Sunday, December 04, 2011

N-dimensional Spheres in N-dimensional Boxes


In an imaginary world of high dimensionality there would be an automatic and perpetual potato famine, for the skin of the potato would occupy essentially its entire volume.
-- Herbert Callen, Thermodynamics and Introduction to Thermostatistics [1985].

Things do get weird in higher dimensions. Brian Hayes has an awesome essay in American Scientist on another one of those weird things -- the volume of an n-dimensional sphere is a vanishingly insignificant fraction of the volume of its bounding cube:

... Both the n-ball and the n-cube grow along with n, but the cube expands faster. In fact, the curse is far more damning: At the same time the cube inflates exponentially, the ball shrinks to insignificance. In a space of 100 dimensions, the fraction of the cubic volume filled by the ball has declined to 1.8×10–70. This is far smaller than the volume of an atom in relation to the volume of the Earth. The ball in the box has all but vanished. If you were to select a trillion points at random from the interior of the cube, you’d have almost no chance of landing on even one point that is also inside the ball.

What makes this disappearing act so extraordinary is that the ball in question is still the largest one that could possibly be stuffed into the cube. We are not talking about a pea rattling around loose inside a refrigerator carton. The ball’s diameter is still equal to the side length of the cube. The surface of the ball touches every face of the cube. (A face of an n-cube is an (n–1)-cube.) The fit is snug; if the ball were made even a smidgen larger, it would bulge out of the cube on all sides. Nevertheless, in terms of volume measure, the ball is nearly crushed out of existence, like a black hole collapsing under its own mass.

How can we make sense of this seeming paradox?

3 Comments:

  1. Digbijoy Nath said...

    I never understand anything beyond 3 dimensions, no matter how beautiful or elegant their description is mathematically ! Unless we can imagine or visualize, abstract things like n-dimensional space are of little 'use' to science or technology. But they are fun to work on mathematically, I agree.

  2. Ankur Kulkarni said...

    @Digbijoy In fact, that is why they are useful! They allow us to address things that we are not equipped to or accustomed to imagine. And that is where the next challenges lie.

  3. ajitjadhav said...

    @Ankur and @Digbijoy:

    At least for the point Abi highlighted here, offhand, I think that it should be a simple exercise to supply a kind of a "visualization"---an abstract visualization, to be sure, but still, a visualization. Give it a try, and see how the logic Hayes presents might become much directly graspable.

    (If you would like to have a hint about this possible "visualization" idea, drop me a line, though, frankly, I don't think that a hint is all that necessary. ... BTW, I haven't actually tried working out Hayes' cases using it, but, guess, it should work out OK.)

    --Ajit
    [E&OE]