[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.

[HappMacDonald]

> (Aside: Strangely, the relationship is non monotonic and is actually maximal for n=6)

For this aside I crave a citation.

When n=1 the sphere fit is 100% as both simplex and sphere are congruent in that dimension. And dismissing n=0 as degenerate (fit is undefined there I suppose: dividing by zero measure and all that) that (first) dimension should be maximal with a steady decline thereafter thus also monotonic.

[ivanbakel]

This looks to have been a conflation by the GP between the volume of the unit sphere itself and its ratio to the volume of its bounding cube (which is not the unit cube.) The volume of the sphere does top out at an unintuitive dimension, but indeed the ratio of the two is always decreasing - and intuitively, each additional dimension just adds more space between the corners of the cube and the face of the sphere.

[sota_pop]

I’m familiar with this example of hyper-geometry. Put more abstractly, my intuition always said something like “the volume of hyper geometric shapes becomes more distributed about their surface as the number of dimensions increases”.

[kibwen]

There's a great 3blue1brown video that demonstrates this unintuitive result in a relatively intuitive way: https://youtube.com/watch?v=zwAD6dRSVyI