While much of what you say is true, it seems to me to miss the main point of the article. It feels like you've read the first part about the sphere inside the other spheres, but not read the rest that talks about the other reasons a sphere is "spikey".

You say:

  > It's not the spheres that are "spikey", the cubes are.

It's true that cubes are spikey, but it's in an obvious and easy to visualize sense. Take an ordinary 3D cube, stretch out the corners and you've got the idea.

The reason high-dimensional spheres are "spikey" is more subtle, harder to visualize, and easy to get wrong when working in machine learning. I've now explained it in several replies in this thread, clearly I need to go back and revise the original to try to forestall this mis-reading.

http://news.ycombinator.com/item?id=3997551

http://news.ycombinator.com/item?id=3997681

http://news.ycombinator.com/item?id=3998259

http://news.ycombinator.com/item?id=3997215

http://news.ycombinator.com/item?id=3998259