I like to study rotations in higher dimensional spaces. It was worked out by Clifford in the late 19th century.(adsbygoogle = window.adsbygoogle || []).push({});

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 factIn a Euclidian 4D space an infinite 2D plane doesA 3D infinite plane or other such thing is necessary to form a partition.notpartition the space.

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.

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# Rotations of 3-spheres

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