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Dimentia in More Dimensions

The program HYPERCUBE had obviously caused the disappearance of my friend Magi. The happy ending to his four-dimensional dementia came with a telephone call. Not surprisingly, he spoke of wondrous things. "You probably think I'm crazy," he said. (The phrase is always a sure tip-off.) "I've just been floating around in the fourth dimension. I saw a cross section of my house sweep by. Then I moved in close and tickled my cat's kidneys..."

I will spare the reader any further details of the conversation. Suffice it that I persuaded Magi to run HYPERCUBE no more and to keep further explorations entirely on the intellectual plane. He has followed my advice, he says, and now he professes to have made many marvelous discoveries through his artificially amplified insight. For example, he has come up with two posers that seem worth passing along.

Think for a moment about the following sequence of objects: a unit line, a unit square, a unit cube and so on. The nth member of the sequence is the n-dimensional analogue of the cube. Now try two mental experiments on the objects: draw the diagonal to the n-dimensional cube and inscribe an n-dimensional sphere within the n-dimensional cube. The diagonal stretches from one corner to the opposite one; what happens to its length as the number n becomes progressively larger? What happens to the volume of the n-dimensional sphere, again as n becomes progressively larger? Magi's answers seem hardly sane; I shall give them in next month's column. [an error occurred while processing this directive]