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.

Mathemetaphor

There are a number of interesting (to me) opportunities for parallel between the pure abstract world of Mathematics, and the messy, sensual worlds of philosophy, art and thought. Like all parallels, they tend to converge toward infinity. Of course we can't just jump right into this feast of parallax without building up to it with a little digestif. So, by way of a cheese platter (and there may be some olives in there as well):

A Sierpinski Gasket is a type of ternary Cantor set, or self-similar set. It is constructed by taking a triangle, removing a triangle-shaped piece out of the middle, then doing the same for the remaining pieces, and so on and so forth, like so:



The result – if an infinite series can be said to have a result – is a pattern of infinite boundary, and zero area. This totally counter-intuitive concept is poetry in itself. To imagine that by recursive Swiss-cheesing, we can arrive at the Infinite, not by adding to the whole, in the gluttonous, possessive fashion of current North American consumerism, but by taking away, after the fashion of Francis of Assisi, Buddha, the Jain Dharmists:

"Trees renounce fruit and keep us alive. The mountains cast away stones and pebbles, which we use for our works and art. One should renounce worldly possessions devotedly within one's power (shaktistyaga)."

Hey. I'm not saying I'm ready to give up my iPod. This is all merely by way of illustrating that the path to enlightenment is multifold. There are many trail heads (We'll talk about Pi next time), and some of these lie outside the province of our personal expertise.

If you're catching what I'm pitching, throw it back in the comments.