Lately I’ve been looking at the Koch curve.
That is, upon reflection, a sentence better written than spoken.
Nonetheless, I have been looking at it. I mean that literally.
I’m not gazing into my inner consciousness and seeing the spare, spiked, infinite shape as I branch towards endlessness.
At least, I don’t think I am. If I am, I’m doing it by hitting the “play” button here again and again and again and again…
The Koch curve is a special instance of another fractal, the de Rham curve, and it shares with other fractals the characteristic of progressing toward an infinite surface area as it iterates, curving further and further in on itself (so to speak) as it repeats. Look at the curve at normal human magnification. Look at it magnified one million times.
It will look the same.
It is, like many other fractals, beautiful.
It is also mind-boggling and elegant and unsettling. Anything that touches infinity is disconcerting- no, perturbing- at least to me. Things which I do not understand scare me, and I know full well that I will never have a firm grasp, nor even the most fleeting, featherlight touch of a conception, of eternity.
And yet the Koch curve also soothes me.
"...one of the strongest motives that lead men to art and science is escape from everyday life with its painful crudity and hopeless dreariness, from the fetters of one's own ever shifting desires. A finely tempered nature longs to escape from the personal life into the world of objective perception and thought..."
Einstein knew whereof he spoke. The idea that this man, too, looked out at the universe rather than around at his messy, painful, cluttered and irreconcilably complicated life is profoundly reassuring.
Mostly because it tells me that it isn’t just me.
It both shakes and solaces me to look out into infinity, into the spaces between the stars, into a Koch curve, into a Möbius band.
Mathematics, the language in which the universe is written, and among whose vocabulary these shapes number, describes not just ours but the set of all possible universes; not just our reality, but everything that could ever be possible, both within the laws of our universe and outside it.
Mathematics and the words it utters are the ultimate transcendence, the final number, in the firmest and most essential way.
So while I look at this shape, this mathematical oration, I’m not just gazing at an infinitely triangulated fractal. I’m also looking at the reality of the curve Here.
I’m gazing into the heart of what exists Here, what can be described Here.
It is profoundly reassuring, in the midst of much change and turmoil, to gaze into such utterly surpassing and unshakeable order, into something anchored so immutably in the substance of the universe in which I live.
Here and Now the Koch curve looks like this. Here and Now it will always look like this. And as the Now moves into a later Now it will look like this, and then this, and then this…
So, paradoxically, by the progression of an infinite shape I sense that I am Here. Now. Nowhere, nowhen else.
And that is something worth knowing.