Pondering Pendula

December 3, 1996

If you treat yourself to a $59 copy of "Chaos: A Program Collection for the PC", --- 1994 offering of Dr. H. J. Korsch and Dr. H. J. Jodl of the Universitaet Kaiserslautern, you can play and ponder at the same time. That is, as you "penetrate unexplored regions of mathematics", you can also "discover unforeseen links between ideas", such as, "Why are the clouds the way they are?", "Is the solar system stable?", "What determines the structure of turbulence in fluids, the noise in electronic circuits, ... and so on."

For me, the double pendulum program, of the ten chaos "games" included, is the most "transporting". A double pendulum is simply two pendulums where one is hung from the bottom of another. As you fiddle with the initial conditions (e.g. the starting energy, angles or etc.) for this dual-natured self-referential system, you can watch the energy and dynamical activity shifting back and forth between the lower and upper pendula, and watch the two behave almost as one from time to time. At other times, they go into very chaotic movement, almost if they are fighting with each other.

All that action appears on the right side of your screen. On the left is the two dimensional trace of the phase trajectory in four dimensional space (pendula are very simple, most of the phase spaces for nonlinear systems have much higher dimensionality than four). No, I'm not referring to "time" as the fourth dimension. Time is the fifth dimension in this case.

Now what's amazing is, that however erratic the pendulum motion appears, the phase trajectory dynamics are very ordered, ... strange maybe, but very ordered. And when you look at the Poincare section (where the phase trajectory intersects with the null point on the pendulum positions --- i.e. the zero energy rest position), the order is even more well behaved and really quite aesthetically pleasing. But this is no fantasy game, this is about real world natural stuff, and it says that much of what we habitually interpret as "noise" and disorder is neither "noise" nor disorder, but very highly ordered behavior.

And herein lies the rub, because our western anal-retentive logico-mathematical cultural tradition has had us ignore and discard all this hidden order for the past 2400 years (since Aristotle, 350 BC).

As we watch the pendulum swing, the thought comes; how the hell could I infer the whole system if I was sitting on the lower pendulum and never opened my eyes to the overall system. In other words, how the hell could I get an overall picture of what was REALLY going on around me, by working the problem "bottom-up"? Clearly, there's no f___g way!

And what about my sense of "time". There's clearly more than tick-tock, tick-tock going on here, there's some complex rhythms that are analogous to those which develop on the scale of astronomy. The lower pendulum, seen on its own, is like the earth's rotation around the sun; tick-tock, back-forth, day-night, true-false, one-zero, .... binary, linear .... B O R I N G! On the other hand, if we look at the system of two pendulums, we can now regard the upper pendulum as the revolution of the earth around the sun, so now we get a more complex system of harmonics. This takes us into the fourth dimension in terms of both system phase trajectory and cognitive mode.

If we think of this as a new reference clock, we are now shifting from "time" as a linear generic directionless quantity, to "time" more as it was seen in its early etymological meaning as a "season". Time now appears to be going somewhere, to have some "purpose". So the clocks seem to run differently in association with this fourth dimension. What did the old testament say, again? ---

"To every thing there is a season, and a time to every purpose under the heaven.

A time to be born, and a time to die; a time to plant, and a time to pluck up that which is planted ..."

Old King Solomon, whose reign is credited with this "seasonal clock" statement, apparently had no problem thinking in the fourth dimension. In fact most of the ancient cultures (Solomon was circa 970 BC) had bootstrapped into this way of thinking, and their pictographic languages and calendars were tied to the fourth dimensional "seasonal", "harmony", "Karma", "Feng Shui" or whatever, clock concept.

It wasn't until about 350 BC that Aristotle came along and said; "wait a minute! that damn "seasonal" clock is too blurry and inherently unpredictable as a base for modelling our reality. Let's focus in on the predictable detail and build our way back up from the bottom so that we'll be able to understand and predict everything in its entirety. ... And ever since, in the west, all we think of when we think of time is; tick-tock, tick-tock, tick-tock. It is ravaging us like the clock-crocodile ravaged Captain Hook in Peter Pan, and it is always there in the background, threatening us.

Ok, what was that again? ... Aristotle said that if we could get the detail right, we could work our way bottom-up and get the big picture solution complete with detail. Now that sounds great ...... but don't we have to neglect the upper pendulum rhythms in order to establish the lower pendulum detail? So, that seems to equate to making the assumption that nature is linear, does it not? .... that we can take the problem apart and re-assemble it (add the solutions) to get back to the overall solution. Trouble is, looking at this damn double pendulum on my PC screen, that linear concept doesn't appear all that realistic. For one thing, there's a continual exchange of energy between the lower and upper pendulum. I guess Aristotle must have constrained the problem to "closed systems", systems that don't exchange energy with the outside world. So this solution of Aristotle's must assume that the world is composed of disconnected non energy-exchanging linear parts.

Let's see now; Prior to Aristotle, we were not satisfied with having an approximate solution to an exact problem. So thanks to Aristotle, and with the further aiding and abetting of Descartes and Newton, for the past 2500 years, we have been able to get an exact solution to the approximate problem ... and the only approximation we had to make was in viewing the world as a linear assemblage of disconnected parts.

What a deal!

We've certainly got a cleaner, neater world out of all this bottom-up thinking. All we have to do now is just look for the big features and trash the small stuff.

... Trouble is, we don't really get away from detail cause we are always working things from the bottom up and that means starting from detailed basics ... and detailed basics are growing exponentially as information and knowledge grow. So we keep having to manage more detail even as we're trashing more and more detail, and we still don't seem to be able to get the big picture right. Maybe we just need more detail. Perhaps we should build the super-conducting super-collider and perhaps we will discover the tiny "missing link" that makes everything right.

After close to 3 millenia of clean, neat and pure thinking and living, it's unthinkable to go back to blurry and ambiguous top-down "astrological" thinking. The idea of seasons and harmonies is kind of quaint and appealing, and not without some nostalgia, but to try to get to detailed answers as to what I should do in this moment, starting from the movement of the earth sun moon and planets is a bit much.

Still, as I watch this double pendulum, .... maybe I could use a combination of both clocks? ... If I used the "seasonal" or "harmony" clock to make sure of the overall orchestration and "phasing" of things in my models of what's going on, then I could use the tick-tock, tick-tock stuff to home in on the local detail. In that way, I would be taking advantage of this higher dimensional consciousness we seem to be capable of, as well as our logico-mathematical capabilities.

Hmmm. 2500 years of solving an approximate problem exactly ... trying to logically infer, on the basis of the movement of the local pendulum we happen to be sitting on, the meaning of the rest of the universe. What did Goedel's Theorem say again; ... logic is not sufficiently robust to handle the case where you stand on your own head.

Overall, I would say that it's time to resurrect Heraclitus. Even covered in cowdung, he makes more sense!

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