Counterintuitive Spaces

Given a unit square, what's the area of the region that is a distance of 0.05 or less from the outer edges of the square?

Finding the answer is simple. Subtract the area of the inner square from 1.0 (the area of the outer unit square).



In three dimensions, you can find the volume within 0.05 from the edges of the unit cube by subtracting volume of the inner cube from 1.0 (the volume of the unit cube) and the answer is 1.0 - 0.9 * 0.9 * 0.9 = 0.271.

The question can be carried up into higher dimensions for unit hypercubes. In n dimensions, the answer is 1.0 - 0.9ⁿ. Note that the larger n gets, the closer the difference gets to 1.0.

The counterintuitive consequence is that in highly dimensional spaces, most of the volume is concentrated near the surface. Even in 20 dimensions, nearly 88% of the volume of a unit cube lies within 0.05 of the surface.

I most recently stumbled on this interesting fact while reading Richard Hamming's Numerical Methods for Scientists and Engineers. While searching the web for more information, I stumbled onto a number of nice posts on the subject by Eric Lippert of Microsoft.