3 ms·

I am not sure I would interpret this observation as spheres that are getting more spikey. Maybe what gets more spikey is the actual box, so the sphere has it ea

• JadeNB 10y ago
  > I am not sure I would interpret this observation as spheres that are getting more spikey. Maybe what gets more spikey is the actual box, so the sphere has it easier to touch or even extend outside of it. I think that this is not correct. In n dimensions, the cross section of a unit n-cube near the edge is a unit (n - 1)-cube, just as your intuition tells you. However, the cross section of a unit n-sphere near its intersection with the unit n-cube is a tiny, tiny (n - 1)-sphere (the radius of a cross section at height z = 1 − 𝜀 goes to 0 as n goes to ∞); this is the 'spikiness' that the author is discussing.
  • js8 10y ago
    On the other hand, it seems to me (maybe I am wrong) that the solid n-dimensional angle of each corner of the n-cube is progressively smaller (as a ratio of full n-dimensional angle). So in that sense, each n-cube gets more spikey, and there is "less space" in each corner for each n-sphere that enclose the central n-sphere.