A Slice of Cliché

"MORE!", I hear you scream, as we crest this foothill on our climb to the peaks of math-metaphor ecstasy. Let us then ponder the infinite, O willing and supple pupil:

The constant pi, denoted π, is defined as the ratio of a circle's circumference C to its diameter d. You probably already knew that.

But pi turns up in a surprising number of places. It rears its head in the cosmological constant, Heisenberg's uncertainty principal, Einstein's field equation of general relativity, Coulomb's law of electrical force, the Magnetic permeability of free space, and Kepler's third law, to name a few.

As a whimsical example, imagine that you are shackled to a cold iron rack, in the cellar of a madman's château, watching a razor-sharp pendulum scythe through the air above your helpless nubile body. The evil Count asks only that you answer one question, and he will set you loose:

"Posit an infinite rectangular lattice of perfect 1-Ohm resistors, just so:


Calculate the resistance R between two nodes in the grid. To one node", cackles the Count, "we will arbitrarily assign the coordinates (0,0). In this coordinate system, the other node lies at (1,2). With each swing of the pendulum, my dear, my revenge draws ever closer."

Well it turns out that there's a whole branch of mathematics devoted to this sort of thing (natch), but the bottom line is this head-scratcher:


Where R is the resistance between the origin node, and the node described by coordinates (m,n). See the pi? No? Well for our current example of (m,n) at (1,2), it all reduces to this:


And so pi has reduced the infinite to an easily solvable, finite-boundary solution space. Well, "easily solvable" is relative here, I guess. I certainly don't understand a word of it.

What I do understand, though, is that here is a number with its hand in the infinite. The digits of pi basically extend on forever, a number with no end. Pi, like beauty, truth, identity and enlightenment, is ever incomplete, ever approximate. It has a head, but no tail. A starving ouroboros.

How sad.

The good news, though, is that it perfectly embodies proof of the human mind's ability to abstract the infinite. Oh sure, there will always be a bunch of literal-minded diehards trying to calculate pi to the umpty-billionth digit, to kill the magic, but the majority of non-insane individuals are capable of reducing it to a symbolic representation of that ratio, and to use the gestalt π as a placeholder for all the strange concepts it represents.

And now there's one more.