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Hyperspace structures

Hyperspace structures
  

The hypercube

Produced by Andy Burbanks

 
Mpeg movies:
For the best results, play the movies with your viewer in loop mode.
 A spinning hypercube
Spinning hypercube
(one complete revolution)
 A spinning hypercube
Spinning hypercube
(alternative projection method)
One of the simplest four-dimensional structures that we can imagine is the hypercube. It is the four-dimensional analogue of an ordinary cube.

To get an idea of what we mean by a "hypercube'', try the following exercise:

  • Imagine a single point in space. We think of a point as being a zero-dimensional object.

  • Now, imagine moving this point in one direction, say Eastward. It will sweep-out part of a line in space. We think of this line segment as being a one-dimensional object.

  • If we take the line, and move it in a direction perpendicular to itself, say Northward, then the moving line will sweep-out part of a plane (in fact, a square or rectangle).
    A square region of the plane is a 2-dimensional object.

  • Now, take the square, and move it in a direction perpendicular to itself (i.e. at right-angles to the plane), say Upward. The moving square can sweep-out the region of three-dimensional space that we would call a cube.

  • Now, we just take the analogy further.
    Move the cube in a direction perpendicular to the three-dimensional space it occupies. It's a bit difficult to imagine where this direction lies, but there seems to be no fundamental reason why we can't extend the process again. So let's suppose we can move the cube in this way. We'll call this new direction Tripward.
    As the cube moves Tripward, it sweeps out a region of four-dimensional space which we call a hypercube.

Of course, there is no reason to stop there. There are no mathematical impediments to concieving of higher-dimensional 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 three-dimensional versions.

But even this apparently simple structure can seem to behave in a complicated and counter-intuitive way. The movies here demonstrate what happens if we view a projection the hypercube into three-dimensional space as it is rotated around.

Rotation takes place about a point in 2-d and about a line (or `axis') in 3-d. Both affect only two of the coordinates present.

It seems only reasonable, then, that rotations in 4-d 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 three-dimensional space can actually be achieved by a rotation of the object through the fourth dimension.

The movies show what the 3-d 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 3-cube may be constructed by folding six squares together, so a 4-cube 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 3-d space is achieved for the first movie by simply ignoring the 4-th 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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