2009

Mon, Jan 26

<22:02 EEST> While doing research on certain ordinary differential equations, I came across a wonderful piece of mathematical insight named Mamikon's theorem. It states that the area swept by a tangent to a curve is independent of the details of the curve; what matters is the length of the tangent itself, and how much it rotates during the sweep. For example, if a tangent of constant length R rotates through 90°, it sweeps an area of the corresponding quarter-circle.

There are some fascinating results you can derive using this, including a novel proof of the Pythagorean theorem. What's interesting is that you can do integrals involving curves whose exact expressions are unknown, as long as you know how the tangent behaves. Tom M. Apostol explains more.

The veracity of the theorem itself is, IMHO, equally intuitive. I found myself thinking of a possible counterexample, but after reminding myself that we're limited to tangents only, the idea turned out pretty solid. For example, a tangent to a straight line sweeps a zero area. The only way to sweep an area is to turn the tangent, and while the motion along the curve inevitably means a lot of movement parallel to the tangent, the parallel sweeping component is always zero.

Speaking of parallels, the theorem reminds me of Kepler's second law, which is traditionally expressed in terms of areas swept by the planetary radii. It's as if the past masters behind physics and calculus were on to something, and we decided to dump the intuitive, geometric approach in favour of number crunching at some point. But with Mamikon's ideas and thing like geometric algebra, perhaps we could regain some of that power.

Sat, Jan 17

<15:27 EEST> Last night I started to write a spindown daemon that I've been needing and planning for a while, and I've just released it. Besides scratching a practical itch, it's also been a nice little exercise in multithreaded Python and ebuilds.