Wednesday, October 15, 2008

Wednesday Math, Vol. 43: The four dimensional cube

In math, dealing with more than three dimensions is relatively simple. A point in two dimensions is simply (x, y), two coordinates, one for left-right and one for up-down. For three dimensions, we add a coordinate (x, y, z), and the z coordinate can be thought of as forward-backward, a direction perpendicular to both our first two coordinates. We can't see any direction that is perpendicular to all three of our physical dimensions, but that doesn't stop mathematicians from blithely adding another coordinate (x, y, z, t). In Einstein's spacetime, t would be time and the two directions would be past and future. But there are other concepts of a fourth dimension as well, and although we have a hard time seeing a fourth perpendicular, the math of it is fairly straightforward.

Here is the method for building a four dimensional cube, known as a hypercube or a tesseract. They are fun to draw and kind of pretty if done carefully.

We start in zero dimensions. Zero dimensions is a single dot, which should have no height or width. In all the constructions of this nature, the zero dimensional thing is a dot.

To make a one dimensional cube, we take two zero dimensional cubes, two dots, and connect them with a line segment.

We follow the same pattern to make the two dimensional cube, which is better known as a square. We take two one dimensional cubes, and connect points to the corresponding points on the other one dimensional cube. When we went from zero to one, we didn't have to worry about "corresponding", since there was only one point on each.

And now we have the three dimensional cube, the thing we just call a cube. (In the 2-D world of The Simpsons, it is named a frinkahedron, named after the scientist Professor Frink, since it is a bizarre and imaginary thing only understood by poindexters.) Two 2-D cubes are connected to each other point by point.

So now we see the pattern, somehow we will have two 3-D cubes in four dimensions and connect corresponding points to make a hypercube.

When teaching this to kids, or to adults who think like kids, it's useful to have graph paper.

The first step is to put points on a 6x6 grid as shown at the left. Every point has four other points that when taken toegther make the corners of a 2x1 rectangle, which reminds me of how a knight moves in chess, two up and one over. These are the points we connect to one another, and this creates two cubes linked together. If you take any point in the picture, and pick any three out of four colors, with a little concentration you'll be able to see the cube that is made out of lines of that color. For me, it's easiest to see the red-blue squares, and I can pick either pink or beige to see the cubes, and the unchosen color is the connector color to the other cube made up of the chosen colors.

Rainy day fun time, and it's not even raining!


Tara Mobley said...

I love hypercubes. I have a Java app on my computer that is basically a Rubik's Hypercube. Those things move very oddly.

I can see all kinds of cubes in that picture.

dguzman said...

It's so liney.

Mathman6293 said...

Now that you have given me dimensional magic, I need some more. Please ... how to teach binomial multiplication, long division, synthetic divsion, addition subtraction, factoring, solving quadratics and simplifying rational and radical expression in 17, 90 minute class periods... Maybe in the nth dimension.

Matty Boy said...

Hey, Mathman, send me an e-mail at my contact info. I'm glad to help.