Friday, November 9, 2012
Sunday, January 30, 2011
It is sometimes said that a straight line is a circle with infinite radius. This statement can be formally understood in several ways, but the notion of curvature is helpful in this department. Curvature of arbitrary curves is defined and calculated using the tools of differential calculus. For lines and circles, though, we can appeal to intuition. Imagine driving a car around a circular track, going at constant speed. If the circle is very large, you barely need to turn the steering wheel. That's because the rate at which the angle your car is pointed doesn't have to be large to keep you on the road. Now if the circle is very small, the curve feels tighter, and you must turn the wheel more to stay on the road. This suggests that the curvature should be inversely related to the radius of the circle. In fact, that is exactly the definition. For an arbitrary curve, we would use calculus methods to approximate the curve with a circle, and the curvature of the point would be the curvature of the circle. Now if you drive on a straight road, you need not turn the steering wheel. Our intuition says the curvature of a line should be zero. The "approximating circle" is obtained by taking circles that get bigger and bigger so that their curvature becomes smaller and smaller. So in that sense, a line is a "circle of infinite radius."
So far, though, I've only talked about lines and circles that live in a flat plane. In fact, lines and circles look very different when we change the space they live in. First let's use the following definition of a line: given two points, it is the curve of shortest length connecting the points. We generalize the notion of a line by calling any curve that satisfies that definition a geodesic. If you were an ant living on the surface, and your Aunt Antonia asked you what the quickest way to the neighboring ant hill, you'd tell her to walk along a geodesic. In the upper half-plane model of the hyperbolic plane, geodesics are actually circles which intersect the boundary of the plane in right angles. Taking each term in the proper context, we can really say that in a hyperbolic world, "the lines are circles."
[Note: I recently revived this blog and changed its title, threw up a few of my favorite columns from the last year, and added a review I wrote of a Rumpus Book Club selection. This post is just to explain the new title.]
Tuesday, October 12, 2010
Sunday, October 10, 2010
Today, I’m eating my words. Last September, I lampooned the “proliferation” of majors, trying to combat the overcredentialization happening in college today.
Now I plead guilty. Last Wednesday, I officially declared a second degree, a Bachelor of Arts in English.
I find ways to circumnavigate my previous stance, rationalizing that back then I was really only lashing out against the bad reasons people use for adding majors. For example, I still find it indefensible to add a major because “it’s only a few more classes.” These evasions, though, still seem hollow, so I’ll just concede the hypocrisy and move on.
While this addition might come as a surprise to the people who know me as The Math Guy, it is not foreign to me.
I come from word people. My father is a word man, as was his father before him. They are English teachers. The love of words is in my blood, and this inheritance evokes a silent stirring which draws me back to the word.
Along with surprise, I often get kudos for being “well-rounded” or using “both sides of the brain.” I find this frustrating. First, it exemplifies the tendency to only treat interests which are legitimized by a degree as genuine.
Second, I dislike the right-brain/left-brain conceit. It assumes that just because the brain is split into hemispheres that the mind is divisible into two meaningfully distinct halves. Based in some science, it has been overextended, overused and oversimplified.
There is one reaction I don’t receive. My math major shields me from the interrogators who demand of most English majors, “But who will hire you?”
It is a common lament among English majors who, after so many rounds of questioning, have resigned themselves to the expectation of post-baccalaureate unemployment.
The imperative of economic value extends beyond individuals. Increasingly, cash-strapped universities and society in general force the humanities to justify their existence.
These trends have followed from the commodification of the college degree.
Because universities are now the gatekeepers to the white-collar professional world, students see the campus as a marketplace and the college experience as an exchange of goods. We pay a (rapidly increasing) price for a diploma, which we then try to trade for a comfy, salaried position.
But as the “Avenue Q” song observes, our technocratic society views a B.A. in English as a “useless degree” because it does not give students a specialized set of skills.
Indiana University should be different. IU’s roots are in the liberal arts, and its core unit, the College of Arts and Sciences, is a liberal arts institution. Moreover, we do not have an engineering school.
However, even here, we have a business school whose careerism is borderline dangerous and attitudes toward the humanities that are inappropriate for a university like ours.
While reality can be harsh, I’ll end on a note of idealism courtesy of Princeton:
“But somehow I can’t shake / the feeling I might make / a difference to the human race.”
[Originally published March 29, 2010 in Indiana Daily Student, "A gap in education"]
Saturday, September 19, 2009
In physics, resonance makes amazing things happen. Technically, it occurs when a force drives a system at its natural frequency, creating large amplitudes of oscillation. Untechnically, it occurs when you coax something into acting just as it wants to. While pendulums and springs are the standard examples, resonance is ubiquitous: MRI, cyclotrons, violins, radios, quantum mechanics. Anywhere you look you’ll see excited oscillators.
Sometimes, one idea can resonate deeply with us and effortlessly explain many things. For me recently, this idea is the simple gift. By simple gift, I mean one without expectation of return, one that simply moves among people and makes them closer together. These gifts can be abstract—knowledge, art, love, life, time—or concrete—a bed, a meal, a shirt, a drink. It’s not about the content of the gift. What matters is the palpable bond which forms between giver and receiver.
On the program for the recent inauguration was a world-class quartet playing “Air and Simple Gifts,” a John Williams arrangement of the Shaker tune by Joseph Brackett. For me, this performance (though recorded) was one of those works of art that stirs. Why? Well, there are many reasons I should have liked it. The tune was familiar to me from my church as “The Lord of the Dance”. As a cellist, I appreciated the technical skill of the musicians. The title captured an idea that had been beavering in my head for a while. I was an Obama supporter and was already excited to be witnessing history. However, while these aspects provided background for the awe of that moment, there is something beyond words which contributed to the power of my response. That is the gift of art. The hidden frequency within me to which this piece was well tuned is a complex and mysterious product of not only those few things I just mentioned but also everything I had ever thought and felt and experienced.
Simple gifts resonate with me. They may not resonate with you. Your self, your echo chamber, may not see these vibrations as coherent. It may be all noise to you. If so, I say go find something that does resonate with you. Find the chord that just feels right. It’s waiting somewhere for you to hear it. Find the idea, the religion, the philosophy, the equation, the proof, the story, the painting, the driving frequency that sets you in motion. I bet if you find it, you will find it is a simple gift.
[The first column I wrote for the Indiana Daily Student. Other columns can be found here.]