Gracious Living

December 11, 2010, 19:32
Filed under: Math, personal | Tags: , , ,

I’ve been really busy with finals and haven’t had time to finish a proper post, but wow is this amazing.  The suggestion (not new, but very eloquently expressed) is to stop using \pi, the ratio of a circle’s circumference to its diameter, and start using \tau=2\pi, the ratio of its circumference to its radius.  (Spelled “tau”, rhymes with “cow”: do you know your Greek alphabet?)  Pretty much every important use of \pi is actually a use of 2\pi: the circumference formula, integration in polar coordinates or around a circle, finding roots of unity, but also Gaussian/normal distribution stuff, Fourier transforms, and zeta function identities.  Where there are exceptions, the insistence on using \pi instead of 2\pi obscures the nature of the equation: writing the area of a circle as \frac{1}{2}\tau r^2 instead of \pi r^2 highlights the fact that it’s the output of an integral, and though e^{i\pi}+1=0 looks beautiful, it obscures its own meaning: complex exponentiation by \pi corresponds to a rotation by 180 degrees or \pi radians.  If you write e^{i\tau}=1 (or, as the author suggests, e^{i\tau}=1+0 to preserve the “beauty”), you immediately see what the formula’s getting at: x\mapsto e^{ix} is a periodic function with period \tau corresponding to a rotation of the complex plane.

And if that weren’t enough, look at this piece of beauty:

A diagram of important angles on the circle, measured in multiples of tau

See how easy things could be?

Although \pi has been around for a long time, changes in terminology do happen and math moves right along. I don’t know what we would do if we still had to do group theory with Galois’ original wordings of “substitutions” and “arrangements.” So I think that among mathematicians, this could catch on pretty easily.  It’s easy to spread the meme, too — when you’re presenting something, just start out with “let \tau=2\pi” and watch the daylight glimmer on people’s faces.

But I’m not so sure that this could catch on in schools, which is where it is most needed.  I’m a math major and I still take a few seconds to remember those godforsaken radian angle measures.  It’s annoying and breaks one’s train of thought.  Why do we expect school-age kids to have to jump through this hoop every time they do a trig calculation, which is pretty much all they’re graded on? (another stupid thing about education, but I digress.)  All it teaches them is that math is about memorization of arbitrary things.  With \tau, there’s nothing to memorize: one-third of the circle is \tau/3.

I just feel that there’s considerable inertia in changing math curricula, especially over something as basic as this.  The only reason we learn math at all is because Eisenhower and Kennedy decided to put a man on the moon.  And because Egypt was troubled by the horrible asp.  Putting \tau into the curriculum would require schools to pay for new versions of these stupid things called “curriculum planners” and “textbooks.”  Quite frankly, that’s not what American schools need to be spending their money on, and it’s by no means even the biggest problem with math education.  The author of the page I linked suggests that teachers introduce \tau to teach kids critical thinking and show them that the best way to do something isn’t always the way you’re first taught.  This sounds awesome — an actually interesting debate in a math classroom?! — but it seems like a teacher that did that would meet interference from higher up, on the grounds that it stifles kids’ ability to communicate effectively (all math education being about communication, to people as diverse as cashiers and accountants).

On the other hand, the textbook-publishing lobby would love an excuse to publish a new edition.  In addition, if \tau becomes accepted in the math community, it’s only a matter of time before education starts changing as well.  Especially if the curriculum started to change so that it had more to do with actual math… wouldn’t that be a treat.

Why do we write like this?
December 6, 2010, 08:20
Filed under: Literature, personal | Tags: , , , ,

Up way too late but…

I notice when I write that I wind up branching out in lots of different directions at once.  I use parentheses and footnotes to calm the struggle but sometimes it just isn’t enough.  Maybe good writing just involves cutting out parenthetical thoughts and arranging everything in a linear order such that ideas flow directly from the ones before them.

Or maybe tosets aren’t the best model for writing? Why shouldn’t our paragraph structure be modeled on a general poset, that is, a tree structure?  This would give the writer considerably more freedom — they could branch out in one numerous directions at once, make different paths for different sequences of thought, and let the paths meet up when the sequences converged.  Footnoting is a very shallow version of this, and perhaps on a scale like Infinite Jest‘s it begins to approximate what I’m talking about.  There’s the possibility of directly arranging the text on the page in a tree-structure, which looks like concrete poetry, but I’m looking at a book about Dadaist and Futurist typography right now and honestly, most of their poetry is still totally ordered, just rearranged, with maybe one or two instances of the piecewise function thingy.

Hypertext is one example of what I’m talking about.  I’m sure there have been tons of hypertext manifestos since the dawn of programming, but seriously — where are our great hypertext novels?  The only good one I can think of is Dictionary of the Khazars by Milorad Pavic.  I also feel like I should count interactive fiction, which is often more game-y and less novel-y, especially in the way that it holds hidden information and doesn’t allow the “reader” to go back in time.  But Emily Short’s work, for example, does stand out as highly literary.  I also seem to recall a webpage I found in middle school that was an array of randomly-generated nonsense paragraphs on brightly colored backgrounds, and when you clicked any of the links, it would send you to another copy of the webpage, but with more frames, a few pop-up boxes, and more randomly-generated text.  Same if you tried to close it, which is something I think most modern browsers prevent.  If anyone could direct me to this page, I’d be eternally grateful.

But is there not a single good book that’s written as a hyperlinked webpage and that excels because of this structure?  Hell, using AJAX or Javascript, you could even have links between individual paragraphs and have the results spool out all smooth-like, like this math website.  Is there such a thing?  Anyone?

The game plan
October 22, 2010, 04:08
Filed under: personal | Tags: , ,

Hello everybuddy!

My name is Paul and I will be your friend.  I’ve written a few things about myself on my about page.  I would like to use this space to talk about things I like — math mostly, but also books and movies and cool ideas.  Though I’m not very good at it, I like to analyze songs and books and stuff and I think this will be nice practice.  As I said, I’m new to blogging and so I’d imagine the look of this place will change a lot until I get settled in.

A few words about the mathematical content.  I was mostly inspired to this by John Armstrong’s excellent blog, which has covered a lot of territory in explaining math to the “Generally Interested Lay Audience.”  I’ll start with pretty basic stuff and work up; since he starts with groups, I think I’ll start with topology, though I’ll probably end up covering a lot of algebra as well.  If you don’t like my exposition, please read his!  He knows a lot more than me and is a good writer as well.

I offer you The Following Promises:

  • I will have a considerably freeform approach to my math discussion.  In particular, if I read a cool paper or learn a cool idea in class, I’ll consider changing tacks in order to teach my readers about it.  If a reader asks me to discuss something, I will do my best to do so.
  • I will also try to keep the math discussion readable.  This means that I’ll start with elementary concepts and work up (and we’re going to cover some cool, higher math eventually).  If you’ve done high school calculus, you should be good — and again, if you ask me to explain something, I will.  Of course, the best way to learn is by doing.  I am not particularly good at writing exercises, but I’ll definitely link to books with good exercises and discussion.
  • Likewise, I’ll link to direct prerequisites in new posts, so you know where to go to learn what you’re missing.
  • I will never call my writing here “musings.”
  • Math is a very beautiful and underappreciated art, and I think of it as an art first and foremost.  Even if you’re not initially interested, I hope to attract you to some part of it.  This isn’t really a promise, but what the hell.
  • Oh, I’ll try to end each math post with a cool example — something that made me really appreciate the theory I’m explaining.  Conversely, my writing process will probably be to write until I reach a cool example.  This is a good and a bad thing.
  • Above all else, I want this to be a blog of ideas.  This means not only the process of math but its philosophy and the reasons we think the way we do.  It means not only analysis of ideas but discussion.  I welcome comments and commenters and really want to hear what you all have to say.

Long first post.  Tomorrow, I’ll get started!