A sphere of course have sum of the squares of all points equal to one. So in high dimensions, it’s clear that most points have to be close to 0. Hence the “spikes”
Yes. I appreciate the article because it gets people to think about high dimensional geometry. But I think the right takeaway is that in high dimensions, cubes have huge volume compared to the enclosed sphere. So the sphere is not spikey at all. Instead realize that the diagonals and volume of the cube get very large compared to the sphere.
The 'spikes' depend on an arbitrary choice of coordinate system.