Constraints on Planes of Rotation

[PatrickPowers]

Here in 3D one may either have an object constrained to rotate only in a single plane or have a freely rotating object that can rotate in any combination of the three planes. You can't have an object constrained to rotate in only combinations of two planes. You can't have something free to rotate in axy + byz but not in cxz. But I don't know how to prove that so maybe this isn't true.

It seems to me that a similar thing holds in 4D. One may either have an object constrained to rotate only in a single plane, constrained to rotate in any combination of three planes, or free to rotate in all six planes. What you can't do is have an object constrained to rotate in exactly two planes. Let's say the coordinates are [w,x,y,z]. If the object can rotate in combinations of the wy and xy planes then it must also be able to add in some of the wx plane. Is that right?

[PatrickPowers]

Here in 3D one may either have an object constrained to rotate only in a single plane or have a freely rotating object that can rotate in any combination of the three planes. You can't have an object constrained to rotate in only combinations of two planes. You can't have something free to rotate in axy + byz but not in cxz. But I don't know how to prove that so maybe this isn't true.

It seems to me that a similar thing holds in 4D. One may either have an object constrained to rotate only in a single plane, constrained to rotate in any combination of three planes, or free to rotate in all six planes. What you can't do is have an object constrained to rotate in exactly two planes. Let's say the coordinates are [w,x,y,z]. If the object can rotate in combinations of the wy and xy planes then it must also be able to add in some of the wx plane. Is that right?

Aha now I see it. In 3D have a sphere with a rod through the center protruding from the north and south poles. Have a ring that's larger than the sphere with a slot on the inner edge. Put the ends of the rod in the slot. If the ring is in the yz plane then the rotational plane of the sphere can have any combination of the xy and xz planes but no element of the yz plane. (What was so hard about that?)

Now for the 4D case...

[Hugh]

I asked o3-high-mini to help:

Let’s begin by recalling the idea behind the 3D case and then see how it carries over to 4D.

The 3D Example

In three dimensions the rotation group is generated by rotations in the three coordinate planes: for example, the xy, xz, and yz planes. These rotations obey Lie algebra commutation relations such as

  [J₍xy₎, J₍xz₎] = J₍yz₎

which tells you that if you’re allowed to rotate in the xy and xz planes (and these rotations are “non-disjoint” in the sense that they share the x coordinate), then the algebra forces the inclusion of a rotation about the remaining plane (yz). In other words, if you try to allow rotations in exactly two non-disjoint planes, closure under composition forces you to include the third. (Note that if you restrict an object so that it rotates only about one fixed axis—that is, in a single plane—you have a one-dimensional subgroup, SO(2).)

Extending to 4D

In four dimensions the rotation group is SO(4) and it has six independent planes of rotation. (In fact, SO(4) is “larger” than SO(3) in that sense.) However, the same principle applies when the two chosen rotation planes are not completely independent (i.e. when they “share” a coordinate).

For instance, suppose our coordinates are [w, x, y, z] and we allow the object to rotate in:
• the wy plane (rotations mixing w and y), and
• the xy plane (rotations mixing x and y).

Because these two planes both involve the y coordinate, their generators do not commute. A quick calculation using the standard Lie algebra commutation relations (where for generators J₍ab₎ and J₍cd₎ one has
  [J₍ab₎, J₍cd₎] = δ₍ac₎J₍bd₎ – δ₍ad₎J₍bc₎ – δ₍bc₎J₍ad₎ + δ₍bd₎J₍ac₎)
shows that

  [J₍wy₎, J₍xy₎] = J₍wx₎.

Thus, if the object is free to rotate in the wy and xy planes, it is forced by the algebra to also be able to rotate in the wx plane. In other words, the set of allowed rotations is not closed if you only include two intersecting planes; you must include at least a third.

The Exception: Disjoint Rotation Planes

It’s worth noting that in 4D one can have a two-parameter subgroup of rotations that is “pure” in the sense that the two rotations occur in completely disjoint planes. For example, rotations in the wx plane and the yz plane are completely independent (they act on entirely different sets of coordinates) and commute with each other. This subgroup is abelian and is often described as a torus (SO(2) × SO(2)). However, in Patrick’s example the two rotation planes (wy and xy) are not disjoint—they share the y axis—and that is why the closure of the Lie algebra forces in the third rotation (in the wx plane).

Conclusion

So to help with the 4D case:
If an object in 4D is allowed to rotate in two intersecting planes (such as the wy and xy planes, which share the y coordinate), then the Lie algebra’s closure property forces the inclusion of a rotation in the third plane (wx). In this scenario you cannot have an object that is constrained to rotate in exactly two such planes; you must allow rotations in at least three planes. This mirrors the 3D situation, where any two non-disjoint rotations generate the third.

I hope this clarifies the issue!