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.

King of the Woods

Structure announces emphatically, “If I could just see one bear.”

Sincere quips, “The bare mass.”

Hat tells me his criteria for a good branch. It must be high enough that a bear standing on its hindlegs can’t reach it, sturdy enough to hold the food of four travellers. It must extend a fair distance from the trunk, so that a bear which has scaled the trunk cannot reach for the food. It must not be too sturdy, lest it tempt the bear to climb onto it.

With rope and branch we fashion a pulley, then struggle with the strength of three men to hoist the food bag to the necessary elevation. Finally, we wound the rope around the trunk of an adjacent tree, and find that Force = mass*acceleration. “Prelim physics works!” Sincere exclaims joyfully. I remember this problem, remember being surprised by the final geometric result: if a rope is wound around a cylinder, the force needed to maintain a certain tension in a rope diminishes exponentially as the length of the winding. Two windings around the adjacent trunk, and I can hold the food bag with a finger.

Bear sightings are rare in the Catskull Mountains. Here, tempestuous rains lash the mountains and transform hiking trails into flowing streams. Rain-slicked sedimentary rocks protrude from puddles, from mud that engulfs, these plate-size rocks serve as stepping stones. Larger boulders are strewn across the landscape, they must be overcome by a combination of hand, foot and graspable tree roots. Footing is treacherous. The pack is heavy. We climb.

Four mountains on the second day. When we finally reach the lean-to, I am debilitated by shooting pains which spike whenever I support my weight with my left knee. I am trailing the group, holding them back. As the rest of the group set up camp, I struggle to a nearby spring to bathe my wounded knee with water as cold as midwinter breeze. The numbing is welcome, but undone by my clumsy exertions as I head back to camp. The rustling of leaves alerts me that somebody is oncoming. Has my group finally noticed that I’ve been gone too long? Why would they hike through brush when there is a perfectly good trail?

A snout protrudes from the foliage, a black-furred head emerges with beady eyes, pointed ears, wide mouth, then its chest and forelegs, I quailed at their girth, they dwarf the limits of human strength, suggest a terrible ease. I still myself, I find once again this familiar sensation, that a moment can be pregnant with death. As this creature is about to cross the trail, it senses me. Its first reaction is surprise, it rears back, as if to retreat. I register a quiet astonishment, that I understand its reaction perfectly, because it is so recognizably human. I sense its fear of me, because its first instinct is to run. My relief is quickly extinguished, because it reconsiders, swivels its snout toward me, and appraises me with alien, undeniable intelligence. How did the bear and I come to be, stumbling across each other in the woods. I recognize you, bear. My human-centered worldview is supplanted by you.

The bear takes a tentative step toward me. This is an awkward time to have a wounded knee. Bear, dare I intuit your intentions? I turn tail and limp briskly back to the spring. I am careful not to run, not to reveal my fear. I am unsure I can run even if I wanted. For ten steps I did not look back, for ten steps I am haunted, my mind narrows to a raw, pulsating sliver, a single thought, that it could be bounding toward me, its claws sinking into my back.

I look back. The bear has not moved. I widen the distance between us.

Until once again the bear feels safe. It crosses the human-made trail and disappears into the trees and brush, a lumbering force of nature, oblivous to branches and leaves. Where it goes, the rustle of leaves follow, alerting every creature, but it heeds not, fears not, it is the king of the woods.