A leap into high dimensional space [series]

curious phenomena in volumes, distances, angles, and separability in high dimensional spaces.

Published

“To deal with hyper-planes in a 14-dimensional space, visualize a 3-D space and say ‘fourteen’ to yourself very loudly. Everyone does it.”

Geoffrey Hinton

Humans are really good at visualizing one, two, and three dimensions. Unfortunately, this doesn’t scale past three very well1. What’s more interesting is that some of the intuition that we have for 2D and 3D really doesn’t extend to higher dimensions as we might initially expect.

Here is the series in chronological order (to be updated with new entries over time):

  1. The volume of the $4$-ball

  1. Although there are apparently methods to visualize a 4D space, I’m not sure how accessible it is for everyone and how effective it is. Reading about it feels like people describing going on spiritual trips. ↩