4D to 3D Projection Animations

by John Dick

(3) Wheel Animation with added WX Rotation

This animation shows an added 180 degree WX rotation of the viewpoint in 4D (WXYZ). The thumbnail shows the configuration half way through the WX rotation where the W dimension is shown and the X direction is "away".

This Animation

This animation may be more surprising since it includes a true 4D rotation of our point of view. For a moment in the middle of that rotation (where the paths themselves are in motion) the W dimension is no longer "away" but perfectly realized: our camera being now pointed in the X direction. This means that, for a short while, X is "away".

You may have noticed apparent collisions between the 3D wheels shown in the animations. This is natural consequence of such a projection. For example, a 2D drawing of a 3D scene in which a nearby object blocks our view (overlays) a distant one might be troublesome to a 2D viewer. However, we 3D viewers would have no trouble interpreting the drawing properly. If fully realized 4D shapes were presented instead of line-drawings, nearby (larger) wheels would "cover" the more distant ones as they (seem to) collide, just as in the 2D drawing.

Answer to the hard question from animation (2): Since all the paths are on the ground at Z=0, they are all the same distance “away” and so there would be none of the foreshortening (changing sizes) that we see in the animations here. Thus the path circles would lie on the faces of a 3D cube.

A Projection instead of just a Slice

A simple way to allow our 2D person to view the picture, though, might for me to draw only dots on the page, so that he/she could see around the dots to those "inside" and could, in a natural way, infer the lines that those dots implied, and then also the 2D forms that those lines contained.

You can see where I'm going with this -- a line drawing is our equivalent to the 2D person's dotted drawing, and in a natural way, we infer not only the surfaces defined by these lines, but also the volumes implied by those surfaces.