by wendy » Wed Apr 20, 2005 2:42 am
I consider the clifford-rotations more fundemental than the great-arrow ones in four dimensions.
A clifford rotation can be made from having equal orthogonal rotation-speeds: this makes every point rotate at the same rate, in great arrows around the centre of the sphere. From any point on the surface, one can not detect any "primary" great arrows, since any pair of orthogonal great arrows make for a parallel set.
In the quaterions, one makes a standard rotation by means of complementry clifford rotations.
In the last message, i discussed the nature of rotations of the glomochoron (4-sphere), in terms of a bi-glomohedric prism and pyramid. I shall now elaborate on what this means.
A glomohedrix is the surface of a three-sphere, it literally means the shape of a 2d cloth (hedrix), bent into positive curvature (globe-shaped, or glomic). We can for any great circle, define two rotation-directions, a N and S pole. This means that any rotation of a 3-sphere (glomohedron) can be represented by defining the unique North pole.
In four dimensions, we have a rather more complex set of rotations, but it is best to start from the clifford rotations. When you "follow" a set of clifford rotations, from going along one great circle, and keeping the orthogonal fixed, the close great circles appear to spiral once around you. If one follows this spiraling further, what happens, is that one gets the shape of a glomohedrix, or 3-sphere surface, where antipodes = orthogonal rotations.
For a set of orthogonal great circle, i can rotate these "left" and "right" by unique values, so that every value arrives one to one at a great arrow. This is used in quaterions, as the basis of their rotation: ie one can derive any arrow from any other by qXq' for some values of q.
I gave earlier coordinates for the shape of great arrows in 6 and 7 dimensions. Let's start with the 7d figure.
At x,y,z,1,0,0,0 you see in the xyz space, a sphere-surface.
At 0,0,0,-1,x,y,z you see another sphere-surface
We now consider the middle three coordinates, where one sees just a polar axis, and the sphere is represented by two point (N and S pole of some rotation). Each of these correspond to a great arrow.
At 0,0,x,y,z,0,0 you see a zigzag line.
This zigzag line, runs like four sides of a tetrahedron, that leave out two opposite edges. Looking down the 'y' axis, we see a square, representing all the rotation-modes derived from these pair of great arrows.
The vertices of this square correspond to the left and right clifford axies, rotation forwards and backwards. The order around the square is, eg Lf, Rf, Lb, Rb.
The progression down a given line corresponds to reversing the speed of the "slower" rotation from +1 to -1 (ie converting one orthogonal rotation from L to R.).
The plane at y=0 cuts this zigzag at four points, the vertices of a square. These vertices correspond to four great arrows, the ones on each diagonal correspond to the same great circle direct and reversed.
The nature of rotations is that from any starting point, the laws of balancing energy wil make the mode of rotation move to a clifford-point, namely, one of the two ends of the rotation (usually the closer end). So if something is rotating freely in a great arrow, tidal friction would tend to make it rotate in a clifford-motion.
W