The necessary abstraction
A rotation in 2D is simple. There is one plane, and we specify an angle of rotation in this plane. We may say that the plane as a whole is invariant, in the sense that the plane returns to itself after any rotation. Trivially.
A rotation in 3D is not so different, if one chooses a natural coordinate system. We pick {x,y,z} such that a rotation affects any vector lying in the x-y plane, but does does not affect a vector lying along the z-axis. In this sense we say there is one invariant line, and one invariant plane.
A 4D rotation is a new animal. The generalization from 3D to 4D is not obvious, but is so simple to state that one can easily deceive oneself into thinking, but that's only natural! One is able to pick 4 orthogonal axes {w,x,y,z}, such that any vector in the w-x plane is rotated by an angle #1, and any vector in the y-z plane is rotated by possibly a different angle #2. Thus, there are two invariant planes, no invariant lines. We may specify a 4D rotation by asking, what directions do these two planes lie, and what is the angle of rotation in each plane.
5D, 2 invariant planes, one invariant line.
6D, 3 invariant planes.
...
For any rotation in 2N dimensions, we have N invariant planes, and N angles in each plane. If the magnitude of all angles are equal, we call such a rotation equiangular. Any equiangular rotation can be represented by a matrix: exp(iJ#), where # is the single angle, and J is a 2N dimensional matrix satisfying: J*J = identity; J is antisymmetric. There is no "succession of rotations of the same angle, in different directions", as you asked. The truth is more remarkable. Pick any vector in 2N-dimensional space. By application of an equiangular rotation, it will be rotated by an angle #, no matter which direction one picks. It is a simple exercise to prove this, from what I have told you about the matrix representation.
As I was thinking of how to explain 4D rotations, I recall our multi-dimensional conversation. It still seems impossible to me to absolutely visualize 4D, yet the mind seems to get around it, by abstraction and piece-wise comparison. I can't simultaneously imagine {w,x,y,z}, but orthogonality is an easy concept, which I can apply to piecewise-comparisons of {w,x}, then {w,y}, etc. When I'm trying to imagine two orthogonal planes simultaneously, I sometimes hold the {w,x} plane in my left hand, and the {y,z} plane in my right hand. Having a different spatial origin for each plane (not actually correct) makes it easier to imagine that these planes are completely non-intersecting. It seems some truths I must let go.
In my paper, I derived the eigenspectrum of a product of two equiangular rotations, each of the same angle. The technique of the proof is beautiful, and I hope it is powerful enough to give a deeper characterization of equiangular rotations, and perhaps any rotation. I didn't think about it during our life-ambition conversation. Here's one for the list. I would like to make an abstract mathematical statement that has far-reaching, if unpredicted, applications. One night out of a hundred, I go to sleep thinking, I should have been a mathematician, theirs are the ideas that are ripe for reinvention, begging for diverse applications, perhaps forgotten for centuries, then reincarnated in unexpected glory. Then I wake up.
A rotation in 3D is not so different, if one chooses a natural coordinate system. We pick {x,y,z} such that a rotation affects any vector lying in the x-y plane, but does does not affect a vector lying along the z-axis. In this sense we say there is one invariant line, and one invariant plane.
A 4D rotation is a new animal. The generalization from 3D to 4D is not obvious, but is so simple to state that one can easily deceive oneself into thinking, but that's only natural! One is able to pick 4 orthogonal axes {w,x,y,z}, such that any vector in the w-x plane is rotated by an angle #1, and any vector in the y-z plane is rotated by possibly a different angle #2. Thus, there are two invariant planes, no invariant lines. We may specify a 4D rotation by asking, what directions do these two planes lie, and what is the angle of rotation in each plane.
5D, 2 invariant planes, one invariant line.
6D, 3 invariant planes.
...
For any rotation in 2N dimensions, we have N invariant planes, and N angles in each plane. If the magnitude of all angles are equal, we call such a rotation equiangular. Any equiangular rotation can be represented by a matrix: exp(iJ#), where # is the single angle, and J is a 2N dimensional matrix satisfying: J*J = identity; J is antisymmetric. There is no "succession of rotations of the same angle, in different directions", as you asked. The truth is more remarkable. Pick any vector in 2N-dimensional space. By application of an equiangular rotation, it will be rotated by an angle #, no matter which direction one picks. It is a simple exercise to prove this, from what I have told you about the matrix representation.
As I was thinking of how to explain 4D rotations, I recall our multi-dimensional conversation. It still seems impossible to me to absolutely visualize 4D, yet the mind seems to get around it, by abstraction and piece-wise comparison. I can't simultaneously imagine {w,x,y,z}, but orthogonality is an easy concept, which I can apply to piecewise-comparisons of {w,x}, then {w,y}, etc. When I'm trying to imagine two orthogonal planes simultaneously, I sometimes hold the {w,x} plane in my left hand, and the {y,z} plane in my right hand. Having a different spatial origin for each plane (not actually correct) makes it easier to imagine that these planes are completely non-intersecting. It seems some truths I must let go.
In my paper, I derived the eigenspectrum of a product of two equiangular rotations, each of the same angle. The technique of the proof is beautiful, and I hope it is powerful enough to give a deeper characterization of equiangular rotations, and perhaps any rotation. I didn't think about it during our life-ambition conversation. Here's one for the list. I would like to make an abstract mathematical statement that has far-reaching, if unpredicted, applications. One night out of a hundred, I go to sleep thinking, I should have been a mathematician, theirs are the ideas that are ripe for reinvention, begging for diverse applications, perhaps forgotten for centuries, then reincarnated in unexpected glory. Then I wake up.
posted by Dr. Broke at 1:26 AM
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