## A Farewell to Symmetry

This past winter a huge storm of a scope greater than the blizzard of 1888 lashed the entire East Coast of the United States with cyclonic winds and buried it under mountains of snow. Hundreds of people died as a result of the storm, and property damage was in the millions. This storm appears to have been born out of three innocuous disturbances that, by themselves, would not have been noteworthy. A cluster of thunderstorms developed over the western part of the Gulf of Mexico; a mixture of rain and snow moved across Texas on winds from the Pacific; and light snow and gusty winds dropped down into the Midwest from the Arctic Circle.

Chaos theory began with speculation about the Butterfly Effect, which holds that so slight a disturbance as the batting of a butterfly's wings at a remote source-- say, an Indonesian rain forest--could cause unexpected effects far from the source, such as a tornado in Kansas. The 1993 storm may have resulted from a triple butterfly effect and, thus, was triply dangerous--and triply unexpected. Chaos theory has excited scientists across the whole spectrum of science and may turn out to be a unifying theory of everything. One day, orthodontists are going to be interested in it. The development of the human body is probably as good an example as there is of the evolution of some kind of order out of chaos.

We do not have an understanding of exactly how genes work. There are not enough genes to micromanage the development of a hand, an eye, or a set of teeth. Differentiation at the cell level is likely a product of biochemical interactions. A gene may say, "Make a dental shelf", and assign the task to a group of cells and amino acids with the capability of interacting and differentiating into the dental arches. The making of a human jaw is a chaotic event with local and remote effects. It has been observed that the left mandibular length of a jaw may at first be greater than the right, and that this may change in time so that the right becomes greater than the left. There is an interactive dynamic going on that is not well understood in terms of linear mathematics. The only effort I can think of to describe this phenomenon--and on a macro level at that--was Donald Enlow's thought about differential bone growth at the sutures; and, as far as I know, it has been permitted to languish there on the pages of his book.

Leaving aside the criticism that orthodontists are stuck in a mode of trying to describe a three-dimensional object with two-dimensional mathematics, there is a gnawing doubt that linear models adequately describe nonlinear forms. Linear mathematical models are based on flat planes, lines, and constant growth rates. Nonlinear models are based on curves, dilations (which sound a lot like Ricketts's gnomons), and variable growth rates. A quote from Stewart and Golubitsky's book Fearful Symmetry is apt: "All too often a nonlinear system has been forced into a linear mold.... The philosophy behind this approach seems to be that a wrong answer, or a right answer to the wrong question, is better than no answer at all."

They continue, "Scientists and mathematicians aren't the main culprits among those who place too heavy reliance upon non-existent patterns. It's a common human failing. A whole breed of financial analysts currently attempts to predict the behavior of the stock market by applying a range of 'patterns', either geometrical or numerological, whose basis is-- to say the least--dubious. The sensible ones make money by teaching the system to everybody else. Several schools of architecture--often respectable and respected--are based on numerical mysticism. Le Corbusier's 'modulor' emphasized ratios based on Fibonacci numbers and the golden ratio. It's not that people don't design good buildings by these methods; it's that their design sense plays by far the greatest role, and the mystical framework is so flexible that any reasonable design can be incorporated into it."

Plato observed that nature tends to produce mathematical patterns, and we tend to think those patterns are basically symmetrical; but nature is never perfectly symmetrical. There are tiny fluctuations down to the molecular level.

In orthodontics, we are dealing with two concepts of symmetry. In one case, we mean balance; in the other, we mean bilateral mathematical symmetry. We accept the first as basic and deal with it in our analyses as if it were the second. Orthodontists impose a symmetry on the dental arches, whether the system within which they will operate is symmetrical or not. The fact is that it is easier to manufacture anything repetitively to a symmetrical pattern. From dinner plates to beer cans, we make symmetrical objects when we mass-produce. Despite human individuality, the making of the species is mass production; and despite differences in appearance that other humans can recognize, probably most humans look pretty much the same to a dog. That is, a dog doesn't look at a human and think "cat".

The human body is not bilaterally symmetrical. It has more of a mirror-image symmetricality, but not a perfect one. This has been amply shown by splitting an image of a face, copying each side in reverse and joining original and reversed halves. The resulting pictures are quite different. It is not surprising that asymmetry may be the rule rather than the exception in the environment in which we work. Why is it, then, that we can impose a symmetrical dentition on an asymmetrical base and seem to get away with it? If Ernst Mach was correct in saying that "in every symmetrical system every deformation that tends to destroy the symmetry is complemented by an equal and opposite deformation that tends to restore it", it is conceivable that a noticeable structural asymmetry of the jaws may be compensated for in time by an equal and opposite deformation of bone or by a reconfiguration of muscle. Or--and it will become a subject to ponder--are we getting away with it?

So far, we have depended on the adaptability of the system, and that is probably correct to a point. How do we get past that point? Orthodontists are probably oversimplifying the orthodontic problem and applying the simplest linear mathematics--Euclidian geometry--to it. But since we have advanced in our time from a measurement world of paper, pencil, and compass to the computer, tools are at hand to apply geometries other than Euclid's , and some of them may better describe the stomatognathic system than Euclid's.

Where does all this leave orthodontists and the symmetrical basis for orthodontic correction? At our present level of understanding, we and our patients are probably better off if we continue making symmetrical arches. We could improve our present evaluations of asymmetry simply by taking and analyzing frontal headfilms in addition to lateral headfilms, and using photogrammetry to produce three-dimensional models from them. We may someday have a better vision of the complicated patterns of growth and the effects of tiny growth changes on adjacent and even remote structures, and of the complex neuromuscular system that drives growth. We may then be able to accommodate both sagittal and frontal asymmetries and produce a system better suited to the environment the teeth and jaws are in, assuming that symmetry is conducive to better function and stability. However, since forces that may remain beyond our control are likely to break the symmetry of our best efforts with butterfly effects we may never entirely account for, it may be that we can only hope for relative symmetry, relative stability, and permanent retention.