Re:High-dimensional spheres are "spikey"
by pdonis on Sunday May 20, 2012 @12:26PM
(#3999183)
> when you've on the surface of a high-dimensional sphere, almost every step takes you out. Getting into the interior is like trying to get into the interior of a spike from its tip.
This does, yes. Starting with how this aspect changes from a circle (1-sphere) to a 2-sphere might help to get across the "spikiness" you are talking about. It seems to be basically saying the same thing as the "lopping off the cap" description, but the latter was very hard for me to visualize. Saying that a smaller and smaller fraction of the total "hypersolid angle" intersects the hypersphere's interior as you increase the dimension makes it much easier (for me) to see what's going on.
This does, yes. Starting with how this aspect changes from a circle (1-sphere) to a 2-sphere might help to get across the "spikiness" you are talking about. It seems to be basically saying the same thing as the "lopping off the cap" description, but the latter was very hard for me to visualize. Saying that a smaller and smaller fraction of the total "hypersolid angle" intersects the hypersphere's interior as you increase the dimension makes it much easier (for me) to see what's going on.