3 ms·

Fun. I love that weird intuition trap you mention - the volume of a N dimensional sphere of radius R sphere goes to zero as N -> infinity. https://en.wikipedi

• jacobolus 10y ago
  This comes down to the definition of (hyper)volume. We use a (hyper)cube as our unit, but that’s somewhat arbitrary. We could alternately define (hyper)volume using a standard n–simplex as the unit, then we would say the volume of an n–ball → ∞ as n → ∞.
• Someone 10y ago
  here is a way to intuit that: - because (1-ε)^n/1^n goes to zero if n goes up, most of the volume of a cube gets concentrated near its surface. - to go from a n-dimensional cube to a n-dimensional sphere, you have to cut away most of its surface (everything except for 2n 'caps' near the points with coordinates (0,0,0,...,0,0,±½,0,0,0,...,0,0,0)), and hence, most of its volume.