[sp332]

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.

[raimondious]

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.

[NickPollard]

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.

[0x0]

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

[cosarara97]

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

[0x0]

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?

[sp332]

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.