4D toys

I'd love to know the math behind the physics engine, particularly angular momentum. Planar rotations are weird in 4d.

Using the 4th Dimension, the two interlocking rings can be separated. However, what would be the equivalent puzzle for the 3rd dimension? What 2D system requires the 3rd dimension to be separable? Is there even an equivalent? Clearly a circle within a circle is one such system, but it corresponds to a ball wholly contained by another ball in 3D, not a pair of interlocking rings.

You can't have interlocking rings in 2D without having some kind of intersection. You have a similar problem with moebius rings and klein bottles: moebius rings are 2D objects but you have to represent them in a 3D world if you don't want them to be self-intersecting. If you attempt to represent a moebius ring in 2D it'll necessarily self-intersect.

Klein bottles are the same thing but with an added dimensions: any representation of a klein bottle in 3D makes it look like it's going through itself, even though in 4D it wouldn't: http://s3files.core77.com/blog/images/2013/06/klein-bottle-0...

It's also true if you remove a dimension: a 1D moebius strip would simply be a circle, but if you try to draw it in 1D you end up with a segment where both halves of the circle are overlapped. So every time we have a N-dimensional object that can only be properly represented in N+1 dimensions.

That's also the same reason you can't solve the problem of connected three objects to three other objects on a 2D planes without intersecting:

     A   B   C


     X   Y   Z

You can't distribute a, b and c to x, y and z on a 2D plane without intersection.

Topology is fun.

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.

If you start from the understandings (such as we have)of physiology and evolutionary psychology I think the answer is "no" (perhaps there's a trick someone can figure out via a different path -- I hope so).

It seems pretty clear that looking at 2D images (and in particular, still ones) is a learned task, like reading, rather than an innate one. Both appear to tie into deep structures in the brain, but both are very recent inventions.

From the perspective of developmental stages described by Piaget, children learn to view in three space primarily with objects within reach (parallax pretty much peters out around the ends of your arms. Once the child becomes mobile, she is able to use semantic understanding to estimate the size of distant objects and get a rough idea of distance. The whole human process of seeing is very different from the way, say, a NN is trained on an image: the whole thing isn't gulped in at once, but we foveate on various parts of the image and assemble / confabulate a whole. You can see this in the structure of Chinese classical painting or pre-persepctive European paintings: distant objects aren't sized in any way proportional to their apparent size. This really maps more to how much attention you pay to various objects in the scene.

You then learn to map that into a 3D model which I believe (but am not digging up refs this instance, sorry) has hardware support.

Thus the 2D->3D process exploits a lot of learned and innate knowledge and technique that you have already developed. With one exception you haven't any 4D experience. That one exception is temporal data -- we can easily extrapolate from, say, a sphere shrinking and growing. Apart from that, there isn't much to work with.