It occurs to me that when I was thinking about smaller circles, I was not only thinking about smaller circles but also very small degrees (say one degree) of a larger circle. I suppose those two patterns can be combined whether into spirals, figure 8’s or some other movement combination. An interesting pattern I’m trying to “wrap my head around” (no pun intended) would be a gyroscopic pattern. From wikipedia at time of post:

gyroscopeA gyroscope is a device for measuring or maintaining orientation, based on the principles of conservation of angular momentum. The device is a spinning wheel or disk whose axle is free to take any orientation. This orientation changes much less in response to a given external torque than it would without the large angular momentum associated with the gyroscope’s high rate of spin. Since external torque is minimized by mounting the device in gimbals, its orientation remains nearly fixed, regardless of any motion of the platform on which it is mounted.

Brilliant. If you are at the center “neutralizing with one turn of the waist” it seems there is really some kind of gyroscope-like motion – ideally your orientation is changed minimally due to the external torque or motion of the “platform” but your “waist turn” “throws out” that externally applied force. That throwing or peng or off-balance direction doesn’t seem to happen the way the waist turns on a vertical axis but in a circle or part of the arc in some other direction or plane. At the simplest, it is easy to imagine the diagram showing the thrower turning on a vertical and a horizontal axis as in a basic hip throw but the throwee going over you in a different circle, as in the diagram above. When you apply force to a gyroscope, it doesn’t flop over – it “borrows your force” and spins a different way and “throws you out”. This idea seems more explanatory than the taiji yin-yang diagram and closer to what the experience of throws can feel like when it seems effortless. Conservation of angular momentum also sounds intriguing. I think that explains why it’s easy to balance and turn on a bicycle when it’s moving (the wheels are spinning) but difficult when it is still. The gyroscopic effect when you lean and turn while moving causes a conservation of the angular momentum – meaning you can balance easily because you are aided by this effect, I think. Another sports example is a figure skater who pulls in her limbs to spin faster (less external torque so momentum is conserved?). I don’t quite understand it but it seems another reason why smaller circles are better.

How Stuff Works explains the gyroscope.
Some more rumination after trying to understand these articles: On page 3 of the article it says “When the section at the top of the gyro rotates 90 degrees to the side, it continues in its desire to move to the left. The same holds true for the section at the bottom — it rotates 90 degrees to the side and it continues in its desire to move to the right. These forces rotate the wheel in the precession direction. As the identified points continue to rotate 90 more degrees, their original motions are cancelled.”

That provides an interesting clue to “borrowing force” and sending it back. People tend to try something like that intuitively albeit clumsily at first or in very specific applications, but here is an explanation that could help one understand how to do that more effectively or all the time automatically. I tend to think of the redirection and nearly simultaneous peng happening in only one plane – like the common analogy about a large ball spinning – you push on it and it spins and a tiny part of it pushes right back on you – but it turns out that must be wrong or incomplete at least – adding the gyroscopic effect (which I guess the ball could have) adds a perpendicular plane that better explains how you are then “thrown out” at an unexpected angle. In the Wikipedia picture, the inner ring always rotates on a vertical axis, the middle ring always rotates perpendicular to that, but the outer ring is propelled in a diagonal plane. Because you are “adhering” to the attacker, his motion becomes part of your gyroscopic motion and follows the outer ring or gimbal. Taking this theory farther, if you have some simultaneous gyroscopic effects all over in small circles all tied back to the main one in the dantien area through silk reeling exercises and “intent”, then combine them in spirals, etc. – wouldn’t you get that nice always-on peng quality you want? Then once you get the “feeling” you test it in some very basic push hands and repeat until you have “internalized” it so it happens automatically and ultimately you can maintain this quality in “applications” and “free fighting”. Maybe that’s why the Chen stylists have so many turns that can seem flowery at times. Add to that the observation that uke is always off balance when moving and that is all the more elaborate circular theory (beyond just push-pull) one needs. I prefer to get to a reductionist principle approach. Then I don’t have to think about move back then forward, move around, move in a figure 8, get resistance and go the other way, match speed, attack first, counterattack, etc. – the tiniest move that works effectively that is still based on some “invisible” motion is what “internal” seems to mean. If there is some way to get this internal spinning gyroscopic effect even though there seems to be nothing actually turning, maybe that would help, at least as a theoretical concept. Many people talk about a turning sensation in the dantien. That is getting into qi and TCM that other people don’t want to go into and seems even more difficult to understand. Back to gyroscopes and humans, according to the Wikipedia article, one possible and experimental application of gyroscopes is as an implant in the brain to help with balance if there is trouble with the inner ear so there is probably something to it. The How Stuff article also links to an article on the Segway that uses several gyroscopes and bases its balance mechanisms on human balancing. There is some other material on gyroscopes for airplane autopilot, space shuttle navigation, etc. Hmm…