The Perception of Intersections

So I was sitting one day, completely sober and drug-free, staring at my screensaver. It was a part of the XScreenSaver collection and called 'Hypercube'. I was rather intrigued by it mostly because we had touched upon hypercubes in my Discrete Mathematics course just a few days earlier and bits and pieces of computational topology were still dancing through my numbed brain. It was a heady feeling.

I ran that screensaver dozens of times and pretty soon a rather convoluted description of the images I saw trickled into my brain; what I saw was a four-dimensional object rotated through three-dimensional space and displayed on a two-dimensional screen with one-dimensional objects rendered with pixels of arguably zero dimensions. Quite a mouthful, huh?

It didn't stop there, though. A hypercube is nothing more than an extension of a cube into four dimensions, just as a cube is an extension of a square into three dimensions and so on. I began to wonder about how it would look if an object of four dimensions suddenly decided to intersect with our mundane old three. I was stumped trying to visualize this, until I started from the beginning.

Pointilism

In the beginning, there was nothing — zip, zilch, nada — and in the world of geometry when something has no dimensions then we call it a point. A point is relatively uninteresting until you do something with it, like use it for target practice. Let's do that, shall we.

Let's say we have a point that happens to be the bullseye of a target on an archery range. Now the point is a happy-go-lucky thing just minding its business until an archer bent on destroying the joy of little dots located a hundred yards from him by application of force to the end of an arrow. Carefully the archer takes aim at our point, pulls back and, with a sneer as farewell, sends the arrow screaming toward our hero. Just as the tip of the arrow reaches our hero, we freeze time.

While also being a neat trick, we freeze time so that we have a chance to put our own observer on the point. (I know, it's physically and mathemmatically impossible, but bear with me for a moment!) So our observer, sitting en pointe notes that the observee is in some neutral state, N. This is a good state because this is the state where the point is happy-go-lucky. Now, let's move things along a few frames.

As soon the arrow touches the target the state of point changes to N'. N', however, is not constant, but varies along the length of the arrow. At the beginning the point experiences the metal tip. As the arrow continues its flight the point encounters the different materials that make up the arrow: wood, plastic and whatever else our archer used. Our observer, observing at the point of intersection, would notice the point changing as the cross-section of the intersecting arrow until it has passed through and the point is back to its neutral state, N.

Simple, nes pas? Okay, what happens when we expand our actors by a dimension each?

Linear Interruption

Now we can give our observer a one-track mind and set him up with a line, a one-dimensional object.

The Amoeba Strikes Back

Bad SciFi Movies

Where to Go from Here