southa wrote:Why? Because if you force the second projection to happen orthogonally to the 4D viewpoint, you get a lot of illusions. For example, if you look down a cubical passage in 4D, the projected image has the 4 side walls as frustums (analogous to the side walls of the cube-within-a-cube projection of the 4-cube). But now you project to 2D parallel to the coordinate axes, and suddenly you can't see 2 of the frustums anymore because their opposite edges coincide. What you really want is to project to 2D using another viewpoint (which resides in 3D), that looks at the retina from an angle, so that you can see the 6 frustum volumes in the 3D retina separately. The cubical wall at the end of the corridor will be much more obviously a cube; if you had projected from an orthogonal viewpoint, it simply becomes a square, and you have absolutely no information about whether it's a square, a cube, or a cuboid in the 4D view.
That's valid if you constrain the orientation of the viewer, but if you allow the viewer to rotate by any angle in all six planes, they can effectively move your viewpoint for the 3D retina to wherever they like. So no matter what viewpoint you pick, the tetronian can reorientate themselves so that the illusions reappear. Put another way, there are viewer rotations that have the same effect as the arrow keys in the 4D Maze Game - you can recreate the illusion view with those too.
The point is that the player should be able to rotate the 3D view freely in order to ascertain what exactly is being seen, without changing his 4D viewpoint. It doesn't make sense that the 4D viewpoint needs to be rotated just so some poor 3D being can understand what the tetronian is seeing.