On the subject of higher dimension mathematics, I recently discovered this video : "The most beautiful formula not enough people understand"

You might find it interesting too.

https://www.youtube.com/watch?v=fsLh-NYhOoU

I was glad to see the cube defined symmetrically around the axis, i.e. abs(x) <= a.

Like a circle, radially, vector length <= a.

I get anxious when encountering the inconsistency between origin centered Hyperspheres and 0-1 bound Hypercubes.

I've always wondered why humans seem to be unable to visualize four-dimensional objects in their heads.

When discussing this with others, the arguments often revolve around the fact that as we experience our reality in 3D, there should be no reason for us to be able to visualize anything in a higher dimension.

This argument seems like an arbitrary limitation of the human mind, which I don't think holds up. It is sometimes associated with our inability to think of new colors, but I think this is a completely different problem.

I think thats a pretty simple answer; when was the last time you interacted with a 4th dimensional object be it representation or not? For the most part I think people base their perception of reality on their experience and so if you have never interacted with something, how could you imagin it?

Reminds me of this youtube video I saw some time ago of someone that posed the question of what would minecraft be like if it was in non-euclidean space. The author said it took some time to get used to it and when they tried to play normal minecraft it gave them motion sickness.

If you have no knowledge of what a car is or its internals are like, could you still imagin what's inside?

"This argument seems like an arbitrary limitation of the human mind, which I don't think holds up. "

And why not?

Evolution did not reward us for thinking about spaced out concepts, but for coming up with new ways to get food, outsmart the prey, build tools.

That thinking in 4D is helpful for building tools is a new thing so we evolution did not optmized for it (yet).

The only reason we can’t fly is that we don’t have wings.

It’s a limitation created by our brains evolving to process 3D environments. If we were LLMs we have no 4D training data.

A great example is the film Arrival.

Our 3D visualization relies havily on photons doing the heavy lifting of traversing 3D space in straight lines, people get, you know, accustomed to it. In fact how we see things is frozen by physics, not brains - they are just accommodated to reality

There are no such utility particles doing any heavy lifting in 4D, so nothing to accommodate to.

I can think of new colors, but I don't have a way to relate them to stimuli I've experienced, which makes it difficult to make sense of them the same way I can recall seeing existing colors. Still, I and many others have a notion of color that encompasses more than what my eyes can actually see, and I even have some friends that see much, much more in terms of color (such as texture, sensation, etc.)

I have a beautiful animation of a tesseract on an old drive somewhere I kept, back in the animated gif days, I'd look at it now and then in wonder. Reminds me of the same extraordinary feelings about the beauty of the universe with this clip: https://youtube.com/shorts/TGuxwgUyu2A?si=knDzBVqTaZ4oqMEj

Pretty immediately my new favorite hypercube-displaying-exploration article. That's a great, very clear step through the problems and common methods, and I really like the end results.

I suspect every attempt will be unsatisfying, but it does a good job of showing "there's more happening here than it looks like at first".

There was some 90s shareware DOS visualizer for various 4D shapes. I think it might have been called Hypercube, and thought this article might have been about it. It wasn’t, but was quite informative nonetheless!

Timecube... Although I'm not sure if that's what you were thinking of.

The simplest hypercube visualization is taking the fourth dimension as time. Then it’s just a regular cube that appears at once, and a unit of time later again vanishes instantly. ;)

Flatland is an interesting way to think about dimensions. If we were living in a 2D world, anything living in the 3D world would blow our minds and do things we would think should be impossible.

https://en.wikipedia.org/wiki/Flatland

There’s also The Planiverse, with a different take on what a 2D world would look like.

https://en.wikipedia.org/wiki/The_Planiverse

“Speaking of ways, pet, by the way, there is such a thing as a tesseract.” -A Wrinkle in Time