Science Porn

[Alex K]

Can somebody help me think/visualize about the 4th dimension as it applies here? Or maybe, point me to some good reading. Is the 4th dimension supposed to be thought of as directional or spatial in this context?

I can follow that '3d objects cast 2d shadows, so 4d objects cast 3d shadows', but why does this come to be a cube within a cube?

I've also seen this cube sort of undulating and inverting in a few animations, what is the significance of that? Is this meant to show the passage of time as being an added dimension?

On the "3d shadow", how is a 3d shadow different from a 3d object? Can it only be "cast" in a 4 dimensional realm, whatever that may be?

Hopefully math isn't the answer here, I don't speak that language very well.

Ok I'll try:
Imagine the usual scenario in one dimension less: a pointlike light source not too far away shines on a wireframe cube. The shadow it will cast on the wall is (roughly) something like two squares of different size inside each other, with the corners connected. In the shadow, you see that in each corner, three lines lead away which in the original object are all orthogonal on each other (but in the shadow, only two are orthogonal). In this method of projection, the third dimension is represented as a scaling: objects closer to the screen are represented smaller, objects closer to the light source larger. Hence, the cube face closer to the light source corresponds to the larger of the two squares in the shadow.
So much for the ordinary scenario.

The same can be generalized to one dimension more, where now the position in the fourth dimension is translated into a scaling: a pointlike lightsource casts a shadow of a 4D cube onto a three dimensional screen which we see as space. The further an elemwnt of the 4D cube is away from this "screen" in the new fourth dimension, the larger it gets represented in our 3D "shadow". In this case, both "faces" of the 4D cube are entire 3D cubes (compare: the faces of 3D cubes are squares, ie 2D cubes), and the "face" further from the screen gets represented as the larger 3D cube, the other as the smaller. Corners are now connected to four lines each, which are all orthogonal in 4D space, but in our 3D shadow, again one is not.

Always go to the analogy in one dimension less and try to draw comparisons one can generalize.