Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: We're not as 3-Dimensional as We Think, published by silentbob on August 6, 2024 on LessWrong.

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 ℝ ℝ function, such as the density of a cloud, or the full (physical) inner structure of a human brain (which too would be a ℝ whatever function). The closest example I've found is this visualization of a ℝ ℝ function (jump to 1:14):

(It is of course a bit ironic to watch a video of that 3D display on a 2D screen, but I think it gets the point across.)

Vision

It is true that having two eyes allows us to have depth perception. It is not true that having two eyes allows us to "see in 3D". If we ignore colors for simplicity and assume we all saw only in grayscale, then seeing with one eye is something like ℝ ℝ as far as our internal information processing is concerned - we get one grayscale value for each point on the perspective projection from the 3D physical world onto our 2D retina.

Seeing with two eyes then is ℝ ℝ (same as before, but we get one extra piece of information for each point of the projection, namely depth[1]), but it's definitely not ℝ (...). So the information we receive still has only two spatial dimensions, just with a bit more information attached.

Also note that people who lost an eye, or for other reasons don't have depth perception, are not all that limited in their capabilities. In fact, other people may barely notice there's anything unusual about them. The difference between "seeing in 2D" and "seeing with depth perception" is much smaller than the difference to not seeing at all, which arguably hints at the fact that seeing with depth perception is suspiciously close to pure 2D vision.

Screens

For decades now, humans have surrounded themselves with screens, whether it's TVs, computer screens, phones or any other kind of display. The vast majority of screens are two-dimensional. You may have noticed that, for most matters and purposes, this is not much of a limitation. Video games work well on 2D screens. Movies work well on 2D screens. Math lectures work well on 2D screens. Even renderings of 3D objects, such as cubes and spheres and cylinders and such, work well in 2D.

This is because 99.9% of the things we as humans interact with don't actually require the true power of three dimensions.

There are some exceptions, such as

brain scans - what is done there usually is to use time as a substitute for the third dimension, and show an animated slice through the brain. In principle it may be better to view brain scans with some ~holographic 3D display, but even then, the fact remains that our vision apparatus is not capable of perceiving 3D in its entirety, but only the projection onto our retinas, which even makes true 3D displays less useful than they theoretically could be.

Video Games

The vast majority of 3D video games are based on polygons: 2D surfaces placed in 3D space. Practically every 3D object in almost any video game is hollow. They're just an elaborate surface folded and oriented in space. You can see this when the camera clips into some rock, or car, or even player character: they're nothing but a hull. As 3D as the game looks, it's all a bit of an illusion, as the real geometry in video games is almost completely two-dimensional.

Here's one example of camera clipping:

The only common exception I'm aware o...