But the cube isn't pegged like that at all. The cube is the set of points where any element of the coordinate vector has a magnitude less than a specified value, whereas the sphere is the set of points where the magnitude of the vector is less than a specified value. Hence the cube gets 'larger' as you make space itself larger by adding dimensions, but the sphere doesn't.
They're totally different objects. Unit balls encode distance by being the set of points within some distance from the center. They are thus, loosely, parameterized by a 1-dimensional measure.
Unit cubes are constrained such that all of their measurements are unit length---something which ensures that their volume is constantly 1. They do this by getting spiky, as xyzzyx mentioned. The distance to cross them shrinks as the vertex distance grows.
So they start to look totally different in high dimensions, evidenced by the ball's vanishing measure.
> It's also driven home by how the Lesbegue measure in d-dimensions assigns zero measure to all (d-1)-dimensional (or lesser) objects. Adding a dimension just makes space immensely larger.
> And so pegging the size of our unit ball to its 1-dimensional parameter, the radius, causes its volume to vanish.
Funny. I was just going to enter a comment how that was the most useful intuition in the whole thread.
There's some inherent flaw here - as 'pegging' the size of unit cube to its 1-dimensional parameter, the length, does not make the volume to vanish.