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Tue, 27 Jan 2009
Amusements in Hyperspace
This evening I tried to imagine life in a 1000-dimensional universe. I didn't get too far, but what I did get seemed pretty interesting. What's it like? Well, it's very dark. Lamps wouldn't work very well, because if the illumination one foot from the source is I, then the illumination two feet from the source is I · 9.3·10^{-302}. Actually it's even worse than that; there's a double whammy. Suppose you had a cubical room ten feet across. If you thought it was hard to light up the dark corners of a big room in Boston in February, imagine how much worse it is in hyperspace where the corners are 158 feet away. There are some upsides, however. Rooms won't have to be ten feet on a side because everything will be smaller. You take up about 70,000 cubic centimeters of space; in hyperspace that is just not a lot of room, because a box barely more than a centimeter on a side takes up 70,000 hypercentimeters. In fact, a box barely more than a centimeter on a side can hold as much as you want; an 11 millimeter box already contains 2.5·10^{41} hypercentimeters. It's hard to put people in prison in hyperspace, because there are so many directions that you can go to get out. Flatland prison cells have four walls; ours have six, if you count the ceiling and the floor. Hyperspace prison cells have 2000 walls, and each one is very expensive to build. So that's hyperspace: Big, dark, and easy to get around. [ Addenda 20120510: An anonymous commenter on Colm Mulcahy's blog observed that "high dimension cubes are qualitatively more like hedgehogs than building blocks". And recently someone asked on stackexchange.math for "What are some examples of a mathematical result being counterintuitive?"; the top-scoring reply concerned the bizarre behavior of high-dimension cubes. ]
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