^k-1 + 1. (Yes, that's the same number, and the function's constant in between.) f(5) = f(6) = f(7) = 3, f(9) = ... = f(13) = 5, and so on.) Oscillations like this -- periodic in the logarithm of the problem size -- happen a lot in problems involving binary trees counted by the number of nodes. Still, it was a bit surprising to see this, because I'd never thought about the problem in the case of "unphysically" large n.

Exercise for the reader: invent a mathematically equivalent version of this problem that doesn't involve urinals.

xkcd on science

Scientific fields arranged by purity, from xkcd.

What about applied mathematicians?

Wow, X suck[s] at math.

See John Armstrong deconstruct Monday's XKCD, which he also talked about when it came out. (For those who haven't seen it: one student writes ∫ x² = π on the board and a teacher says "Wow, you suck at math"; another student, with longer hair, writes the same thing and a teacher says "Wow, girls suck at math".)

Other people have responded to this as well: Ben Webster of Secret Blogging Seminar, Reasonable Deviations, ZeroDivides, and probably some others that I'm missing. It's not my intention to react here, just to point to other mathoblogospheric commentary on the matter. But you really should read John Armstrong's post, because it's funny. (John, I think you should try deconstructing some actual mathematics. I, for one, would be amused.)

Incidentally, is the correct way to cite a blog informally like this by author, or by blog-title? In other words, if you were writing a blog post and linking to me, would you refer to "God Plays Dice" or "Isabel Lugo"? (Secret Blogging Seminar is a group blog, so I had to put Ben Webster's name in.)