4D to 3D Projection Animations

by John Dick

Introduction

It is amazing how little has changed in the past 100 years in our understanding of how we might meaningfully interact with a world that has 4 dimensions. In particular, even though the technology for realizing and presenting 4D environments has advanced in wonderful ways, the perception persists that it is not possible to view such a world in its entirety.

This is most definitely not the case! I hope to convince you that, even though a complete view of such a world may be technically challenging, an accurate 3D view of 4D line drawings (instead of 4D solids) can be both substantial and evocative.

What is not so clear, though, is whether such a view can ever make real sense to us 3D'ers. The animations here will give you a chance to test this for yourself. I hope that maybe it will.

Please view the animation pages for more discussion.

The Animations

The animations here were created by projecting 4D line objects onto the 3D space that you see. This is the exact equivalent of drawing or painting a landscape scene (3D) onto paper (2D). When we look at that drawing or painting we see "into" the picture, correctly understanding the original scene. The big question is: Can we learn to look "into" these 3D animations in the same way?

The 4D environment here likely makes more sense than some that you may have seen. This is due to the introduction of familiar elements that we implicitly understand: wheels that are (mostly) like wheels that we know, and, more importantly, a floor or ground that is defined by one special "up/down" dimension (Z), and which also defines the "upright" dimension for the wheels.

Most of the animations include a rotation of the point-of-view either in 3D or 4D. Think of walking around the scene while filming it. As you will see, the "extra" dimension (W) is completely equivalent to two of the other 3 (X and Y). For example, some of the rotations that you will see (those in the WX or WY plane) will cause the "extra" dimension (W of WXYZ) to come directly into view, either X or Y becoming the dimension we must infer; the "away" dimension.

Click on any of the drawings below to show the associated video and discussion. While the video will repeat indefinitely, please be patient the first time through: depending on your internet connection it may take some seconds to download and play smoothly. Click on any animation to view it full-sized. (<alt><left-arrow> should get you back)

Some viewing hints: Follow just one of the wheels with your eyes if the whole picture is confusing; Viewing at arm's length may help to get "into" the picture.

---------

(1) Wheel Animation with W "Away"

This animation shows six wheels negotiating circular tracks in 4D (WXYZ). All tracks are at "ground" level (Z = 0). All wheels are upright with respect to Z.

(2) Wheel Animation with added XY Rotation

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

(3) Wheel Animation with added WX Rotation

This animation shows an added 180 degree WX rotation of the viewpoint. The thumbnail shows the configuration half way through the WX rotation where the W dimension is shown and the X direction is "away".

(4) Wheel Animation with added WX and XY Rotations

This animation shows added 180 degree WX and XY rotations of the viewpoint in 4D (WXYZ).

(5) Wheel Animation with added WY and XY Rotations

This animation shows added 180 degree WY and XY rotations of the viewpoint in 4D (WXYZ). The thumbnail shows the configuration half way through the WY rotation where the W dimension is shown and the Y direction is "away".