J-Soft Blog

Monday, October 06, 2003

Dunce

On saturday I was visiting a friend in residence and noticed a 4-dimensional cube(or hypercube) on his monitor(it was made with pipe cleaners). This reminded me of my attempts to draw such an object on the bus in Europe this past summer. Another trumpet player in the band had drawn a 4-d cube as a cube within a cube. This is correct, and I think is the most common way it is drawn. At the time it didn't seem right to me. I tried to find some systematic way to go about drawing a 4-d cube. What I did was start with a basis for the space in which the cube exists, and then try to connect all the basis vectors orthogonally. We both ended up concluding that we were drawing the same thing from different perspectives. His was a head on view, and mine was a side angle view, but mine never came out right. Seeing the hypercube on the monitor prompted me to try again. I looked at my original drawings. They were definitely wrong, but I was certain my reasoning was sound. I began again by comming up with a rigid definition of a general cube. Here's my definition:

A cube is a closed object which has 2^n vertices, each of which forms a basis for the n-dimensional space in which the cube exists.

I then noted that that this definition lends itself to an algorithm for drawing an n-dimensional cube.

1. draw a basis for an n-dimensional space
2. at each vertex or end point draw any basis vectors that are not present(keep them short)
3. extend each vector until it intersects with another vector, or a vertex
4. loop steps 2 and 3 until you have 2^n vertices



This algorithm is easily verified for 3-dimensions. I used it to draw a 4-dimensional cube. What I ended up with looks like two overlapping cubes, joined at the corners. It's exactly the same thing as the cube within a cube, but rotated slightly. The cube within a cube is a head on view. Thinking about this I then realized that if you rotate a 3-d cube to view it head on you end up with what looks like a square within a square. I had just wasted a whole bunch of time and effort to come up with the most convoluted way to draw a 4-d cube, but at least it works.
posted by Jesse at 9:55 AM #