Welcome to the fourth Dimension! This program is an implementation of an animated Tesseract (sometimes called a Hypercube). This program lets you see what a three-dimensional shadow of this four-dimensional object looks like. The three dimensionality is simulated by showing two versions of a cube, turned a few degrees away from each other, so that by looking through two paper tubes held like binoculars, the cube looks three dimensional.
Why a shadow? Well, think of a normal, three dimensional box with edges made out of wire. If you held such a box above a table, under a light, you'd get a two-dimensional shadow. It would look like a square with another, larger square around it, with lines connecting the corners. Similarly, if you held a four-dimensional box with wire edges under a light, you'd get a three-dimensional shadow. This would look like a cube with another, larger cube around it, with lines connecting the corners. This is the thing we want to look at. The nicest thing about this program is that it allows you to look at the cube in three dimensions, since the cube is drawn in perspective, and allows a stereopticon effect.
Instructions
To use this program: Fire it up. You'll see the cube after a small wait, and several controls below it. The buttons labeled X+, Y+ and Z+ control rotation around the normal X, Y and Z axis lines (the X axis is a line that goes from left to right, parallel to the floor. The Y axis goes straight up and down (unless you tilt your Mac!), and the Z axis goes into the screen. They correspond to our ideas of width, height and depth, respectively.) You can play around with these buttons to get an idea of what a two-dimensional shadow of a three dimensional box looks like - for example, what a two-dimensional person would see if she wrote a program to simulate a cube in two dimensions.
The T+, V+ and W+ controls are the interesting ones. They represent rotation 'around' planes. This requires some explanation. In two dimensions, you can only rotate around a point. A line, for a two dimensional being, corresponds to a wall, and they would have some difficulty imagining rotating around a wall. For us three-dimensionites, 'normal' rotation can be around a line. To us, a plane is a wall, and we have just as much trouble as the two-dimensional person when we try to imagine rotating 'around' a wall. You'll see this effect if you fire up the program, and before touching any of the other controls, touch one of the T+, V+ or W+ buttons a few times.
