Comments by "Jkauppa" (@Jkauppa) on "The Puzzling Fourth Dimension (and exotic shapes) - Numberphile" video.

  1. try using time as the 4th dimension, then you have x^2 + y^2 + z^2 + t^2 = 1 for a sphere, it just means you have x^2 + y^2 + z^2 = (1 - t^2) time dependent size of a sphere in 3 dimensions over time
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  2. from that equation you have complex solutions to the (x,y,z) because at |t|>1 you have negative sum of squares
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  3. so you get 7-dimensional sphere when t is real, and (x,y,z) is complex
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