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Wednesday, October 20, 2010

One Dimensional Loops in 3d

Imagine a point, infinitely small and at that point is a big red ball and you can't miss it because you're in an otherwise empty room and the wall is painted hospital white.

Suddenly, the ball disappears, and then quickly reappears left of your left shoulder. It disappears again and then reappears right of your right shoulder. It does this repeatedly and you follow it with your eyes, as it jumps from one spot to another, without traversing the distance, even though you're beginning to get a cramp in your neck.

And the ball seems to be "moving" faster and faster, and the time it takes between appearing and disappearing gets smaller and smaller.

This happens until the ball appears and disappears so quickly that you cannot readily make out if it's to your left or two your right at any given moment.

Stop. The room is gone. What properties are being described here?

#1. Perception

The first tool of animation was our eyes. The simple insight that if you wave something fast enough in front of your face it seems to blur into one thing. Flip books are an obvious example of this, but the perception is a trick of our eyes, right?

#2. Space

Yes, but consider this. The smaller the amount of time between disappearance and reappearance the closer the ball looks to being in both spaces at the same time.

If the time between disappearance and reappearance is Zero, the "velocity" of the ball is Infinity.

If the time between the disappearance and the reappearance of the ball is Infinity, the the velocity is of course, Zero. Infinity in this instance maps to always and never. The ball is always at both locations at once if the velocity reaches infinity, the ball is never at both locations at once if the time it takes to get from one location to the next is infinity.

This example works the exact same way in continuous space.

Let's say we're in an 1980's Atari video game and we shoot a pixel at the bad stick figure. It slowly makes its way across the screen, barely missing the evil stick man. It goes and goes until it leaves the screen, but instead of leaving the screen, it reappears on the other side of it...

If the same pixel is shot at a speed of infinity, and it will continue moving indefinitely, it will appear to an observer that the pixel exists at every point in space along its trajectory, simultaneously, and in fact, that would literally be the case if the universe permitted such a thing as an infinite velocity.

What's the Point?

We have just constructed a 0 dimensional spatial object that has all the same properties as a line. The Point is a Line.

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