Visualising Higher Dimensional Spheres

I often try to think of how one might imagine spatial objects that have dimension greater than three. That this problem is nontrivial seems to come from the basic fact that the human mind developed to interpret and analyse the lowly three spatial dimensions that we appear to inhabit–our ancestors never had to navigate 5-dimensional mazes to escape from 4-dimensional tigers and therefore visualising in 3-dimensions was perfectly sufficient for successfully passing on our genes.

Despite this physiological shortcoming, there are various clever ways of trying to get a grip on four-dimensional shapes–see here, for example, for some nice visualisations. I recently learnt of a neat way to construct an n-dimensional sphere (henceforth, n-sphere) from an (n-1)-dimensional sphere by “glueing cones together”. (The relevant article also mentions two other visualisation methods and tries to link the method described below to Dante’s description of the universe in his Divine Comedy.) The basic construction goes like this (try it with n=1 or 2):

  • First fill in the interiors of two (n-1)-spheres to get two n-balls. These n-balls, which will be the two hemispheres of our n-sphere, should be thought of as being made of an infinite number of nested (n-1)-spheres of decreasing radii.
  • Now superpose the two n-balls, identifying their coincident boundaries–this boundary is the equator of our n-sphere.
  • Finally, mentally drag the insides of the two n-balls in opposite directions and away from the equator, such that an interior (n-1)-sphere further from the n-ball’s boundary gets dragged further away.

If you try using this to construct the 4-sphere from two 3-balls, you’ll realise that there’s no fourth dimension in which to drag the insides of the 3-balls. You didn’t expect to be able to actually visualise a 4-sphere all in one go, did you? Nevertheless, it seems like there’s something to be gained from this perspective, even if it won’t prevent you becoming a snack to that 4-dimensional tiger.

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