4D to 3D Projection Animations

(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. 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.

Here's a little crazy-hard stuff--sorry! Feel free to skip over it for now. . .

The simple XY rotation here preserves not only 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.

As a consequence, in addition to XY rotations we can now also have WZ rotations, and these can both be done at the same time! This gives us the same kind of headache that a 2D'er would have trying to think about 3D rotations for which a cube doesn't look much like the (familiar) square. 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.

A hard question: What would the paths (not the wheels) look like if we did perform a 90 degree WZ rotation of our view so that Z was "away" instead of W? (The 4D camera would be looking down at the Z=0 floor where the paths are drawn.)

Discussion--2D Slices of a 3D World

It is interesting that Marc ten Bosch, author of the breakthrough 4D game Miegakure credits a book published in 1884 as inspiration for his approach. The book he refers to is Flatland, by English schoolmaster and theologian Edwin Abbott Abbott. Upon examination we can see that what ten Bosch has adopted from Abbott is a deep trust in an analogy to a 2D person trying to understand a 3D world, a case that we can completely visualize, being familiar both with the world we live in and (e.g.) the world of images on paper. This is a wonderfully rich analogy on which to build.

Furthermore, ten Bosch also borrows from Abbott an approach that represents the view of (e.g.) a 2D'er perceiving the 3D world as a "slice", or an intrusion into his world. The 2D'er can't really "see" me, but if I would push my finger through his world, he would see an oval shape and, looking carefully (say, using a 2D Xray machine!) might perceive a structure including surface skin and central bone. If the 2D environment contains gravity, he might also understand a rolling ball as a rolling circle, and might come to understand, after some thought, how, when the ball is not centered on his "slice", it seems to roll "above" the ground or floor. This is all beautifully explained in this video by ten Bosch.

The animations in that video are immensely edifying by virtue of their technical competence and wizardry, but also due to a feature hinted at by Abbott, and which has been scarcely treated in other work to date. That feature is an effective reduction of dimensionality by breaking the symmetry in one or more dimensions. Watch for the “downward” pull of gravity and a “floor”, an apparent plane that limits motion in one direction (say “Z”).

This bring us to what's new in the approach here. Might we not do one better than a slice or an intrusion? Why don't we just draw a picture of our world for the 2D'er and show it to him? And maybe we could do a little more to reduce the symmetry--to make that world a little more like our own. . .

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