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.

This Animation

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. 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!)

Too Much Symmetry!

Take a look at the Wikipedia entry Four-dimensional space to view an animation of a 4D "cube" that seems to be designed to boggle the mind. We have no problem seeing a 3D object that is undergoing a beautiful inside-to-outside transformation, but the idea that there's something constant there, a rigid object that's supposedly spinning much the same way that we actually do understand? Hardly.

In principle, the Wikipedia figure seems to be the much the same sort of thing that we are creating here--a 4D wireframe object projected into 3D. However, we find the cube object that we see there pretty much impossible to get a grip on. This is primarily due to the very high amount of symmetry that regular solids have in 4D. We can hardly manage to keep our eye on one part and; Is that part a plane or a volume?

It's also because we haven’t had it explained to us that the reason parts change size is that they’re moving "towards" or "away" from us due to perspective effects that we are perfectly familiar with.

The contrast with the world of our experience could hardly be greater. "Reduced symmetry" might seem like an esoteric concept, but it's really pretty basic: In our personal space, up is not the same as down and forward is not the same as backwards. We are creatures that stand "up" and look "forward". Another way of saying this is that the symmetry in two of our three dimensions is broken. Think of driving a car--the gas and brakes only change how fast things happen: it's the steering wheel that negotiates our world. And that's just a single direction or dimension!

The world created by ten Brock in Miegakure, and the worlds created here, make 4D more accessible by reducing the symmetry with the introduction of "up" and "down"; one special dimension. The animations here further reduce the symmetry with the introduction of wheel objects that define a second special "forwards/backwards" dimension. In our own world, a steering wheel is enough to control the direction of such a wheel: If you were to drive a wheel to negotiate the 4D paths defined here, what would you need? A joystick would do. . .