Comment by asboans

“Fill less than 1% of its space” becomes a very counter intuitive statement in any case when discussing high dimensions. If you consider a unit n-sphere bounded by a unit cube, the fraction occupied by the sphere vanishes for high n. (Aside: Strangely, the relationship is non monotonic and is actually maximal for n=6). For n=100 the volume of the unit 100-sphere is around 10^-40 (and you certainly cannot fit a second sphere in this cube…) so its not surprising that the gains to be made in improving packing can be so large.