Up: Math 53 Selected Course Notes

# The Cube and the Hypercube #3:

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This movie shows a cube being folded out of a flat "cross-shaped" collectino of six squares. The ball at the top is supposed to be a light source, and the shadow of the cube on the plane below is shown. Note how the changes in chape of the shadow correspond to the changes in position of the faces of the cube above. Portions that are closer to the light sorce are larger below.

Once the cube is folded, our viewpoint switches so that we look straight down on the plane and see only its shadow as it unfolds again. This is the information available to a Flatlander who is trying to understand the three-dimensional cube by using only two-dimensional pictures.

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This movie is analogous to the previous one in that it shows the three-dimensional shadow of a hypercube folding together in four-space. We start with the unfolded hypercube made from eight cubes in a flat three-space at the bottom of four-space. The central (yellow) cube stays in place as we rotate the other cubes through the fourth dimension. The portions of them that move closer to the light source appear to get larger, and the cubes appear to distort into pyramid shapes. Note how this corresponds to the two-dimensional shadows of the sides of the cube folding in th eprevious movie.

Once the sides are folded together, we fold the top over. It turns inside out (just as the shadow of the top of the cube did) and closes up by double-covering the rest of the hypercube. Compare this to the end result in the case of the three-dimensional cube in the previous movie.

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