by quickfur » Wed May 06, 2015 12:08 am
What's even more cool, is that when the two rates of rotation are equal, all points on the glome travel in great circles instead of spirals. When this happens, the number of stationary planes (2D planes that do not change under the rotation) become infinite, instead of just two. Each circle traced out by each point on the glome interlocks with adjacent circles in an interlocking, spiralling way, forming what we call in maths a "fibre bundle". This is, in fact, the well-known Hopf fibration of the glome.
In this state, the rotating glome loses one degree of orientation, and yet it still retains a sense of the rotation! What I mean by "losing one degree of orientation" is this: in 3D, when a sphere (e.g. the earth) rotates, it has a single stationary plane, which also unambiguously defines a unique north-south axis. When you look at any rotating sphere in 3D, you can always identify this axis and the stationary plane perpendicular to it. So you may say that the axis defines a fixed orientation for the rotating sphere. Now, in 4D, we know that an object, say a glome, rotating only around a single plane does not have a unique axis. However, it does have a unique stationary plane, that lies perpendicular to the plane of rotation. So even though we can't assign a single directional axis to the rotating glome, we can still identify its stationary plane and the perpendicular rotational plane, both of which have a fixed orientation in 4D space. Even when you introduce a second rotation around the stationary plane of the first, as long as the two rates of rotation are unequal, you can still identify a unique rotational plane for the first rotation, and a unique rotational plane for the second rotation lying perpendicular (in fact, orthogonal) to the first. So you can still "orient" the double-rotating glome with these two planes. These two planes are distinguished from other planes because points of the glome that lie on them trace out circles, whereas points outside of them trace out spirals.
However, something strange happens when both rates of rotation are equal: while the original two orthogonal planes still remain as stationary planes, now a whole bunch of other pairs of planes also become stationary planes -- an infinite number of them, actually. This makes it impossible to distinguish the original two orthogonal planes from the infinite number of other orthogonal pairs that also behave like stationary planes! So now you can no longer use the two planes to fix an orientation for the rotating glome. So it has "lost one degree of orientation". All of these stationary planes are completely congruent to each other, so none of them stands out as being a special landmark that you could use to orient the glome. So in a sense you could say this rotating glome is "non-orientable" (not in the mathematical sense, though).
The bizarre thing, though, is that even this "non-orientable" rotating glome still has a distinct "sense" of rotation. In 2D, there's a distinction between clockwise and anticlockwise, and in 3D, while technically this distinction is absent, in its place we have the direction of the rotational axis which we can use to discriminate between different orientations of rotation. In 4D, however, the double-rotating glome does not have this distinction, because the infinite number of stationary planes means that no matter how you roll the glome around, it makes no difference at all to its rotation! However, at the same time, this crazy orientation-less double rotation actually comes in two breeds, which are distinct from each other! Their distinction is not in clockwise/anticlockwise, nor in orientation ('cos all orientations are equivalent), but in the way the great circles spiral around each other. One breed has the great circles spiral around like a left-handed helix, while the other has the circles spiral around like a right-handed helix. And no matter what you do, it's impossible to rotate one breed into the other -- they are distinct double rotations, yet they have no orientation either!
Furthermore, this strange effect only exists in 4D: in 5D and beyond, you can always rotate the glome through the additional dimension(s) such that a left-handed double rotation becomes a right-handed double rotation. It's only in 4D where this distinction exists.
Weird, huh?