Rotations of 3-Spheres: Exploring 4D Space & Beyond

[Hornbein]

I like to study rotations in higher dimensional spaces. It was worked out by Clifford in the late 19th century.

A 3-sphere is a sphere in 4D space. A 3-sphere -- or any other object in 4space with non-zero volume -- can have two planes of rotation. These planes are always at right angles to one another, and may have entirely independent periods of rotation.

To make this more down to Earth I like to imagine being on the surface of a rotating 3-sphere planet. What path would a fixed star appear to follow? If only one plane is rotating then it would look somewhat as it does here on 2-sphere Earth. The difference is that instead of a fixed point for a pole star, it would be a fixed plane, a sort of great circle of fixed stars that the other stars rotate around. Things rotate around axis planes, not axis lines.Weird! It may seem impossible, but recall the fact In a Euclidian 4D space an infinite 2D plane does not partition the space. A 3D infinite plane or other such thing is necessary to form a partition.

If both planes are rotating with equal periods then the stars still move much as they do here on Earth, with the exception that then every point on the 3-sphere is an equator. Every point on the surface moves in a circle of maximum diameter.

If both planes are rotating with a unequal periods then the stars appear to move basically in a circle modulated by a transverse sine wave. If the ration of the periods of the two planes is irrational then the pattern doesn't repeat. The amplitude of that transverse sine wave depends on the location of the observer on the surface of the planet. On one of the equators the transverse has amplitude zero. This amplitude increase with the distance from that equator. Eventually the transverse sine wave takes over and becomes the circle, while the original circle becomes the transverse.

Telling time on such a planet would require two clocks. The rising and setting of stars would be rather complicated, depending on the location on the planet.

Each point on one equator is the same distance from each point on the other equator. The equatorial plane divides the planet into two hemispheres, but each point not on the equator divides its time equally between the two hemispheres. Strange, eh, for a rigid rotation?

It turns out that there is free software available to simulate such things, though that was not the original intent. The apparent star paths are a subset of "3D lissjous figures" in which two of the three cycles have the same period.

Going to even more dimensions, n-spheres have n/2 planes of rotation rounded down. Odd-dimensional spheres have an axis of rotation as well. A 5D planet has two planes of rotations and an axis. A 6D has three planes of rotation and no axis. A 3D lissjous figure with three cycles all with unique periods is more or less a simulation of paths of stars as seen from a 6D planet, projected from 6-space onto 3-space. A Google image search reveals many such figures.

One thing I don't get is that I read that 4D rotations have chirality. That I don't get. It seems to me that it would depend solely on the point of view, but I read that this not so.

 

[mfb]

Telling time on such a planet would require two clocks. The rising and setting of stars would be rather complicated, depending on the location on the planet.

That's an interesting point. We have incompatible cycle times here on Earth and can handle them without a second clock, but two periods for the daily star cycle are certainly more complicated.

One thing I don't get is that I read that 4D rotations have chirality. That I don't get. It seems to me that it would depend solely on the point of view, but I read that this not so.

Project the 3-sphere onto two balls in our usual 3D space where the surfaces are identified with each other. Let the equator of the faster rotation plane be along the z-axis, consider the ball where the rotation goes towards positive z. Then the equator of the slower rotation plane should be the circumference in the x/y-plane. It has two possible directions, and we can distinguish them. Mirroring space along a line or a 3D-plane (which we cannot do in 4D) would invert chirality.
I hope I got the mathematics right. Hard to imagine 4D-spaces and I didn't calculate it.

 

[Hornbein]

Even our own Earth doesn't really have a chirality. Float over the north pole and you see a counter-clockwise rotation. Float over the south and you see clockwise.

The only way to see it as chiral is to compare it to an arbitrary standard, like north or the Sun's rotation.

Even a clock doesn't have chirality. Look at if from the back and the hands move counterclockwise.

To get a truly chiral object you have to have something like a tetrahedron with four distinct vertices. Even that isn't chiral if it's embedded in 4 space.

My guess is that the 4D Earth is the same way. You can tell what chirality it has only by comparing it to some artificial standard. But in that case it might have four "chiralities." ++ +- -+ --. I dunno.

 

[mfb]

Even our own Earth doesn't really have a chirality. Float over the north pole and you see a counter-clockwise rotation. Float over the south and you see clockwise.

Right, you need 4 dimensions to have chirality. Or 2. Probably every even number will work.
Note: In a 2D-world, you can try to "view it from the opposite side", but to do so you would have to embed it in a 3D space which we don't do.

 

[Hornbein]

Right, you need 4 dimensions to have chirality. Or 2. Probably every even number will work.
Note: In a 2D-world, you can try to "view it from the opposite side", but to do so you would have to embed it in a 3D space which we don't do.

So we are confined to a 2D space. To conclude that a rotating object has an inherent chirality we have to assume that the observer has an inherent chirality that he/she/it can compare with. Usually these flatland things assume that, but it doesn't have to be the case. The observer doesn't have to be rigid. The observer could be an amoeba that can reverse to its mirror image without leaving the plane.

But the observer can compare two 2D objects and see whether or not they are rotating with the same chirality.

 

[mfb]

The observer could be an amoeba that can reverse to its mirror image without leaving the plane.

Then the amoeba is not chiral. So what? It can still distinguish two chirality classes using external tools.
It cannot label those in a unique way unless the physics violates parity (it does in our universe), but it can still sort all rotating objects into two classes, that is sufficient.

 

[Hornbein]

Then the amoeba is not chiral. So what? It can still distinguish two chirality classes using external tools.
It cannot label those in a unique way unless the physics violates parity (it does in our universe), but it can still sort all rotating objects into two classes, that is sufficient.

Right. So if we were to observe a planet in 4D, I think it would be necessary to have some arbitrary standard to classify that planet as to the chirality of its two rotations. In terms of linear algebra, the eigenvalues of the rotations either have the same sign or different signs. (I hope. It's been a long time since I did linear algebra.) But this labeling is arbitrary.

 

[mfb]

Right. So if we were to observe a planet in 4D, I think it would be necessary to have some arbitrary standard to classify that planet as to the chirality of its two rotations.

Sure, in the same way our chirality assignments to objects in our 3D space are arbitrary. Why do we call the proteins (most of them) produced by life on Earth "left-handed"? A completely arbitrary assignment.

 

[Hornbein]

4D rotations exist in the real world in the Minkowski geometry of relativity.

 

[Hornbein]

I like to study rotations in higher dimensional spaces. It was worked out by Clifford in the late 19th century.

A 3-sphere is a sphere in 4D space. A 3-sphere -- or any other object in 4space with non-zero volume -- can have two planes of rotation. These planes are always at right angles to one another, and may have entirely independent periods of rotation.

To make this more down to Earth I like to imagine being on the surface of a rotating 3-sphere planet. What path would a fixed star appear to follow? If only one plane is rotating then it would look somewhat as it does here on 2-sphere Earth. The difference is that instead of a fixed point for a pole star, it would be a fixed plane, a sort of great circle of fixed stars that the other stars rotate around. Things rotate around axis planes, not axis lines.Weird! It may seem impossible, but recall the fact In a Euclidian 4D space an infinite 2D plane does not partition the space. A 3D infinite plane or other such thing is necessary to form a partition.

If both planes are rotating with equal periods then the stars still move much as they do here on Earth, with the exception that then every point on the 3-sphere is an equator. Every point on the surface moves in a circle of maximum diameter.

If both planes are rotating with a unequal periods then the stars appear to move basically in a circle modulated by a transverse sine wave. If the ration of the periods of the two planes is irrational then the pattern doesn't repeat. The amplitude of that transverse sine wave depends on the location of the observer on the surface of the planet. On one of the equators the transverse has amplitude zero. This amplitude increase with the distance from that equator. Eventually the transverse sine wave takes over and becomes the circle, while the original circle becomes the transverse.

Telling time on such a planet would require two clocks. The rising and setting of stars would be rather complicated, depending on the location on the planet.

Each point on one equator is the same distance from each point on the other equator. The equatorial plane divides the planet into two hemispheres, but each point not on the equator divides its time equally between the two hemispheres. Strange, eh, for a rigid rotation?

It turns out that there is free software available to simulate such things, though that was not the original intent. The apparent star paths are a subset of "3D lissjous figures" in which two of the three cycles have the same period.

Going to even more dimensions, n-spheres have n/2 planes of rotation rounded down. Odd-dimensional spheres have an axis of rotation as well. A 5D planet has two planes of rotations and an axis. A 6D has three planes of rotation and no axis. A 3D lissajous figure with three cycles all with unique periods is more or less a simulation of paths of stars as seen from a 6D planet, projected from 6-space onto 3-space. A Google image search reveals many such figures.

A planet in an even number of equivalent dimensions would have no Coriolis effect. Would every point on the surface travel the same distance per rotation as every other point? I don't know how to prove or disprove it.

An evenD Universe wouldn't have magnetic poles. Would a magnet have magnetic planes where the flux is perpendicular to the magnet? I guess Clifford worked this all out, but I'm not going to learn his algebra, elegant though it is.

So what would the path of a star look like on a 6D planet? It would be a semicircle with two sine waves superimposed on it. That is, if we had 6D sight we would see the two sinusoidal motions as orthogonal. We can't do that, but we could imagine the sinusoids as different colors. Then we can scale up to any number of dimensions.

 

[Hornbein]

Right, you need 4 dimensions to have chirality. Or 2. Probably every even number will work.

The experts say that only dimensions 4 or 2 have chiral rotations.

 

[mfb]

There are left-isoclinic and right-isoclinic rotations in 4 dimensions. And the same should apply to general "double" rotations.
Wikipedia article

 

[Hornbein]

There are left-isoclinic and right-isoclinic rotations in 4 dimensions. And the same should apply to general "double" rotations.
Wikipedia article

The expert said that given another two dimensions it was possible to transform either of the Clifford rotations into the other. Me, I don't know.

 

[Hornbein]

The expert said that given another two dimensions it was possible to transform either of the Clifford rotations into the other. Me, I don't know.

I worked through the math. Yes, 4D rotations are not chiral in 6D. But 6D rotations are chiral in 6D. And so forth for N dimensions.

You can prove this by looking at the parity of rotation matrices. By parity I mean looking at the signs of the elements of a rotation matrix. We normalize the matrix to be diagonal. Rotating that matrix 180 degrees changes the sign of two of the elements. If all of the diagonal elements are non-zero, then the signs of the elements can be changed only in pairs. So +-++++ can never be rotated into +--+++ in 6D. But +-++00 can be rotated into +--+00.