I am by no means an expert, but I have thought about trying to visualise the fourth dimension. The issue is that we don’t really see 3D. While in 2D we can view every pixel simultaneously, when we view in 3D we have the issue of aspect and concealed voxels. You can’t visualise simultaneously all voxels of a cube, as most are under others or otherwise obscured. This therefore limits how much we can use 3D to mimic 4D. Perhaps if we were to spend more time with semitransparent objects our brains could learn to visualise 3D better and then maybe 4D.

What an interesting idea! Have you ever produced anything for a representation of 4D with translucent 3D? I'd be very interested to see it...

I'm not sure that could practically work. The problem, is that translucency would allow only a partial view of concealed voxels. As more voxels cover something, the translucent effect compounds.

Additionally, most voxels would appear different with different view perspectives. Due to more or less voxels covering them.

The problem is that in 4D all voxels are visible to the viewer. So viewing a 4D apple would allow you to see the apple from all possible view points simultaneously, including interior views.

To me it just doesn't seem possible to replicate this concept in 3D VR.

What would be the benefits of the 3D VR? There will always be a 2D bottleneck on your eyes' retina. The most important perspective clue I would say is temporal sampling. For dynamical systems, Takens theorem says that a high dimensional topology can be restituted by the sequential sampling of a single variable alone. It seems likely to me that our perception generally works by this principle. That being said, I think we are quite heavily hardwired to 3D perception. I suspect these circuits cannot be wholly overridden.

What would a 3D eye even be like? Is it something like having independent depth perception per eye?

Photons travel in 3D space and may only be occluded by 2D surfaces. There is no getting around that fundamental aspect of reality. In this universe.

This is why as an armchair physicist I'm excited about the new neutrino observations and research going on, mainly because neutrinos aren't affected by gravitational lensing and other effects other particles encounter as they 'roll around' on space-time. Would it be possible to understand conceptually how other dimensions extend out of space time? I'm imagining a Y-axis perpendicular to the space-time fabric....

Gravitational lensing is a distortion of spacetime. It affects all particles, including neutrinos.

I'd say we have two eyes and compile to something 3D, which is mostly possible because of that and nothing is hardwired, just takes a lot of responsive feedback to rewire itself.

We see a 2D projection and infer some 3D information from stereo parallax and other depth cues. But it’s still a 2D projection in the end. A real 3D ”vision” would not have a ”viewpoint”, no perspective, no foreshortening. It’s difficult to even imagine what it would be like to perceive that.

Stereo parallax, great term. In other words how cross-eyed we are.

The reason one can represent 3D "quite comfortably" on a 2D monitor can best be explained using linear algebra. The 3D space in the video is "projected" (linearly) onto a 2D space the same way a single one of our eyes project 3D space onto a 2D space. This is just one choice of projection though. For instance, when he shows the 2D beings perspective, he's chosen a different projection from 3D -> 2D and it looks quite foreign to us. The reason you feel comfortable with the normal projection is because your eyes do it all the time. Your eyes don't however, project 4D -> 3D so you don't feel it as natural.

Of course you can, and it would be different from the cross-sections seen in this video. I doubt it would ever become intuitive though, I suspect our brains are hard-wired for a 3d world.

Here's a video of a rotating 4d hypercube in a 3d perspective:

https://www.youtube.com/watch?v=t-WyreE9ZkI

It's called a tesseract, and as each face of a 3d cube is a 2d cube (a square), each face of a 4d cube is a normal 3d cube, that we see skewed by the perspective.

I'd say our brain is anything but hardwired. Sense-substitution works beautifully too and even a tool like a hammer or a mouse becomes one with your body through frequent use.

Your linked video maps the 4th dimension to time, it doesn't project to 3D. Projecting through time (especially non-interactively) lacks the immediate feedback needed for the brain to grasp it as intuition.

> Your linked video maps the 4th dimension to time, it doesn't project to 3D

No, it projects it to 3d. The movements you see are rotations, not movements of an intersection plane. You can clearly see at any point in time each of the 8 identical cubes making up the tesseract, skewed and resized by projection to a 3d perspective.

Just like how 3D objects make a 2D shadow (or a 2D projection on a screen), you can make a 3D shadow of a 4D object. https://www.fourthdimensionapp.com walks you through this.