4D to 3D Projection Animations

by John Dick

(1) Wheel Animation with W "Away"

The animations here show six wheels negotiating circular tracks in 4D (WXYZ). All tracks are at "ground" level (Z = 0). All wheels are upright with respect to Z. The lighter colored tracks are at constant W values: Since the upper track is farther "away" the wheels appear smaller there.

This Animation

Ground level (Z=0) in 4D is actually a full 3 dimensional space (WXY) instead of the 2D surface (XY) that we naturally expect. So here (and in subsequent views) there are more kinds of circular tracks on the "ground" than we're used to.

"Away" here means exactly what you see in a 2D photograph of a 3D landscape--nearby objects are larger and nearer the bottom of the picture, while distant ones are smaller and tend to be higher, up near the horizon. What you are looking at is a 3D photograph of a 4D line drawing. This photograph was taken by a 4D camera pointed in the W direction.

With this in mind, it's not hard to understand the two lighter orange tracks (the upper one being farther "away"; that is, it has a larger value for W). When moving on the other 4 tracks (two each in the WX and WY planes), the wheels show varying sizes; becoming smaller as they move "away".

In this animation each wheel always stays on the same track--you'll find more variety in subsequent videos. Question: Can you figure out at what point a wheel is moving directly in the W (away) direction?

Discussion--Is 4D Real?

While a 4D world may seem inconceivable to us, realizing such a world in a computer is not particularly more difficult than for a 3D environment.

Computer games already create rich 3D environments for us to explore. These environments do become real for us as we negotiate the tasks of the game. With ever-increasing PC capability, an expansion to 4D is a very real option. It would make possible a richer environment than anything possible in 3D due to greater possibilities for connectivity and shape.

A 4D environment also has "more" space than 3D does. Hyperbolic 3D geometries (also with "more" space) have been getting some press recently for use in virtual reality (VR) environments. What does it mean to have more space? The VR users report that hyperbolic geometry presents a great many more "nearby" elements than is possible in ordinary 3D. Again, this makes for a richer environment.

Worlds with many different dimensionalities are commonplace in science and are often graphically visualized, 2 or 3 dimensions at a time. The game-like environment shown here gives us a chance to see whether we can learn to make sense of worlds, real or created, 4 dimensions at a time.

Please continue reading the discussion on subsequent animation pages.