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Comment by viewer5 | Hacker News Reader

What would the 4th dimension coordinate of your models even represent, though? I'm having trouble wrapping my head around it.

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It's naturally confusing, because a 4th spatial dimension isn't something we naturally experience. We can't really say what it represents because it's not real... it's not a model of reality.

We see in 2 dimensions and we infer the third from various cues. (The quote that we are "dimly aware of a fourth", which you may have heard, refers to time and is a different issue than a fourth spatial dimension.)

We have three dimensions, all orthogonal to each other. We can translate, rotate, and project things in between them.

So all we can do is apply those same principles and see what comes out - we shouldn't expect it to match anything recognizeable though.

Mathematically, they're identical to any other dimension. You can move a little bit into a fourth dimension just like moving a little bit left-to-right. Or you can rotate things around the fourth axis just like turning something clockwise. In the case of this game, he only shows a 2D projection of a 3D cross-section of a 4D world, so that might be adding to the confusion. What you see in the video is only 3D "slices" of a 4D object rotating.

Would something like an Oculus Rift VR help reduce the number of projections from being "4D->2D" to be "4D->3D"?

No, Oculus Rift's screens are still 2D surfaces.

Sure, but at least you get stereo vision and headtracking that makes it less obvious you're just looking at a flat screen showing a mono projection?

It might help mentally, but it won't reduce the number of projections. You'd have to be able to see every point in the whole 3D volume, not just the surface.

One of the axes introduced by the fourth dimension. There are 3 in 3D, 6 in 4D. XYZ space has axes perpenicular to the planes xy, xz, yz. WXYZ space has axes perpendicular to the planes xy, xz, xw, yz, yw, zw.

No. There are only 4 axes in 4-dimensional space - every point is described as a 4d vector, i.e scalar positions along each axis: (x,y,z,w).

xy, xz, etc. are not planes in 4d space - they are 3d-hyperplanes, and perpendicular to them are planes, not axes. Imagine this: you have a 4d vector and you hold x and y constant. You still have 2 degrees of freedom: a plane, not a line.

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