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Folding Cubes and Hypercubes

[Cube Folding]

This movie shows an unfolded cube (red) in space, together with its shadow (pink) on the plane below. The white dot at the top of the image represents the light source. As the red cube folds up, we can follow the results in the shadow below. As the sides of the cube begin to fold up, the shadows of these squares become distorted (one edge is closer to the light than the opposite edge, so one edge has a larger shadow than the other). As the edges of the squares come together in space, so their images come together below. Then the top folds over to complete the cube. In doing so, we see the image get larger (as it moves close to the light), and turn inside out as we go from seeing one side to seeing the other side of the top square. As the top closes up, its shadow forms the well-known "square within a square" view of the cube in perspective.

At this point, we rotate the whole arrangement so that we see only the shadow of the cube and must imagine the three-dimensional cube unfolding that causes these shadows. This is good practice for visualizing the hypercube folding up, as seen in the movie below.

[Hypercube Folding]

This movie shows the analogous sequence of three-dimensional shadows of a hypercube folding up in four-space. Just as we can visualize the cube folding in space using just its shadow (as is done at the end of the previous movie), we must use these thre-dimensional shadows to try to imagine the hypercube folding up in four dimension.

We begin with eight cubes forming a cross-like shape. Some faces are partially removed to make the interior structure easier to see. The central (yellow) cube will be the bottom of the hypercube, and the purple one will be the top. The remaining six cubes form the faces of the hypercube that join the bottom to the top. As these begin to fold up in the fourth dimension, we see their shadows become distorted in three dimensions (as one face of the cubes moves closer to the light source, its shadow get larger). Eventually, the faces of the cubes come together and are joined, just as the edges of the squares that form a cube are glued when they are folded together. This leaves just the top remaining to fold into place.

When the top folds over, it gets larger (closer to the light source), and eventually seems to turn inside out (we go from seeing one side of it to seeing the other). As it closes in to join the six other faces, we are left with the well-known "cube within a cube" view of the hypercube in perspective. The small yellow cube is farthest from the light source, while the large purple one is closest. The remaining six faces appear as truncated pyramids joining these two; these are views of cubes in (four-dimensional) perspective.

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Last modified: 14 Sep 2000 11:59:37
Comments to: Thomas F. Banchoff
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