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In an Aikido Journal article about Terry Dobson, http://www.aikidojournal.com/?id=3936, he related his one and only question to O Sensei in his ten years of training with him -- to please explain circle, triangle, square. O Sensei's considered response was: "Find out for yourself."
I have gone with that -- looking for an objective reality and not merely a symbolic or spiritual description behind that traditional geometric composite image. What I have gotten to, so far, fits the both the traditional description, a more rigorous physical definition as well as richly connecting to the shape and feel of the dynamics I experience in practice.
This blog entry, on topics I have been mulling over for some time now, was prompted by a discussion that was tending to elevate subjective "feel" over objective understanding of physical action. The analogy initially used was in driving, and the upshot was that driving did not require one to design or alter the basic configuration of the car. I like flying as a better analogy. It melds into one thing what the driving analogy broke into two parts-- flying involves both altering physical configuration of the vehicle of the skill as well as the skill of employing it in any given configuration. In the case of aikido, the vehicle or tool of the skill is the body (or bodies) involved. In training we gain skill in use but we also alter the way the tool functions by that use .
I know for fact that I would be quite dead now if I had not been taught the counterintuitive physics necessary to recover from inverted spins and in my later experience with helos, that of vortex ring state. Both involve overcoming a powerful "feel" for certain "normal" reactions acquired in more ordinary flight, largely induced by kinetic illusion.
Kuzushi, among other things, occurs for most people who have not been trained, because their body and mind react automatically to the "feel" of something with action that in their ordinary experience would salvage their balance, but the simultaneous in-yo dynamic in of aiki being actively exploited by someone with training, actually compels those same unconscious changes to make their situation far worse. Illusions of perception are powerful without an objective reference. Sometimes they are dangerous even when there is one. In the famous ambiguous image, at one moment you see the faces, and then the next you see the goblet.
Another good example of tricks the perceptual system plays with orientation I was recently shown is here:
Depending on how one views it, the animated figure of the dancer rotates clockwise -- or counterclockwise. The kinesthetic system is just as susceptible as the visual system to responding with overly definitive assumptions from ambiguous sensory cues as is exhibited with this animated image.
"Feel" is necessary but also is not sufficient for complete understanding -- or safe operation. In regimes where counterintuitive action is frequently or critically experienced, feel uninformed by objective principles of action is a bad gauge for action -- sometimes a very dangerous one. On the other hand no one can practice aikido with a slide rule or laptop. But aiki frequently and critically operates in regimes where it is actively exploiting intuitive cues of "feel" that are deceiving the percipient about the objective reality of the dynamic.
That is the power of aiki over those untrained in its use. Only knowledge of the objective determinant of the events occurring enables one to shift perception from one's initial intuition of orientation in a regime of ambiguous sensory cues. Because of this, we cannot stress too much the proper understanding of objective principle as well as intuitive feel in structuring and understanding what we are doing when we practice it.
Now, to the physical harmonics. I will bet the attached image has a fairly familiar "feel" for movements typical of aikido for many of you.
For visual orientation of the third dimension and dynamic of the curve, it begins as red and then slowly transitions though purple to blue along its length of travel (or vice versa if you are contrary).
The image linked is called a harmonogram. It shows an intuitively related "feel" for the dynamics of actual practice. The shape of the harmonogram given describes a complex Lissajous curve. It is the three notes (three interacting, independent oscillations) of a dominant major seventh chord.
Some of you will dismiss this as being too esoteric or even more esoteric that the traditional circle-triangle-square. I assure you it is not, and has far more possibilities for analytical and descriptive use. The actual math is fairly simple. If you want to work on them graphically, there are several free and shareware Lissajous applets and programs available to playing with. The resulting mechanical possibilities of description are richly complex and deeply interconnected.
A harmonogram is simply the combined path formed by the interaction of two (or more) independent pendula.
It has a fairly simple physical basis belied by the apparent complexity of the path that results. Mathematically, the Lissajous family of curves are 2d representations of one or more sine waves meeting at right angles. They, can in more complex versions, also involve oscillations on two or more coordinate planes, and also more than one frequency oscillation (or shifted phase) superposed on the same plane. But basically they are just superposed oscillations in different orientations.
A sine wave is described by triangular (trigonometric) relationships which map onto revolutions of the unit circle. At the point where the sine waves intersect, two unit circles are defined. That also defines a unit sphere centered at the point of connection. Lissajous curves in 2D projection are all bounded by the unit square (+/-1, +/-1) and in 3d by the unit cube centered on the same point. This purely physical description is of a piece with the traditional formulation of underlying principles. In it may be found precise and deeply meaningful mathematical relationships of circle, triangle, and square -- joined in spherical rotation.
I am coming to the conclusion (prove me wrong, please) that this description of harmonic motion (the proper term for the physics involved) is the fundamental description of aikido's exploitation of physical laws flowing from the structure of the body. Dynamically, it operates in conserving or manipulating the conservation of angular momentum contained in these harmonic oscillations.
Those oscillations (in-yo joined) are free to occupy and reverberate in any part of the space defined by the interaction of all linked parts in perceptible connection with one another. Since it is governed by perception of motion, and actual physical contact is not necessarily required to obtain that mind/body interaction that reaches kuzushi. Perceptible motion in properly connected relation (regardless of actual contact) or the nature of the connection, can achieve this in regard to an aware and reactive opponent.
Each limb forms a double pendulum (potentially more than double, actually, depending on the use). They therefore exhibit coupled complex harmonic motion. The body is poised on two legs forming two linked but independent pendula, another harmonic couple. The body at the waist and point of support on the ground form a double inverted pendulum, another harmonic couple. Two people connected in aikido, swiveling from their points of support on the ground also form two linked but independent inverted pendula, a further harmonic couple.
By my count, summed over all scales, there are at least 5 or more double-linked oscillation interactions that can flow into one another harmonically -- in one person's body. In a system of two people interacting there are ten or more. At any given time, motion can shift imperceptibly from dominance by any one of these twenty or more possible contributors to the multi-phasic oscillations in play, all of which exhibit complex harmonics when coupled to another oscillator in the system, and most of which have large regions of supercritical stability ('top of the hill' dynamic).
The various oscillations travel or communicate throughout the body by the flexibly linked rods of the body's component parts. Different oscillations can travel throughout and past one another in this manner. Different couples can be formed of overlays of complementary relative motion of certain parts if their motion is made appropriate (various kokyunage for example), or by or intentional isolations of intermediate linkages (such as in nikkyo or shihonage).
Oscillations can be progressively damped by the use of the joints as viscous dampeners ("rooting" behavior) or progressively overdriven harmonically at each stage for strikes or throws. Because they are all connected the biggest oscillator in the sysatem (center of mass), can drive the smallest component, with momentum concentration advantages, and conversely the smallest in the system can use the mass dampening of the largest component to dissipate or convert applied momentum.
Balance and attacking power generation depend on the intuitive orchestration of these complex harmonics. Aikido is the art AND SCIENCE of disrupting the intuitive operation of those harmonics, without changing the nature of the harmonic interactions that are occurring. That allows normal reactions in this harmonic concerto to be exploited.
Each oscillation is a form of periodic physical rotation in space. The intersection of two such oscillations induces precession or alteration of rotation from one axis of orientation to the other, as gyroscopes do, and as Foucault's pendulum famously did to show the rotation of the Earth. The perceptual counter-intuition that is involved in precession motion, is famously seen in those devices.
By changing the underlying physical frame of reference for perception of these rotations, the alteration is often not fully perceived until the original frame of reference is completely lost and the departure of motion is too great from the intended path to be salvaged. Any attempt to do so at that point, typically wrecks the necessary harmonic complementarity of their own motion, which then makes any counter using the same dynamic mechanisms impossible.
This is a rigorous physical model that closely fits in mathematical terms the traditional exposition of aiki principle in "triangle-circle-square, joined in spherical rotation." It is framed from the fundamental physics of the linked harmonic pendula of the body, its limbs and any other body in connection. It results in a visual depiction of interactions that are both mathematically correct for a valid physical model and intuitively correct for the objective elements or shapes of typical aikido movements as they are experienced.