Sunday, 20 April 2008

Notions of Convexity

When I was younger I was kept locked in a empty room for weeks on end. The walls were painted beige, and the floor was a soft, dark green, plastic-like substance that indented gently wherever I walked on it, and slowly returned back to it's original shape. At some point I started thing of the flatness of the floor as its normal form. Food was provided for me, and liquids to drink, but always while I was asleep. I saw no-one in all that time, heard nothing other than sounds I made myself, and had only my own thoughts to entertain me.

I worked out after a while how to create self-sustaining dimples in the floor. It wasn't perfectly elastic and would initially overshoot when trying to return to its original state, and I could use that to set up a cycle that would retain dimples for what felt like long periods of time. That allowed me to write, in a primitive fashion, and helped me put my thoughts together.

And slowly, my mind became more convex. I think it became star-shaped from the centre at first, and as my thoughts turned more and more on themselves, the set of all star-shaped points expanded. Because that set is necessarily convex, so my whole mind became convex.

There are odd things in my mind that I found there, that seemed to have always been hidden behind corners before. I wrote down how I'd found them using my floor notation so that I wouldn't forget, and then I came back to them as often as I dared.

The problem with convexity is that it's not the highest level notion. I soon found myself increasing the sophistication of the floor notation, and realised that my mind was becoming subharmonic. And the odd things I found in my mind no longer seemed so odd. When I successfully applied the maximum principle to myself, something inside me seemed to turn through 90 degrees in a direction I'd not seen before, but which had surely always been there, and then there were whole new things to look at.

I used the fork that came with my next meal to prick my fingertips and draw on the floor in my own blood. The design was found in part of my mind that now lived in that extra direction I'd found, and even as I drew it on the floor, parts of it seemed to move by themselves. When I was done there was something organic on the floor. I punched the obvious spot in the design, and the floor rippled, and kept on rippling, the energy for it being supplied from outside. And with the energy came something else.

My subharmonic mind stepped up another level and became plurisubharmonic at that point, and yet more unsuspected directions opened to me, and the something else that I had summoned took up residence in the new directions and helped me see things that hadn't been visible before. This time when the food came I was only feigning sleep. My captor, a woman wearing a pale grey veil, dropped the tray of food as I moved, but my helper had moved faster and was invisible to her, and she became physically completely convex very briefly, and then was no longer my captor.

But the world outside seems so limited and convex, and I long to find a way to enhance its dimensions and show everyone the potential that it has. My helper will then be able to bring its friends and families through to live in this world as well, which can only be good for all us. I am sure of that.

This whole world needs a new, convex, normal form.

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