Comments by "John Gottschalk" (@JohnGottschalk) on "Strange Spheres in Higher Dimensions - Numberphile" video.

  1. I just wrote a similar thing, but didn't connect it to the nD square. What I realised from this experiment is spheres become less efficient to stack in higher dimensions. Which I guess is equivalent to saying the bounding box in which the spheres need to be stacked is becoming larger and larger in ratio to the spheres. I guess the nD squares are spikier, if you consider the corner of a 3d square (a cube corner) spikier than the corner of a 2d square. I thought about the efficiency of filling the square (even with a single circle), at 2d the circle fills more of the square's area, than in 3d the sphere fills of the cube's radius. Same goes for 4d and higher!
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  2. Wait... In the 3rd dimension, the distance between 2 spheres diagonally is no longer the radius of the sphere but the radius of a cut somewhere lower than halfway the sphere, so the actual radius can be bigger. So it not that the sphere is getting spikier... its just that the radius in 3 dimensions is only a cut at a point that is "lower" than halfway in 4d, so again there's a more optimal place for the radius to be centered in. The 4d sphere is still equidistant at all points from the centre. Just as a 3d sphere no matter how you cut it with a 2d plane always looks like a circle, the same is the case for 4d, no matter how you cut it out in a 3d frame, it will always appear as a sphere. So actually, it's not that spheres are spikey in higher dimensions, it's that spheres become less efficient to stack in higher dimensions allowing bigger gaps for other spheres. The same way that 2D spheres are more efficiently stacked, than 3d spheres. I think the experiment doesn't actually suggest the end conjecture. In fact all it seems to tell me is 4d and higher d spheres are actually so smooth in a way, that they leave even more space wasted. Comparing the ratio of area of a square to a circle vs the ratio of volume of cube to a sphere maybe shows the same thing? I'm too lazy to do it right now :b
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