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Multidimensional Mind Benders

To the Editors:

In “An Adventure in the Nth Dimension” (November–December), Brian Hayes notes the surprising fact that the volume of a unit ball in n dimensions peaks at n = 5 and decreases thereafter. He later asks “whether it really makes sense to make volume comparisons across dimensions.” After noting that comparing a 10-square-centimeter disk to a 5-cubic-centimeter ball is “like comparing apples and orange juice,” he goes on to assert that “the obvious standard is the unit cube ... which has a volume of 1 in all dimensions.” But this begs the “apples and orange juice” question: Why does it make sense to compare a 1-square-centimeter square with a 1-cubic-centimeter cube? It seems to me that it does not.

I think a better comparison would be the length of one side of an n-dimensional cube that has the same volume as the n-dimensional ball. This would be equal to the nth root of the ball’s volume, and would have the same units (centimeters) in all dimensions. When one computes this value as a function of n, it is monotonically decreasing, and Hayes’s “surprising fact” goes away.

Gordon Hazen
Northwestern University

To the Editors:

I enjoyed Brian Hayes’s column, in which he examined the problem of placing spheres inside cubes in n dimensions. If you invert that idea, and ask about cubes inscribed inside spheres, you get a different insight into the problem that the volume of the sphere vanishes as n goes to infinity. Since the length of the main diagonal of a cube scales as the square root of the number of dimensions, it follows that, in an infinite-dimensional space, no cube of any size, no matter how small, can fit inside a sphere of any finite radius, no matter how large. The diagonal of the cube is just too long to fit into the sphere. Therefore the volume of the sphere must vanish.

But the problem is deeper than just the vanishing volume of the sphere. In fact, the very concept of volume loses all meaning in an infinite-dimensional space because the volume contained within any finite radius of every possible origin point is always exactly zero. This is the really counterintuitive point here, that volume in n dimensions (length in one dimension, area in two dimensions and so forth), which is one of the fundamental ideas behind all of geometry, has meaning only for finite values of n. There is no problem with lower-dimensional volumes, 2-volumes, 3-volumes and so forth, in an infinite dimensional space.

Doug Robertson
Cooperative Institute for Research in Environmental Sciences
Boulder, CO

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