4D to 3D Projection Animations

by John Dick

(4) Wheel Animation with added WX and XY Rotations

This is the same as animation (3) but with an added 180 degree XY rotation of the viewpoint.

You may notice that the wheels change their physical shape as they negotiate the paths. This is a consequence of their not being objects in themselves, but images of 4D wheels.

This Animation

(Continuing from above) While the "tire" that you see seems to have "thickness", it is in fact the image of an entirely "sideways" 4D wheel rim. The 3D equilavent would be a completly flat tire, nothing more than a "tee" (T).

Looking at a wheel on the bottom path directly in "front" and going in the X direction, the rim extends in the Y direction as we might expect, but also has a (somewhat smaller) extention in the other "sideways" (W) direction. (In 4D we would need a rim with width in both sideways directions to keep from cutting a groove into the ground!)

Discussion (continued)

Placeholder text! A simple way to allow my 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.

Placeholder text! The 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 were semi-transparent, allowing her/him 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.

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