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Mathematics

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Apr. 26th, 2012 | 11:28 pm

It's amazing how one thing can lead to another sometimes, and how you can come across all sorts of fascinating bits of mathematics just by finding the right starting point.

For example, today's xkcd contains a table of "slightly wrong equations useful for approximations and/or trolling teachers", which was apparently created "using a mix of trial-and-error, Mathematica and Robert Munafo's ries tool".

There was a short thread about this on the SeqFan mailing list (dedicated to the discussion of Sloane's Online Encyclopedia of Integer Sequences); this prompted me to actually take a look at ries. It's a nifty tool to say the very least; but what's more, the "See Also" section had a link to another page that Robert Munafo maintains, one dedicated to large (mostly finite) numbers. (There is a bit of discussion of infinite ordinals in the end, but not much, and it doesn't mention most of the interesting and/or peculiar ones.)

Large numbers are highly interesting, of course, but I haven't kept up with the more recent developments (the last time I really looked into these matters was a good ten years ago[1]), and I was happy to see that there were new things there that I hadn't encountered before: Bowers' extended operators and array notation and Friedman sequences[2], for instance. I recommend reading it all; it's fascinating material.

Beyond this, there was another link to another long document that Robert Munafo maintains, discussing notable properties of specific numbers; I only skimmed through this, but the footnotes led me to a series of three lectures that John Baez held as the Rankin lectures in 2008 in Glasgow, talking about his favorite numbers.

The first two of these, discussing the numbers 5 and 8, are very interesting. The third one, however, is downright amazing: a wild ride that highlights all sorts of amazing connections and that'll have you on the edge of your seat the whole time. I recommend not just looking at the slides but watching the video, BTW — it's an hour, granted, but it'll be one of the best hours you've ever spent, incorporating everything from a bizarre "proof" of Euler's that 1 + 2 + 3 + ... = -1/12, cannonball stacking in a square pyramid, why bosonic string theory works best in 26 dimensions, elliptic curves[3], the Leech lattice, and what is known as Monstrous Moonshine.

I couldn't possibly hope to sum up any of it, of course (and I must admit that as much as these things fascinate me, I'm neither a mathematician nor a physicist), but I encourage anyone who's got an interest in these things to check these out. It's better than vegging out in front of Youtube! :)

And it still never ceases to amaze me how something as simple as a passing mention in a webcomic strip can ultimately lead to discoveries of all sorts of nifty new (to me) things.

  1. I don't recall where, alas; the only site I distinctly recall due to its domain name was Infinite Ink, aka ii.com, but it appears that the article on there that I remember was this one about the continuum hypothesis.
  2. That's Friedman as in Harvey Friedman, a very prolific mathematician who I mostly know as a regular poster on the FOM mailing list. Incidentally, he's very interested in large cardinals.
  3. On a side note, it seems that elliptic curves are another example of those "there's two things you need to know about X" things: they're not elliptic, and they're not curves.

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Comments {2}

Robert Munafo

John Baez

from: mrob27
date: Apr. 29th, 2012 08:27 pm (UTC)
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I agree, the Baez lectures are about the best hour I ever spent. And for what it's worth, most of these maths are above my head too (-: - Robert Munafo

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Schneelocke

Re: John Baez

from: schnee
date: Apr. 29th, 2012 08:57 pm (UTC)
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They're fascinating, really. :) I only wish there was another one where John'd explain the "it all boils down to the fact that 4 times 6 equals 24" bit from where he talked about ℤ² and A2.

Thanks for your comment!

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