Mar 19, 2009 12:42
Imagine a hypercube with side length 1. For the common definition of 'volume' (length times width times height times hyperlength times etc.) the volume of the cube is 1. That's true no matter how many dimensions this strange beast inhabits.
Now imagine a hypercube of side length 1+dx, where dx is some small parameter. Computing the volume of this cube, we obtain (1+dx)^n (^ means "to the power of"), which when expanded out gives 1+n*dx+ (some other terms with dx^2, dx^3...). If dx is small, we can ignore those other terms and just get 1+n*dx. The difference in volume between this new cube and our original one, then, is n*dx. It doesn't matter how small a dx we pick; if n is large enough, we will always get a number bigger than 1, the volume of the original cube.
So we are left with the conclusion that most (in fact, it can go arbitrarily high with increasing dimensions) of an object's volume is concentrated near its surface. Isn't that freaky?
Anyway, I guess what I'm really trying to say is your mom is fat.
Hyperfat.