4D to 3D Projection Animations

(3) Wheel Animation with added WX Rotation

This animation shows an added 180 degree WX rotation of the viewpoint in 4D (WXYZ).

The thumbnail on the right 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 would all be the same distance “away” from a 4D camera looking towards the ground. There would be none of the foreshortening (changing sizes and weird angles) that we see in the paths here. Thus the path circles would lie on the faces of a 3D (XYW) cube.

Discussion--Show me a Picture!

Think about the problems that a 2D'er would have if she were looking at a 3D landscape drawing that we have helpfuly prepared as an introduction to our world. And, consider the difficulty of viewing that drawing (let alone understanding it).

Looking (sideways) at our drawing, the 2D viewer might very well only see the picture's frame, or leaving off the frame, only the edges of every shape that we've drawn, the interiors being blocked from view. It's now clear that a complete view could only be possible if everything could somehow be made transparent, allowing her to perceive shading, shadows, and all the aspects of that drawing that allow us 3D folk to view into that drawing to infer what is happening away from the viewer, in the 3rd dimension.

A simple way to allow our 2D person to view the picture, though, might for us to draw only dots on the page, so that 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 probably see where this is going -- a line drawing to us is the 2D person's dotted drawing. In a natural way we infer not only the surfaces defined by these lines, but also the volumes implied by those surfaces. And what's more, we can easily see around the lines to what's behind. So 4D'ers might well do drawings like these to help us to see their world. They may not be able to do much more to help us understand it though--that would be on us.

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