by wendy » Thu May 18, 2006 10:09 am
A "great arrow" rotation is a simple rotation in say, wx plane, holding yz constant. This is the sort of rotation that a wheel would do, since the wx gives "height + forward" while the yz space gives "steering". It is hard to steer your car if the steering axis is rotating!
If one sets the yz plane also rotating, then one gets a "double-rotation". In simple terms, consider a point where y=z=0. This still rotates in a wz plane as it did before. As one goes further out, one starts rotating separately in the wx and yz planes. You can for example, imagine a circle in the wx plane as a cylinder that has been bent end to end (like a torus). If you unfold this, then
rotation in wx plane = going higher on the cylinder
rotation in yz plane = going around the cylinder.
A general double rotation then corresponds to going around the cylinder n times as you go upwards. It is kind of like a screw, really.
Regardless of how big you make the cylinder, or how squat, there is exactly the same number of revolutions in yz for each revolution in xy.
When you make these all equal (ie +1 turn or -1 turn per height), something magic happens. All rotations go around the centre, and you can shift the wx and yz axis to wherever you please.
CLIFFORD ROTATIONS
This magic is the clifford-rotation. Under this sort of rotation, all things appear to revolve around the centre point (rather than some kind of extended axis).
If you were sitting on a planet that was revolving clifford-wise, the heavens would not only go from east to west, but would rotate around the horizon. A bright star that rose over a hill, would reach its zenith at an angle 90 degrees from that hill, and set at an opposite point (antipode) of the horizon (remember, the horizon is now a surface sphere).
Because there is no way of deriving a wx or yz axis system from this, it is more fundemental than a single arrow rotation. This particular rotation corresponds to the multiplication by quaterions, for what this is worth.
Clifford rotations come in two flavours (left and right), which amounts to the horizon rotating during the day, such that the direction 'east' is either the north-pole or south-pole (ie it goes clockwise or anticlockwise around the line from east to west).
THE PHASE SPACES
The clifford-rotations form a lot of arrows that each represent a point on a sphere. If for example, some bright star rises at the east point here, then it will reach its zenith directly overhead, and set to the west point. The places on the planet that have this same bright star form a circle. You can map all of the stars in the heavans onto different or (same) circles.
A second star that rises say, 50 degrees from the first, will rise for all points of the globe, the arc between their rising points will be 50 degrees. If one makes a model of rising distances, one gets a sphere whose diameter is not 180 degrees, but say, 90 degrees. This makes a rising sphere.
This sphere surface forms one of the "glomohedrices" mentioned on the product.
The thing with the left and right rotations, is that for a given great circle, there is only one other great circle, which appears in both the left and right rotations. In terms of our cylinder, it is kind of like going up the cylender clockwise or anticlockwise. These are only the same when the cylinder is flat (no height), or infinitely thin (no width). For any other route, there are two distinct paths.
Therefore, a given great arrow that is left-parallel to arrow X, is not left-parallel to any other arrow that is right-parallel to X. That is, every great arrow has a left- and right- parallelness to a right- and left- parallel of X.
This maps onto phase-space as follows:
a great arrow is a point. Left-parallel-ness is in x1x2x3 space, and right-parallelness is in y1y2y3 space. For a given point X1X2X3Y1Y2Y3, one can go first right-parallel to X1X2X3y1y2y3, and then left-parallel to x1x2x3y1y2y3, or go otherwise via x1x2x3Y1Y2Y3.
Replacing x1x2x3 with x, and y1y2y3 as y, we can then see the four special points:
x,y is the great arrow itself. (ie in wx plane)
-x,-y is the great arrow reversed (ie spinning backwards)
x,-y and -x,y is the completely perpendicular arrow (yz plane) spinning in different directions.
This is john conway's model. It only covers great arrows, and not double rotations, etc.
The phase space for double rotations, has to some how include a mode for dealing with a great arrow at speed x, and a reversal at speed y.
So what we have, is then a% x,y and say b% x,-y, that is, different speeds on different axies. (a+b)/(a-b) is the number of times you go around the cylinder as you go up.
We can then say that one is the faster, and have a tetrahedron, forming four sixes of the square, where
1,1,1 = x,y -1,1,-1 is x,-y 1,-1,-1 is -x,-y and -1,1,-1 is -y,x.
The top and bottom edge are the diameter of the sphere representing clifford rotations. Going from say, 1,1,1 to -1,1,-1 slows down the orthogonal rotation y until it becomes 0 (at 0,1,0) and then speeds up to -1. Any faster speeding would then make y the promary axis, and we go down the next edge of the tetrahedron, ie from x,-y, we make x more -x.
The points at 0,x,y are the great arrows rotations, that is, that which has a static inverse. The values at, say 1,x,0 or -1,0,y are the clifford-rotations, and all other point are "double rotations".
The pyramid product is then in seven dimensions, the points
2,2x1,2x2,2x3,0,0,0 to -2,0,0,0,2y1,2y2,2y3
which passes through the six-dimensional prism (like the mid section of a tetrahedron edge 2 is a square edge 1), gives
0,x1,x2,x3,y1,y2,y3
removing the 0 gives the six dimensional bi-glomohedric prism.
We can replace a range of +2 to -2, for perpendicular axies, by an line from 2 at x1,x2,x3,0,0,0 to -2 at 0,0,0,y1,y2,y3. This gives a tegum, which in turn can be expanded out to a glomopetix (sphere in 6d, literally, globe-shaped 5-fabric).