## Buddhagram Rotations

One can produce series of Buddhagram projections representing a rotation in 4-space and arrange the images into animations. The math for the rotation and projection is done exactly as is done from 3D to 2D. The resulting animations will sometimes appear to be simple rotations of 3D objects whereas others will appear to turn a 3D object inside-out. The key again is to remember that all the animations are valid projections of the identical underlying 4D object. The mind wants to seize on what appears to be a rigid 3D object and assume it's the entire object but one must remember that any apparent 3D object is really just the shadow of a 4D object. Looking at this object from a number of different directions and rotations will never give you a solid understanding of the full 4D object but it will allow you to become very familiar with portions of it and also with the range of possible 3D projections that can be generated from it.

The following set of images are each linked to an animation in MPG format. Each image is a Buddhagram rendering of the 4D Mandelbrot/Julia Set object projected onto one of the six major planes in 4-space. (as opposed to the 3 major planes in 3-space.) Each animation begins with the same standard projection onto the {Zr,Zi} plane and rotates from there to the one in each picture. Each animation shows a complete 360° rotation through the linked image and back to the standard projection. In other words, each image you see below can be found as one of the frames in the animation it links to.

 Zr,Cr Zr,Ci Zi,Cr Zi,Ci Cr,Ci

The first 4 animations all appear to be simple 3D rotations, but again, remember that they are all renderings of the same single object. The last animation is very strange with the standard Zr,Zi image appearing to constantly turn inside-out and upside-down while passing through the Cr,Ci image.

There are 5 images above because from the Zr,Zi major 4D plane there are 5 different 90 degree rotations that will take it into each of the other 5 major planes (denoted by the labels). By analogy, imagine a 3D coordinate system with X, Y, and Z axes. There is one major plane defined by each pair of axes X-Y, X-Z, and Y-Z. So in 4-space with the dimensions named Zr, Zi, Cr, and Ci, there are 6 ways to choose 2 axes, Zr-Zi and the other 5 listed in the table above.

You may be wondering whether there are any other 90 degree rotations that don't involve the standard Zr,Zi plane? You bet! There are 10 more. Here's one that rotates from the Zi,Ci plane through the Zi,Cr plane and back again:

The frame shown above is rotated exactly halfway from Zi,Ci to Zi,Cr. Notice that you can see features of both those major planes in it.

Note that the visual center of the 4D Mandelbrot/Julia Set object is not at the origin. The visual center appears to be centered at {Zr,Zi,Cr,Ci} coordinates = {0,.4,0,.4}. All the rotations on this page rotate about that point in order to keep the figure roughly centered at all times.