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Projections of the Hypercube

Now for the visual (and psychological) complications. There are two standard methods for presenting the third dimension of the hypercube. The orthographic method simply ignores the third dimension, and all the vertexes are projected directly onto the flat surface of the display screen no matter how far they are behind it. In one-point perspective the vertexes are projected onto the screen as though they were shadows cast by a point source of light centered on the screen and some distance behind the hypercube. Viewing the shadows on the screen is equivalent to viewing the hypercube from behind, but visually it is indistinguishable from a front view.

To achieve the effect of one-point perspective in HYPERCUBE one assumes that the third coordinate of a vertex is equal to the distance between the vertex and the display screen, in the direction of the imaginary point source of light. By solving for the sides of proportional triangles the program determines a multiplier needed to convert the first two coordinates of a vertex into screen coordinates. For example, if the imaginary light source is 20 units behind the screen, a vertex at (5,-7,11,8) can be projected onto the screen by multiplying each of the first two coordinates by 20 and dividing each result by 20 - 11, or 9.

I had dreaded including in the small space that remains a complete description of the process for creating stereoscopic images portraying the fourth dimension of the hypercube. There is a general technique for making stereoscopic images, and I hope to devote a future column to the subject. For the hypercube program, however, Banchoff and his colleagues have adopted a much simpler method. For each position of the hypercube make a new pair of images by applying rot14 through an angle of three degrees in one direction and three degrees in the other. Dimension I is the direction parallel to the horizontal alignment of the viewer's eyes, and dimension 4 is the target of the exercise. The two small rotations nicely approximate the views of the hypercube from the eyes of the viewer: merely imagine the two lines of sight converging near the center of the hypercube at an angle of six degrees.

Readers who want to capture the thrills of 3D movies can make stereoscopic viewing glasses out of red and blue cellophane. In this case HYPERCUBE is run twice, once for each small rotation. The result of the first rotation is colored blue by the program and the result of the second is colored red. Readers need not be concerned about which is which if the eyeglasses are made to be invertible.

Personally I prefer not to struggle with cellophane, and I have learned to fuse stereoscopic pairs by sheer force of will. The technique requires that the two rotated images be reduced in size and then translated to horizontally adjacent and nonoverlapping positions on the screen. They should be the same color, and so a monochrome screen is sufficient, and they should be no farther apart than the distance between the viewer's eyes. Do not stare at the images; look instead at some point between them and infinitely far beyond. The two hypercubes will appear to drift and jiggle toward each other like a pair of shy lovers until they fuse.

Even if the third and fourth dimensions get no special treatment, HYPERCUBE can generate images much like the ones shown in Banchoff's graphic sequence. With successive rotations through small angles in the third and fourth dimensions, readers may see the two crude toruses balloon, pinch off and regenerate much like their smoother cousins in the illustration of the rotating hypersphere.

Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area.