You gotta think outside the hypercube

I've got a rotating tesseract demo up on CodePen[1]. At least, I think its a rotating tesseract - I have no idea if it's an accurate representation because tesseracts lead me into thinking about quaternions and then my brain shuts down.

[1] - https://codepen.io/kaliedarik/pen/yLbQpKq

"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).

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.)

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.

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?

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.

The idea of embodied cognition might explain why we struggle to intuitively visualise objects in four-dimensional space. Basically the argument is that our visual imagination is grounded in sensorimotor systems that are optimised for three spatial dimensions. But reasoning is not limited to sensory experience so we can abstractly figure out how spatial systems in higher dimensions would work if they actually existed. Like we do for most mathematical concepts that don't easily map to what our bodies experience.

So by this perspective, it's not arbitrary, but is the result of our physical embodiment and how we interact with the world.

People can solve a 4D Rubicks cube, so some parts are possible I guess.

Are you sure we can't visualize 4D objects? Or rather, what exactly does that mean?

Can we visualize 3D objects? Or do we reason about 3D objects, but only visualize a 2D projection of 2D boundary surfaces embedded in 3D space? I'm definitely not thinking about the inside of my desk lamp, or even its back side, even though those are as much a part of the 3D object as the front surface.

Can you visualize a 1D projection of a 2D object? Probably, but is it a little tricky? How about a 1D projection of a 3D object? And when that 3D object moves? How about a 2D projection of a 3D object, but seen from the inside out with a Hammer retroazimuthal projection, instead of viewed from a distance with the embodied camera eye and wonderfully simple rectilinear(ish) projections that we're so familiar with?

Arguably, I think we can visualize 4D objects. They just look the same as 3D objects, because the visualization is itself a 2D projection. If they move, they look like wobbly 3D objects, as we pick up a different "slice" or "surface".

Now, we don't know very many 4D shapes, because we don't encounter them in our lives. But I think that's fully explained by familiarity, without invoking the idea of an arbitrary limitation. We've all seen lines, sheets, boxes, balls, pyramids— Try describing to a random stranger what a Strandbeest or a Klein bottle looks like.`

> It is sometimes associated with our inability to think of new colors, but I think this is a completely different problem.

As an aside, thinking of a new color is relatively easy. You can sorta actually see new colors just by making a really good pigment[0], or shooting cone cells with lasers[1].

But there's also the possibility of having a new photoreceptor. In which case, you don't just gain 1 new color. You gain 3 new secondary colors, plus an entire new type of 4 "tertiary colors", each of which is as visually distinct as cyan, magenta, and yellow are. And the colorspace itself becomes a 4-dimensional volume, with every existing color able to blend into smooth graduations of the new cone cell signal.

[0] https://www.youtube.com/watch?v=_NzVmtbPOrM [1] https://en.wikipedia.org/wiki/Olo_(color)

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

It's not so easy any more if you try to rotate the hypercube and have to visualize intersections with arbitrary hyperplanes. Already in 3 dimensions the intersection of a cube and a plane can be a non-regular polygon with 3-6 sides.

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