Rotation Phasespace

Higher-dimensional geometry (previously "Polyshapes").

Rotation Phasespace

Postby wendy » Sun Mar 16, 2014 7:29 am

This is neato-stuff i wrote elsewhere, but i put it here because it puts in all the missing links. Just wish i understood it. :(

Imagine you had a thing in 2d space. It just basically rotates, and the nature of this rotation can be controlled by a point on a line. You move right if you want clockwise, and left if you want anticlockwise, the distance is the speed of rotation.

In 3D, the rotation-space is 3d too. A sphere can turn in any direction, but you can use the right-hand-rule to determine a 'north pole'. When you plot a point in 3dpsce, you are passing a ray through the north pole, and the length of the ray is the speed it rotates at.

In 4D, it is possible to have a 'double rotation', different speeds in WX and YZ spaces, for example. It's a little more complex, but you get in the realms of left-clifford parallels and right-clifford parallels: in essence, if WX=YZ, the you have one set of parallels, if WX=-YZ, you have another set.

You can easily prove that all points go around a common centre in a circle, if you are familiar with complex numbers. Put X=w+xi and Y=y+zi. Now, you have all lines passing through the point (0,0) are of the form aX=Y. If you multiply both sides by eiωt, then the (complex) gradient does not change, but over a cycle of t

Since the gradient does not change, one can map the argand diagram onto a sphere by way of placing a sphere at centre (0,0,1). A ray from any (x,y) to (0,0,2) will cross this sphere at a particular point. The polar point represents a gradient perpendicular to the plane (ie the Y axis in the previous example).

Since complex numbers only demonstrate a left-turning, there is a matching right-turning set. All rotations in 4D are represented by a sum of a left-rotation and a right-rotation, of potentially different intensities. The left-rotation represents a 3d space, the right rotation also a 3d space, and the product makes a 6d space.

In none of these examples, do we admit more than 360 degrees to a circle. We merely admit the possibilites of multiple circles rotating at the same time.

Unlike 2d and 3d space, the rotation-space for 4d is 'granular', in that a point anywhere in the 6d space, will by the law of equipartition of energy of the modes, will tend to migrate to one of the two orthogonal 3d spaces representing 'clifford-rotations'.
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