Comments by "" (@willemvandebeek) on "Pi hiding in prime regularities" video.

  1. I have been struggling to come to grips with how an area of a disc is exactly 4 times more than the area of a sphere. I have been imagining cutting 4 discs into into infinitely many thin rings and rebuilding a 3D sphere out of those 2D rings... But with this video could it somehow be proven the area of a sphere is exactly 4 times more than an area of a disc? I have also been looking at the area of a 3-sphere in 4D space, which has a hyperarea of 2pi^2 r^3, but that is a volume... differentiating this would give me an actual area of 6pi^2 r^2, which is not what I would have expected, because; why would pi not square itself going from 2D space to 3D space, but pi suddenly does start squaring itself from 3D space to 4D space? I would have expected that, if the area of a disc in 2D space multiplies by 4 when it inflates into a sphere in 3D space, then the area of a sphere in 3D space multiplies by 6 (or some other round number) when it inflates into a 3-sphere in 4D space (just like 4 1D lines can be folded into a 2D square, 6 2D squares can be folded into a 3D cube, 8 3D cubes can be folded into a 4D tesseract, etc.), and we would have an area of 24pi r^2 for a 3-sphere in 4D space (which is wrong aparently). Hope you can help. :)
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