I want to mention of one of my favorite mental exercises: thinking in higher dimensions. I find it’s useful to expand your ability to visualize difficult things. (I should do a list of brain workouts!) These nerd kings will take you on a journey, and I want to go over the 4D part. (You might watch it more than once when you have time, until you can grasp it!)
Try conceptualizing 4D objects. It’s not easy to handle higher dimensions, but I encourage you to try the fourth specifically. One of the common ways of describing the fourth dimension is to use time as another axis. This has the benefit of showing you an animation of something changing shape to describe all the 3D “slices”, showing you the whole thing. The problem is it introduces a bit of a crutch, in that it feels a lot more like regular 3D, hiding the nature of that extra dimension.
If you are going to imagine something changing shape over time, you need to also imagine that all of the vertices on the object are connected through time to each other. You have to imagine that a certain subset of the frames, even if just the first and last, are all connected by lines you can’t see. See the problem? That’s why it’s easier to imagine the fourth as something like scale instead, when introducing it. Just pretend you can go smaller forever like you can go bigger.
Here is what a 4D rotation looks like on its fourth axis. It’s fun.
Make sure you remember that 4D objects are built out of 3D objects. You need to think of the tesseract as 6 squished cubes connecting the inner and outer cubes. Those 8 cubes are what a tesseract is made of. Don’t think about it as points and lines too much, or you’ll miss what you’re trying to visualize. Here’s a way to fold and unfold it that might help.
I really think that’s the one you ought to have down by now. Now go back and watch the first Avengers movie and you might look at the blue cube different. If you’re religious, you can go see Salvador Dalí’s Corpus Hypercubus.
Once you have that down, you can begin to think about the other five regular convex polytopes. You might know that the platonic solids mentioned in the video are the D&D dice (minus the d10 diamond). Well, you can think of these as the main six 4D dice.
What fun are 4D dice if you don’t roll them?
Nobody would expect you to grasp 120 squishy dodecahedrons. I still recommend trying to study the first four to expand your mind. The first three should be reasonably within reach. For fun, we’ll end with the 120-cell from the inside.