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.

Local Connectedness and Some Weird Counterexamples
November 10, 2010, 14:25
Filed under: Math, Topology | Tags: , , , , ,

Last time we talked about connectedness and path-connectedness and showed that though all path-connected spaces are also connected, connected spaces are not in general path-connected. There is a partial remedy to this problem, found in the local versions of the connectedness properties.  Recall that a local property says something about neighborhoods of each point, rather than the space itself or its open sets.  Below the fold, I talk about local connectedness and local path-connectedness, and bring up a couple examples and counterexamples, including one of my favorites, the Conway base-13 function.

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November 6, 2010, 08:17
Filed under: Math, Set Theory | Tags: , , , ,

When we talked about cardinality, we defined “standardized” finite cardinal numbers as the set \mathbb{N}, which we modeled as 0=\emptyset, 1=\{0\},2=\{0,1\}, and so on.  We’ve since noted certain special properties of this model:

  • the set exists by the ZFC axioms
  • because of this, it is “pure” — everything is a set of sets, there are no ur-elements
  • the “successor function” S(n)=n\cup\{n\} is well-defined, injective, and its image is everything but 0
  • because of this, if a statement is true for 0 and its truth for n implies its truth for S(n), it is true for all elements of \mathbb{N} — this is the “inductive property”
  • n\subset m iff n\in m, and these synonymous relations are total orders on \mathbb{N}.

In the discussion of the Axiom of Choice, we defined a “well-order” as a total order in which every subset has a least element, and proved that every set can be well-ordered if we assume the AC.  In fact, the subset/element ordering on \mathbb{N} is already a well-order: given M\subset\mathbb{N}, the set \bigcap M is a least element, and you should prove that this is always an element of M (induction might help).

Cardinalities tell us everything about sets up to bijection.  But when sets also have orders on them, this isn’t enough.  If we care about the orders on X and Y, the only functions we should be caring about are those that are order-preserving: that is, that f(a)\le_Y f(b) when a\le_X b.  Likewise, rather than all bijections, we care about the order isomorphisms: bijections that are order-preserving and have order-preserving inverses.  We’re “pairing off” the sets again, but in the same order.  None of the bijections with \mathbb{N} in the post on countability did this, and it’s pretty clear why: any order isomorphism has to preserve the type of ordering, and \mathbb{N} is well-ordered while \mathbb{Z} and \mathbb{Q} aren’t.

The order isomorphism classes (or order types) of general posets or tosets are many and difficult to talk about.  But the order types of well-ordered sets are easier to study.  Below the fold, let’s take them on.

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October 26, 2010, 21:45
Filed under: Games | Tags: , , ,

I really wanted to say something about topology last night — maybe tonight.  I had a take-home midterm last night that I just barely got done.

The co-op house I live in is doing a reorganization of its library.  First of all, it’s awesome that we even have a library.  Second, this means more useless books for my private stash.  I’m not so much a bibliophile as an ideophile.  I’ve been tempted to pick up books whose subjects I already know thoroughly just to get a different perspective.  I don’t really have much time for personal reading at school, but whenever I see free books I have dreams that “well, I’ll get around to this someday” and “I can give it back when I’m done.”

This time around, picked up a physics textbook, a book on computability, and this crazy little thing called SIMSOC.  Created by the sociologist William Gamson, it looks to be a social experiment/role playing game/business retreat exercise focused around the challenges of maintaining a coherent society while trying to satisfy your individual goals.

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