While thinking about high-dimensional spaces and their less intuitive properties, I came to the realization that even three spatial dimensions possess the potential to overwhelm our basic human intuitions. This post is an exploration of the gap between actual 3D space, and our human capabilities to fathom it. I come to the conclusion that this gap is actually quite large, and we, or at least most of us, are not well equipped to perceive or even imagine “true 3D”.

What do I mean by “true 3D”? The most straightforward example would be some ℝ³ → ℝ function, such as the density of a cloud, or the full inner structure of a human brain (which too would be a ℝ³ → whatever function). The closest example I’ve found is this visualization of a ℝ³ → ℝ³ function: