I was having a conversation with two fellow bloggers (listentomethunder and IrishKatie) about infinity and I cited an example which I could not adequately illustrate in the comments section. So here it is. Ladies and Gentlemen, the Koch Curve:
I first saw the Koch Curve in James Gleick’s book, Chaos: Making A New Science. For the full technical explanation of its significance, check out Wikipedia’s Koch Curve article. The image above is constructed by drawing an equilateral triangle inside a circle and then adding triangles ad infinitum as illustrated below:
I have always been puzzled by the seeming contradiction of a finite physical existence and an infinite universe (or soul or time or anything else humans have not been able to find the beginning and end of). All physical laws that I experience indicate that nothing is infinite except in the most poetic or mathematical sense. But the Koch Curve inside a circle convinced me that finiteness and infinity can co-exist neatly. It is a mathematical example, but for some reason it opened a door in my mind and I felt I could understand on an intuitive level the infinite processes of existence occurring within the finite space of my life. Maybe because it’s geometry. I always got geometry whereas trigonometry and calculus left me in the dust.
The multiplication of the triangles within the circle can continue infinitely, causing the outline around them to be infinitely long. Yet that outline will never cross to the outside of the circle. Infinity inside a finite space. Yup.
You may stumble, as I did, over the fact that the outline around the triangles at any given moment is not infinite. But the more I thought about it, the more I speculated that perhaps infinity is a process rather than a static state. Hee hee! Think about that for a minute. That means time is a necessary component of anything infinite, at least as far as human comprehension is concerned. If outer and inner space are indeed infinite, this implies that time is part of the substance of things. Part of matter. I could think about that for more than a minute.
I realize that most people will feel this proves nothing. Just to be clear, I believe everything, even reality itself, is unprovable. When it comes down to it, there is nothing that proves I am not hooked up to a machine somewhere that is feeding me the various stimuli that I think define my daily existence. I could be dreaming. All the time. Since my perceptions are the only reference point I have, I choose to function according to what they tell me, but that does not mean they are “real”.
Scientists posit theories about how the universe works, theories which for a while are accepted as fact. And then we go into space and discover things that move and shift in ways that defy those “facts”. So scientists posit new theories and we believe in those for awhile, until we discover some subatomic particle that behaves in a completely contradictory way. More questions. More answers. And then again, more questions.
I am not a scientist or a mathematician, but I am interested in what both have to say about my life and the world I live in. I try to understand the concepts even if I can’t understand the numbers. Because sometimes they light up my psyche and I experience an almost shamanic comprehension of the universe and my tiny place in it. The Koch Curve inside a circle granted me one of those moments.
I am a constantly increasing multitude of triangles in a simple circle. I will most likely never transcend my physical boundaries, but my experience of myself is infinite.
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