Illustrating Definitions with the Wikipedia-verse


#1

This occurred to me while reviewing my notes from last class.

Wikipedia is kind of its own weird universe. It’s got articles on all sorts of random stuff. You can play games with it. It even has a bizarrely “attractive point”, the article “Philosophy” as illustrated in the article Getting to Philosophy, originally detailed in some XKCD comic I can no longer remember the name of. The basic “function” you need to iterate is as follows:

  1. For some x_0 (a random wikipedia article in this case), click the first link in the article that is neither (in parenthesis) or italicized.
  2. Keep doing that.

Most of the time, you’ll get dumped in the article Philosophy, which is kind of weird.

HOWEVER, I didn’t write this just to illustrate an attractive point. I wrote this because there are also periodic orbits and pre-periodic points to be found amongst the many articles of the Wikipediaverse!

Okay, so don’t start from any random article. Start (appropriately enough) from the article Orbit (dynamics). Follow the same procedures I detailed above. Your path should start as follows, unless somebody’s gone and edited the articles along the chain:

Orbit (dynamics) \rightarrow Mathematics \rightarrow Quantity \rightarrow ...

Keep doing that. Keep a list of the articles you go through. (Hint: You can probably fit everything on a post-it note if you don’t write especially large.)

If nobody’s gone through and edited the articles in the chain, you SHOULD end up in a periodic orbit through wikipedia articles that, in this instance, began at mathematics. Orbit (dynamics) is a “pre-periodic point” in this case, while Mathematics is a “periodic point”.

So, here are my questions: Are there any “fixed points” in this “function”? Are there any other good orbits we should be aware of? What would a “repulsive point” look like and could we find such an article without just guessing? Have I understood all of these definitions and am I using them correctly?


#2

Very fun! Here are a few observations.

  • The domain of the function is discrete. That makes things quite a bit different. We can certainly talk about notions such as fixed point, orbit, and periodic orbit. While I can see how one might be temped to call the Philosophy page “attrractive”, I don’t think that the notions of attractive or repulsive really make sense in this context, since there’s no notion of distance between the points in the domain.
  • I would not be surprised if there were no fixed points. It would seem strange if the first term you used to describe X was X.
  • Every point in the space must be periodic or pre-periodic. This is a simple consequence of the fact that the space is finite. Thus, every orbit must eventually repeat.
  • The Philosophy page itself is periodic. Unless I made a mistake, the orbit has period 15 and looks like so:
    Philosopy -> Problem_solving -> Ad_hoc -> List_of_Latin_phrases -> English_language -> West_Germanic_languages -> Germanic_languages -> Indo-European_languages -> Language_family -> Language -> Communication -> Meaning_(semiotics) -> Semiotics -> Semiosis -> Action_(philosophy) -> Philosophy
  • I guess that any page that eventually leads to “Philosophy” must also lead to one of these. Somehow, “Ad_hoc” doesn’t carry the same cachet, though.
  • Iteration on discrete spaces like this is notoriously hard because the underlying space has nowhere near as much structure. The most famous mathematical problem along these lines is the Collatz conjecture, which seems hopelessly intractable.

Finally, here’s a graph illustrating the idea that I found on Wiki Commons: