N Dimensional Rotations and Magnetism

[PatrickPowers]

How many distinct rotations are possible in ND? This is the same thing as asking how many distinct magnets are possible.

There is a crucial distinction. Is it possible to identify the different rotations? It makes a difference. This is true even here in 3D. If you have no way to orient yourself then there is only one possible rotation in 3D. If you look at the Earth from above the north pole then the earth is rotating counterclockwise. If you look at the Earth from above the south pole then the earth is rotating clockwise. If on the other hand there is some way for the observer to orient themselves, by using the magnetic field or stars or whatever, then there are two possible rotations.

In 2D there are two distinct rotations, as the 2D observer can't help but have an orientation. So let's exclude that and consider the non-orientable observer cases. In 4D there are also two, as the two rotations can be compared against one another. In 5D only one. Essentially the distinction is that in odd dimensional spaces there is always an axis, a null dimension that isn't rotating. One can use this dimension to rotate so that any given rotational plane appears to change direction.

So how many distinct rotations are there in a 2N dimensional space without orientation? There are always two. The only invariant is the parity of the rotations. Any rotation of the observer may change the sign of two rotations, leaving parity the same.

Now suppose it is possible to order the planes of rotation. Perhaps the rotational periods give us an ordering. Then there are 2^(N-1) distinct rotations for 2N > 2. Same with the possible number of types of magnets. If the magnetic planes are of differing strength then there are the same number of possible different magnets.

If you have some way to orient the first plane in the order, then you get a full 2^N distinct rotations.

[wendy]

There is not a lot of what i could find, but i given up searching the literature, because of the wrong approach.

In even dimensions, there is 'planetary rotation', which can be view in complex space, by the equation of a line. This means, in CEn, which maps onto E2n, the equations that generate a straight line in En, but treated as complex numbers, gives CEn. The lines that pass through the origin, can be multiplied by \(cis\ \omega\t\) which, for the mapping of CEn onto E2n, every point lies on a great arrow that orbits the origin at the same angular speed.

The contention is then that every rotation in E2n is comprised of n possible sets of these, each one uniquely distinct. So in six dimensions, we ought find three 'orthogonal' planetry rotations, in eight, four such rotations. The planetry rotation in any given dimension has the same rank as not exceeding that dimension, so in 3d, it is 1 3-d rotation, in 4d, it's 2 3d rotations, in 5d it is 2 5d rotations, and so forth. In 2d, it is 1 1d rotation.

That is to say, the full field of rotations, can be imagined by x controls each of y dimensions. x is half the even number not exceeding n, and y is the largest odd number not exceeding n.

In two dimensions, you have 1 1d control. A slider, that goes from -u to +u representing its rotation speed.

In three dimensions, you have 1 3d control. In essence, you move a point in 3-space, the radius represents the speed, and the position is the north pole.

In four dimensions, you have two 3d controls. These control the left and right isoclinal rotations. Whether the resulting rotation is left- or right- handed, depends largely on which control has the bigger speed. When the two speeds are set equal, you get a 'wheel' rotation, where all axies except two are fixed.

In a motor car, one has the wheel as in the hedrix (2space) of height and forward. The wheel works by swinging the axle against a succession of points at the rim, which produces motion. Any other rotation at the axle will turn the cabin of the carriage around, making travelling a dizzying experience.

Five dimensions is still foggy, although the controlls are known to be 2 5d controlls. I have seen one of these.

[wendy]

Magnetism in 3d is a curl, but it can also be regarded as the outcome of retarded potential.

Part of the reason that i am looking at circulation in higher dimensions, is because of eddies in fluids, including magnetism.

If one supposes that magnetism is the interaction between moving charges, then \(\vec H\) is a ray pointing from the source in all directions, but modified by the the direction of the motion, and \(\vec B\) appears to a moving charge in the plane (n-1 space) orthogonal to its motion.

The study with the out-vector tells us that any loop (boundary of an n-1 patch in n-space), bounds a specific vector area, there is an area moment given by the vector area by the intensity of circulation. This generalises the magnetic area dipole \(\vec m = I \vec A\), where A is n-1 space, and I is a numeric.

This particular relation derives from the definition of volume = moment of area, and that the sum of the area-vector is zero, because the volume is not dependent on position. This means that if we take an n-balloon, and cut a hole into its interior, the necessary patch for it is the same vector area regardless of shape.

[PatrickPowers]

Aha, I neglected to mention that I assumed the number of rotations is always maximal. If you allow some of the planes to be still then the number of "rotations" is greater.

What I have in mind is the signs of these rotations, either + or -. So in 8D the signature of a maximal rotation would be something like +--+. If the observer can't tell which rotation is which then in even-D spaces rotation of the observer can transform this signature to any other signature with the same parity. Signature +--+ can go to +-+- but can't become +-++. So there are two equivalence classes of these maximal rotations.

[PatrickPowers]

If one supposes that magnetism is the interaction between moving charges, then \(\vec H\) is a ray pointing from the source in all directions, but modified by the the direction of the motion, and \(\vec B\) appears to a moving charge in the plane (n-1 space) orthogonal to its motion.

I would say that electromagnetism is the interaction between moving charges.

By relativity we can always say that one of the charges is motionless. That point combined with the velocity vector of the other charge defines a two dimensional plane.

Then we can say that the other charge is motionless and get a different 2D plane. The two charges see things differently.

[wendy]

I have plotted out the number of different rotations in 2N and 2N+1 dimensions to be N!.

I'll play around with the model tomorrow, to see if it all makes sense.

[wendy]

The phase space for rotations can be thought of as the background for a slider, where every possible rotation and speed is represented as a different point. This includes those rotations at different frequencies.

The model is as in radiant space, each point of which represents a cartesian product of the shape given at the sizes on the axies, against whatever the axies represents. So the point (1,2) would represent a product of sphere-surfaces (that being the stated axis here), of a size 1 sphere in prism product with a size 2 sphere. The solid product would require all the points in the rectangle 0,0 to 1,2 be included,

The altitude space is essentially space divided by direction.

For dimensions 2N and 2N-1, the sphere on the axis is the surface of a 2N-1 sphere. In odd dimensions, this corresponds to an polar axis to a 2N-2 rotations. In even dimensions, it corresponds to a swirlybob, such as taking the slope of a line in CEn, This has n-1 defining equations, (eg z=ax, z=by, ... in CE3), and the sphere is generated by projection from the point where the line from (1,0,0,...) to (0, a, b, ..) crosses the sphere whose diameter is (0-1, 0,0,...).

For 2N and 2N+1, there are N separate forms of this sphere, each in a sense chiral to the others. This is represented by a coordinate system, where each axis represents one of these spheres, the coordinate of the axis represents the intensity or speed of rotation. The combined coordinate then represents a sum of these rotations, which can add and subtract the relative rotation in a given direction.

The point representing any rotation is a single point in this space, each axis supplying further a direction in space for its range of directions. Thus in six dimensions, the altitude space is 3d, ie x, y, z, and these are mapped onto 15 axies as x1, x2, x3, x4, x5, y1, y2, ..., y5, z1, ..., z5.

Where x=y=z=..., this corresponds to a great arrow or wheel rotation. This is where two coordinates rotate, and the rest remain static. generally we can suppose x>y>z..., where all points are in motion, often in varying helix-on-helix motions rather than a simple orbit. This is the motion in 4d, when one speed is faster than the other is for the faster one to spiral around the slower one. If x, y is such a spiral, then y, x is the opposite hand of that same spiral.

The thing is that we can tell a left-screw from a right screw in 4d, which is the outcome of x>y vs y>x, The thing then matters that where x>y>z against all other combinations, represent distinct rotations (free from coordinates). Also, does x>y=z represent a case with non-turning axies.

[mr_e_man]

Five dimensions is still foggy, although the controlls are known to be 2 5d controlls. I have seen one of these.

Why not just say there's one 10D control? Can the two 5D controls really be separated?

The point representing any rotation is a single point in this space, each axis supplying further a direction in space for its range of directions. Thus in six dimensions, the altitude space is 3d, ie x, y, z, and these are mapped onto 15 axies as x1, x2, x3, x4, x5, y1, y2, ..., y5, z1, ..., z5.

What rotations do these 15 axes represent, exactly?

In 4D space, with axes e1,e2,e3,e4, the rotation phase-space has axes x1,x2,x3, y1,y2,y3, where x is left-isoclinic and y is right-isoclinic. The x1 axis represents rotations generated by the bivector e1e4+e2e3; that is, a point v=(v1,v2,v3,v4) gets rotated to

(v1 cosθ - v4 sinθ, v2 cosθ - v3 sinθ, v2 sinθ + v3 cosθ, v1 sinθ + v4 cosθ).

The other axes correspond to these bivectors:

x1, x2, x3:
e1e4 + e2e3, e2e4 + e3e1, e3e4 + e1e2

y1, y2, y3:
e1e4 - e2e3, e2e4 - e3e1, e3e4 - e1e2

So what would this look like in 6D space, with a 15D rotation phase-space?