A trick to visualizing higher dimensions


Perhaps it's worth saying a few words on why those interested in math care about higher dimensions.  Why care about 20-dimensional space when the world we live in is not 20-dimensional?  Broadly speaking, spatial reasoning is not just useful for describing the physical world, it's a sort of problem-solving tactic for any scenario where you want to think about collections of things (like lists of numbers) as individual points of some object.

For example, viewers of the "Who cares about topology" video will remember how nice it was to associate pairs of points on a loop with individual points on a Mobius strip.  But this is only associating collections of two things with individual points.  As the relevant collection gets larger, the corresponding object is harder to hold in one's mind, since it lives in a higher dimensional space.  For example, there is something analogous to a Mobius strip that represents unordered quintuplets of points on a loop, but seeing it would require being able to visualize 6 dimensions.  That doesn't stop mathematicians from working with such objects, though, it's just that the work becomes a bit more abstruse.

The tactic I provide in this video is not some magic make-you-feel-like-neo-from-the-matrix visual that lets you hold a full high-dimensional object in your head.  Instead, its main function is to emphasize that what we really mean by "a point in n dimensions" is "a list of n real numbers", and moreover that it can still be fruitful to think visually about that idea even when you can't conceptual the list of numbers as an individual point in space as we can for 1, 2 and 3 dimensions.