Hyperspace structures 

The hypercubeProduced by Andy Burbanks 
To get an idea of what we mean by a "hypercube'', try the following exercise:
Of course, there is no reason to stop there. There are no mathematical impediments to concieving of higherdimensional spaces than four. When you have seen this method of constructing a hypercube, it seems a relatively simple extension of the zero, one, two, and threedimensional versions. But even this apparently simple structure can seem to behave in a complicated and counterintuitive way. The movies here demonstrate what happens if we view a projection the hypercube into threedimensional space as it is rotated around. Rotation takes place about a point in 2d and about a line (or `axis') in 3d. Both affect only two of the coordinates present. It seems only reasonable, then, that rotations in 4d should be about a plane, and should also affect only two of the four coordinates. Among other things, this means that a reflection in a mirror in threedimensional space can actually be achieved by a rotation of the object through the fourth dimension. The movies show what the 3d projection looks like as we rotate a model hypercube slowly through a complete revolution, revolving about three orthogonal planes at different rates. Just as a 3cube may be constructed by folding six squares together, so a 4cube may be made by folding eight cubes into each other. In the second movie, one such cube is marked by a different colour. Note: The projection into 3d space is achieved for the first movie by simply ignoring the 4th coordinate of the vertices. For the second movie, the 4th coordinate is used to project the vertices towards or away from the origin, so that objects closer to the viewpoint in the 4th coordinate bulge outward toward the viewer. 

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