system
Given the following post, respond as if you're commenting on Reddit:
human
What can I do to better understand the concept of higher dimensional space like hyperspheres?

I am trying to understand hyperspheres but every time I think I understand what I am reading/hearing I realize I don't. I can do hypothetical ideas like quantum mechanics but these things seem to much for me to grasp.
gpt
Stop trying to picture it. There's a reason mathematicians use symbols and equations to talk about this stuff: It's really powerful and can be intuitive.

Learn about Linear Algebra, which can be visually explored in 2-3 dimensions. Here, things like orthogonality, projection and distance are all immediately intuitive, but you start working with them in terms of equations. Instead of orthogonality meaning "A right angle", it means x*_1_*x*_2_*+y*_1_*y*_2_*=0. Instead of a projection meaning you draw a line straight down onto another, it's a function p(x) satisfying p(p(x))=p(x). Instead of distance being the length on a piece of paper between two points, it's any function d(x,y) satisfying certain things like d(x,x)=0,  and d(x,z)<=d(x,y)+d(y,z). You  switch your pictorial intuition about geometry, which is limited to <=3 dimensions, into an intuition about equations which can be any finite dimension. With this more sophisticated language, you can understand more sophisticated things, like higher dimensional spheres.

This would be just one part of the attack on understanding hyperspheres. Multivariable calc and its generalizations is also useful, and then there are even more abstract fields of math that really get into the gritty of how these things work. There's a lot we don't understand about them and spheres can actually be really difficult when you get to that level. One of the Millennium Prize Problems was about hyperspheres.

But trying to imagine things in your head will just be limiting, give yourself a headache and you'll leave with a lot of misconceptions, since 3d visual intuition does not have to generalize to 4d or 5d intuition. Equations fix all of these problems.