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Tesseract

© Copyright 2001, Jim Loy

A tesseract is a four-dimensional "cube" or a hypercube. Just what is that? On the left, we see a sequence of square, cube, and a question mark where our tesseract should go. We may have a little trouble visualizing the tesseract, as it is a four-dimensional object. Well, we can unfold a cube into two dimensions. Perhaps we can do something similar with our tesseract. Let's unfold it into three dimensions. Its faces are made up of cubes, how many? On the right, I have drawn an unfolded cube and an unfolded tesseract. Luckily, a tesseract is a very regular object; each "fold" will be identical. Because of that we end up with eight cubes.

Here is another way to draw a tesseract in two dimensions. It is a wire frame tesseract. We see two undistorted cubes, one in the middle, and one enclosing the entire wire frame object. And we see six distorted cubes connecting them.

One of the characters of Piers Anthony's new book, Reality Check, asks how many cubes are in a tesseract. He answers his own question saying that there may be any number, maybe infinitely many. Well, the answer is eight. Higher dimensional hypercubes turn out to have twice the number of hypercubes (of the next lower dimension) for faces as there are number of dimensions. A five-dimensional hypercube would have 10 four-dimensional hypercubes for faces. Maybe that is what the character had in mind. In infinitely many dimensions, a general hypercube would have an unlimited number of faces. But the faces of all but the smallest wouldn't be cubes, but hypercubes of many dimensions.

See Time, The Fourth Dimension and Klein Bottle.

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