Formula for rotations in n-space

Higher-dimensional geometry (previously "Polyshapes").

Formula for rotations in n-space

Postby quickfur » Mon Apr 18, 2005 5:15 am

In the course of writing a vector calculator program with arbitrary dimensions, I came across the need for a formula for rotating vectors in arbitrary dimensions about arbitrary planes, and derived a nice formula involving only vector operations:
http://quickfur.ath.cx:8080/~hsteoh/math/genrot.pdf

Just thought I'd share. :-)
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Postby wendy » Mon Apr 18, 2005 5:39 am

Even in four dimensions, there is more than one way to rotate a vector. This is because the "hold n-2 axies" rule does not give all rotations, but rather a particular one: the one i call the "great arrow" rotation.

A great-arrow rotation might be implemented by a pair of reflections. Not all rotations in higher dimensions are great arrows, and hence so implemented.

In four dimensions, the space of great arrows forms a six-dimensional figure that i gloss as a "bi-glomohedric prism". This can be formed by the cartesian product of the surfaces of two orthogonal 3d spheres. Any point on this surface represents a great circle + direction (ie a great arrow).

But in 4d, we have n-2 = 2, and so it is possible to rotate also in a second set of axies. The space of rotations when one accounts for this is a 5d surfae (petix) in 7d, the name for this is a "bi-glomohedric pyramid".

This makes for a product of surfaces of spheres in axies 1,2,3 and 5,6,7, separated by a distance in axis 4, ie

s1: x1^2 + x2^2 + x3^2 = 4, x4 = +1, x5 = x6 = x7 = 0
s2: x1 = x2 = x3 = 0, x4 = -1, x5^2 + x6^2 + x7^2 = 4.

The balance of the points are those points P, that fall on a straight line between points on s1 and s2.

In five dimensions, the space of great arrows makes for the Spin5 group, which i have not figured out what this manifold in 10 dimensions looks like. On the other hand, the admissible rotations in 5d is apparently a manifold in 11d.
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Postby quickfur » Mon Apr 18, 2005 2:59 pm

wendy wrote:Even in four dimensions, there is more than one way to rotate a vector. This is because the "hold n-2 axies" rule does not give all rotations, but rather a particular one: the one i call the "great arrow" rotation.

A great-arrow rotation might be implemented by a pair of reflections. Not all rotations in higher dimensions are great arrows, and hence so implemented.

But aren't the other rotations equivalent to a combination of great arrow rotations? E.g., in 4D, there are two orthogonal planes in which an object can undergo great arrow rotation, so it seems that a fully arbitrary rotation in 4D can be expressed as a composition of great arrow rotations in the two planes.

I've always found this interesting, in that if planets in 4D are indeed possible, then in the general case they'd have two simultaneous and orthogonal spins (or a single "complex spin", if you consider the combination as a rotation proper). Depending on the relative rates of these spins, it would mean very interesting spiralling or twirling paths of the sun across the sky. Fascinating.
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Postby 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.

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Postby wendy » Wed Apr 20, 2005 2:47 am

Yes, indeed, one can compose all rotations from great arrows, and all great arrows from pairs of reflections.

It is important to note here that in 4d and higher, great arrows are not the normal for rotations, and that some modes of rotation require more arrows, usually as many as INT(n/2).

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Postby quickfur » Wed Apr 20, 2005 4:27 am

wendy wrote:Yes, indeed, one can compose all rotations from great arrows, and all great arrows from pairs of reflections.

It is important to note here that in 4d and higher, great arrows are not the normal for rotations, and that some modes of rotation require more arrows, usually as many as INT(n/2).

W

Ah I see. So you're saying clifford rotations are more "normal" for higher dimensions?

Also, your point about 4D rotations tending toward a clifford point is interesting. I've never thought about that before, but it does make sense. If a hypothetical planet is rotating along two orthogonal great arrows at different rates, there would be an energy gradient between the two rotations, which would lead to a transfer of energy from the higher energy component of the rotation to the other, lower energy component. So eventually it will converge on a clifford point. Now, I'd think this same principle should also apply to smaller-scale objects in rotational motion as well; so one would expect that a 3-sphere (glome) would also rotate this way when set in motion! And in fact, any object sent into a tumbling motion would tend toward this kind of motion as well... which means spinning objects in 4D are harder to imagine, but a lot more interesting than I'd previously thought!
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Postby wendy » Thu Apr 21, 2005 11:06 pm

I suppose you could bookmark my polygloss. Every now and then i am adding theme pages, on different aspects of dimensions.

It's http://www.geocities.com/os2fan2/gloss/index.html

The next theme is for clifford-rotations, and how the sky, the starry vault, the rising-spheres, etc work in 4D. Rising-spheres are relatively interesting development in 4d.

While free objects tend to some mode of clifford-rotation under four dimensions, as in three dimensions, the transfer of modal energy is easily stopped, although it generaltes a torque. Unlike 3d, the torque is not something of the kin of v cross w, since there is no cross product in 4D. But this does not mean that there is no torque in that dimension.

One might note, for example, that something set spinning would continue to spin around an axial plane, even as the earth rotates. That is, the axial plane would itself appear to rotate to an observer, rather like Corolis force. But the nature of this force is different, and i suspect it does not make "water swirl backwards in different hemispheres" (the force is way too weak for that to happen in a bath tub or toilet-bowl.)
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