> 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.