4D to 3D Projection Animations

by John Dick

(2) Wheel Animation with added XY Rotation

This animation shows an added 180 degree XY rotation of the viewpoint.

Periodic slowing occurs as the wheels reach a point where another path may be chosen, making it easier to follow any individual wheel.

This Animation

The added rotation here is easy for our minds to make sense of, being (apparently) a simple XY rotation in 3-space about a Z axis. But this rotation preserves not just Z, but both Z and W. Thus it is actually not a rotation about the Z axis, but about the WZ plane! This is an idea that is hard to wrap our minds around--don't worry if it doesn't make much sense.

It might help to know that this leads to the fact that we can now also have rotations in (not about) the WZ plane, giving us the same kind of headache that a 2D'er would have trying to think about 3D rotations. But we won't be going there (for now) because in the environment here, the Z axis is special: It defines up and down. So, we will always present the Z axis as vertical; and this would be undone by (e.g.) a WZ rotation.

In this view, the wheels take a more complicated route; follow one wheel and you'll explore more of the path environment.

Answer to the question from animation (1): The wheels are going directly in the W direction at the midpoint of the darker red paths, that is, where two red paths just touch.

A hard question: What would the paths (not the wheels) look like if we rotated the view 90 degrees in the WZ plane so that Z was "away": our 3D photo being taken by a 4D camera looking "down"?

3D Slices of a 4D World

Consider the example presented more than a hundred years ago by English schoolmaster and theologian Edwin Abbott in his book Flatland, and wonderfully realized by Marc ten Bosch in this video for the upcoming video game Miegakure: The example of a 2D person trying to "see" our 3D world. That person can't really see us, but could, conceivably, see a (2D) slice through our world. This example has been richly presented by both Abbott and ten Bosch.

But consider that same 2D individual looking at a (2D) drawing of a 3D landscape that we might introduce into that world, and consider the problems for viewing that drawing (let alone understanding it). 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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