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The Cube and the Hypercube Rotating:

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This movie shows a cube rotating in three dimensions together with its shadow on a plane below. We can use the shadow to try to understand the cube above, and it helps to have several different views of it. As the cube rotates, we can gain more information about the relative locations of its parts.

The end of the movie shows what we would see if we only viewed the shadow of the cube. We can reconstruct the original cube in our heads from these two-dimensional images quite easily. The goal is to do the same for four-dimensional objects by looking at their three-dimensional shadows.

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This movie shows a cube rotating in three dimensions. You should think about this picture in two ways: first, as a thre-dimensional object rotating in three dimensions; and second, as a two-dimensional shadow of a rotating three-dimensional object. The connection between the two is important: how the two-dimensional image tells you about the three-dimensional obejct is what you need to try to grasp.

Note how the large red square is closer to you in three-dimensions, and the blue one which seems smaller in the shadow, is entirely within the red one. As the cube rotates in three dimensions, we see the blue square shift to the left and distort; then it puses through the side of the red square, evetually pulling entirely through. As the cube rotates further, the blue square becomes the side face of the cube, and we see it as a trapezoid in two dimensions. You can fallow the modtino of the cube further as the blue square moves to the front, and then to the other side, and finally back to its starting location, noting the changes in shapes and positions in the two-dimensional shadow.

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This movie is analogous to the previous one in that it shows the three-dimensional "shadow" of hypercube rotating in four-space. To begin with, we see a large red cube containing a smaller blue one. The red cube is closer to the light source, so its shadow appears larger. As the hypercube rotates, we see the blue cube move to the left and distort, just as the saw the blue square do in the previous movie. Eventually, the blue cube moves throught he side of the red cube, and becomes one of the side subces of the hypercube. It looks to us like a truncated pyramid, but it is actually a regular cube in four space. We just see it distorted because one face of it is closer to the light source, so its shadow is larger than that of the opposite side.

As the hypercube rotates further, we can see the blue cube move with it and eventually become the larger "front" cube of the hypercube.

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