Three-Dimensional Shadows of the Hypercube

Reasoning by analogy, we can imagine a sun in four-dimensional space casting a three-dimensional shadow of a hypercube. To make such a shadow in three dimensions, we can proceed as we did when drawing shadows in the plane. We start with four edges coming out of a vertex, no three of which lie in the same plane. We complete the pairs of edges to form six parallelograms, then complete triples of edges to form four parallelepipeds, distorted cubes with all six sides parallelograms. Finally, we put in the last four edges to obtain the four groups of eight parallel edges each. We can make such a model from sticks or wire, or we can instruct a computer to show us what such a model would look like if we filmed it rotating around in three-space. Even though the images on the computer screen are two-dimensional, the computer can produce animated sequences simulating the form of three-dimensional shadows of higher-dirnensional cubes.

Even before the emergence of graphics computers, artists and designers were constructing three-dimensional images of objects from the fourth and higher dimensions. Two remarkable examples come from nonmathematicians who became fascinated by the challenge of visualizing these strange objects. The wire models of projections of higher-dimensional cubes and other objects built by Paul R. Donchian are part of a permanent exhibit at the Franklin Institute in Philadelphia. (One model is shown in Chapter 5.) David Brisson, professor at the Rhode Island School of Design and founder of the Hypergraphics group of artists, produced sculptures of cubes in four, five, and six dimensions. (A watercolor of two views of a hypercube is shown in Chapter 6.)

Movies of the shadows of the cube and hypercube as they rotate.

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