GUEST EDITORIAL

Chaos theory and fi'actals R e c e n t l y , the natural sciences have shown an unprecedented interest in two new fields--Chaos Theory and Fractal geometry. This is because they show promise as a means of explaining and describing irregular patterns and forms in Nature in mathematical terms. As a corollary, they will raise the status of day-to-day clinical experience because many observations can now be substantiated. LITERATURE REVIEW The main source for Chaos Theory for this editorial has been the book by Gleick (1990). Clinicians may prefer this source to physics texts on the subject since the language of physics is somewhat foreign to them such as the chapter by Ford (1990). ttowever, the introductory chapter by the editor, Davies (1990), is easier to read with less technical jargon, but it does not have sufficient detail. Gleick's book provides an interesting history of Chaos Theory but is unfortunately quite severely marred by journalese. For fractals, Jurgens et al.'s article (1990) is easier reading than Goldberger et al.'s (1990) despite the latter being on human physiology. According to reviews (the book is not available locally yet) clinically relevant information on fraetals may be gleaned from papers given at a conference in London and edited by Crilly et al. ( i 99 I). As will be discussed with important orthodontic examples later, the term fractals is a abbreviation of fractional dimensions. Previously we have considered the patient in a number of whole dimensions (Mollenhauer 1985) but fractional dimensions allow for consideration of similarities rather than exact duplicates. This is the mathematical equivalent of pattern recognition used by clinicians.. But the publication which makes the greatest impression is Chaos: The Sofnvare. It is the companion computer software to Gleick's book. The manual alone is worth reading. The set of six programs are for playing with on ones' personal computer, to show the manifestations of Chaos Theory. They show the difference between the mathematical approximations of the past and the near-real thing in the world of Nature. This is similar to, but more sophisticated than, a frequent theme of past editorials on the 'loneliness of the mean.' While the patterns produced by the Mandelbrot and Julia sets and their variations are interesting and beautiful, because of our scientific background we are more attracted to the Magnets and Pcndtdum, Strange Attractors and Toy Universes modules. These modules clearly show that we are now freed from the bondage of restrictive clinical theory tied to the old dogma of the physical sciences which put procedures and outcomes into neat little boxes. For example, with the software one can

From the "'Fditor's Newsletter" in the March 1991 issue of the Australian Orthodontic Journal, Melbourne, Australia. 811131709

play with a simulation of a pendulum which is controlled by magnets. Yet, despite starting the swing at the same point on consecutive trials, it is impossible to predict the final resting place. And this is the simulation of a physical system! ttow much less are our chances of predicting mandibular rotation responses to appliances in the more complex biological systems of our patients? At last mathematicians and physical scientists have caught up to our way of looking at things . . . . so we have not been so unscientific after all by acknowledging incredible variation or chaos, rather than being tied to approximations. In fact, clinicians have not only have had to acknowledge this variation but have had to work with it. Oddly, Chaos Theory provides a mathematical basis for our ethics as per the last editorial on The need for a Science of Philosophy for Orthodontics. That is, we can now mathematically justify the long held precept that clinical treatment cannot be guaranteed, because Chaos Theory proves that it is impossible to predict the outcome. Equally significant for clinical observation is a point made in a paper on The Physiology of Perception by Freeman (1991). The paper reported a formal animal study of olfactory recognition. It suggests that chaotic patterns are involved in cognition: 'We think the olfactory bulb and cortex maintain many chaotic attractors, one for each odorant an animal or human being can discriminate. Whenever an odorant becomes meaningful in some way, another attractor is added, and all the others undergo slight modification.' Does not this ~xplain clinical experience'?. Those of us who have done thousands of tracings glean some insight into features of the craniofacial complex. One hears other professionals say this of histological examinations, radiological assessments, etc. Underlying chaotic patterns require samples of thousands for discernment. In Freeman's section on Further Reading a reference is titled: 'How brains make chaos in order to make sense of the world . . . . ' Unusual cases, which are called outliers, are generally neglected by statisticians as discussed in an editorial on Cognitive Science (Mollenhauer 1989). Patterns of response of largish groups of patients tend to be disregarded by the literature because they have not fitted the somewhat reductionist approaches of traditional statistics and research. Most clinicians have responded by a psychological state of passive aggression to the reductionist dogma of the physical sciences. That is, they mentally walked away, they gave up trying to explain their perceptions and hunches to researchers, because it was considered unscientific not to reduce data into neat classifications or discrete batches with numbers attached. But, on too many occasions, clinicians have not taken the time to agonize over their problems and observations. They felt it was not their responsibility. On the other hand, a few research supervisors also find refuge in past paradigms by, for example, resisting newer statistical techniques. This latter has led to the rule that research supervisors have to sign forms to say that the statistical techniques used on postgraduate students' research by the Statistics Department will be acceptable to the clinical departments concerned. 163

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Interestingly, Gleick's book, on about five occasions, uses the term qualitative in reference to mathematicians' comments. This puts a new connotation to Leibniz's famous statement 'God is a mathematician.' Traditionally, clinicians have used the case report method of documenting unusual individual cases. But, qualitative trends or patterns of subgroups have not been encouraged in the literature. On occasion they have been derided (Little et al. 1981), yet qualitative research can be done just as rigorously as quantitative research (Mollenhauer 1984). But even in quantitative research, Davies et al. (1991) have shown that results need to be interpreted by stating in the Abstract and in their Discussion that the improvement of oral hygiene after orthodontic treatment probably derives from constant admonition by the clinician during treatment appointments rather than the dental alignment per se. This reasonable conclusion arises from a comparison with the posterior teeth which were aligned before treatment and a smaller (presumably unpublished) study of removable appliance cases. Later a skeptical approach will be applied to Chaos since, after all, it is still in its infancy. But the exciting thing about it is that it will give practitioners confidence to talk to mathematicians about their honest observations. Thus all can contribute now to the orthodontic knowledge-base. This may encourage practitioners to document clinical observations in the literature. Lack of this has led non-practicing researchers to design inappropriate research based on our literature. For example, many of us tried the carboxylates many years ago for band cementation but twelve months later rapidly gave them up due to massive decalcification presumably due to a delayed breakdown of the chelation bond. In conversation with a materials scientist who did much dental research, this observation was not accepted by him because it was not in the literature! Understandably, people do not rush into print with their mistakes or disasters--at least in the fornml literature. Such things tend to be restricted to study group newsletters which are not usually read by non-practicing researchers. The tone of Gleick's book is one of constant disparagement of the scientific world. With this in mind and following on from the previous point, obviously there is a need to review professional publishing. It is important that the formal journals have their original articles refereed, but this is not a panacea for establishing the truth. More and more national journals are stressing the need for constructive comments by their referees rather than the negative and sometimes vindictive comments of past years. Journal editors need time to educate themselves, and thereby their referees, rather than being totally engrossed in the publishing process continually. That is, they should be philosophical leaders, not just paper shufflers. As well as the formal original articles and case reports, this journal has several sections whereby new approaches and observed patterns can be documented. One is the Suggestions for Research Design and the other is Caveat Communiques. The American Journal of Orthodontics and D~nirfacial Orthopedics has two such sections--guest editorials and Viewpoint. But it is important that editors ensure that con-

Am. J. Orthod. Dentofac. Orthop. Augu.~t 1992

tributors do not 'fire from the hip.' That is, they need to be well thought out and researched, and need plenty of sympathetic consultation by the editor--which again necds time, since such sections do not lend themselves to passing on to referees.

TERMINOLOGY AND CONCEPTS The most significant concept to come out of Chaos Theory and the one most widely used in the popular press is the Butterfly Effect. This term arose from a paper given in 1979 by Dr. Edward Lorenz, titled 'Prcdictability: Does the flap of a Butterfly's wings in Brazil set off a tornado in Texas?' Lorenz, a mathematically oriented meteorologist, was one of the first to study chaos theory in depth after attempting to model the weather by computer in 1960. More than any branch of medicine and dentistry, experienced orthodontists can relate to the principle of sensitive dependence on initial conditions which is the formal term for the Butterfly cffect. Unless a clinical case is started from the outset with a rational treatment objective (preferably a VTO), good separation, a good banding/bonding and correct initial archwires, but above all, good rapport with the patient and parents, a good result may not be achieved. The thesis, the sensitive dependence on initial conditions, is not new and is found in folklore from which Gleick quotes a traditional poem on p23. To emphasise the point, the following is an orthodontic parody of that poem. For For For For For For

want want want want want want

of good separation, a molar band was not seated; o f a correct fit, the band became loose; of a fired band, the arehwire became distorted; of comfort on holiday, the father cut the archwire; o f an intact archwire, there was anchorage loss; of proper anchorage, the case failed.

Let us extend this theme a little more. Would not the neophyte say simply that the anchorage could be recovered by headgear? The experienced practitioner, on the other hand, could foresee future compliance problems from line 4 - - d u e to a possible loss of rapport. It is a common observation that disputes between the parties about other matters can influence the effectiveness of headgear and elastics. Also, why is headgear not usually effective when attempts are made to recover anchorage in failed serial extractions? That is, was there an initial attitudinal problem of the family in not accepting the correct mainstream treatment in the first place, which manifests later as non-compliance with the headgear? In other words, does the experienced clinician, by being observant of his procedures and results become sensitive to their dependence on hfftial conditions and thus avoid certain problems in future treatment? Over a period of time, a clinician may consciously or unconsciously read messages, verbal or nonverbal, before treatment and relate them to such things as treatment time. Knowledge plus judgment must eclipse knowledge alone. On the other hand, an orthodontist is limited in questioning about domestic and personal background. This fact is

Voh,me 102 Number 2 highlighted in an article by Jacobs and Wright (1990) wherein it is shown that a social worker can elicit domestic influences. One case report brought forth the information about a problem-patient who had been enticed from home by a paedophile. The whole article highlights the research problems in studying failed appointments and compliance. It must be born in mind that all studies on compliance in the health fields indicate that only 30% cooperation can be expected. That is, the majority of two thirds will not comply with instructions. If this is added to the bruxers who smash their appliances, and the clenchers who triple the time to do torquing movements (presumably due to ischemia of the periodontal membrane), it is easy to see why treatment time is a non-linear phenomenon. That is, one cannot measure the distances the teeth have to move and compute the treatment time. Yet there seems to be consistent patterns of treatment durat i o n - c o u l d this be an underlying chaotic pattern? Vig et al. (1990) showed that the duration of orthodontic treatment time for 438 patients from a number of practices with very different extraction rates (25-84%) was remarkably similar whether the cases were treated on an extraction or non-extraction basis. It is becoming more and more obvious to this clinician how important parafunction is to treatment times. Adult treatment emphasises this point since the effects of parafunction are more obvious. The problem of studying parafunction 'prechaos' was that the effects of clenching even in younger patients comes in various degrees, as compared to the smashing of appliances from bruxing which can be quantified readily. Bruxing and clenching use different muscles. On rare occasions, a patient exhibits both. As mentioned in the interview with Dr. Alan Parker (1988), simple orthodontic cases can only be labelled as such in hindsight--not in foresight. This is a casual qualitative observation but a more rigorous qualitative observation comes from Associate Professor T. Freer in another interview (1989). Therein he said that current quantitative occlusal indices were found wanting and that they were less reliable than the judgment of experienced specialist clinicians. All of these consideratio~s give more weight to the studies indicating that the best orthodontic delivery system is via orthodontists. As discussed in the last editorial (Mollenhauer 1990d) some orthodontic delivery systems were based on political ideology rather than clinical reality. It is surely time that democracies used a more rational approach in this area based on past experiences--even from other countries if necessary. In the terminology of Chaos, we need to study such variables in terms of Strange Attractors which graphically show their effects in the Chaos software package. Leshan and Margenau (1982) also discuss our inability to predict the human conditions by formulae. In research, rather than a 'pecking order' where prospective studies are rated above retrospective studies, and the lowly anecdotal below that, each good study should be viewed as an evolutionary stage. Otherwise it is like saying adtilts are more important people than teenagers and children. A s . . children grow into teenagers and thence into adults, so may anecdotal observations lead to retrospective studies which may lead to prospective studies. This was considered in more

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detail under Cognitive Science and Spiral Thinking (Mollenhauer 1989 and 1990b). When people started to look seriously at chaotic patterns, it became obvious that it is the rule rather than the exception in Nature. Chaos Theory started with determining the underlying patterns from extensive observations of weather. A parent, patient or ancillary may focus entirely on one aspect of a problem to the exclusion of other important features. Such myopic vision may look only at the upper laterals in a Class II Division 2, whereas the major management is overbite reduction and torque. Somewhat similarly, the traditional scientific method has been to break up a subject into parts or to look at the factors separately. Chaos people would call this reductionism and criticise the old established protocols. Conversely, the clinician, the Chaos person and the detective must put the pieces together, to synthesise t h e m - and to put them into perspective. Of course, Chaos is not only relevant to biology, since Davies (1990) has noted that there are three ultimate frontiers of the new physics. They are the very large, the very small and the very complex. These allude to cosmology, quantum physics and Chaos, respectively. One of the first requirements to a study of Chaos is to differentiate between linear and non-linear systems. But the latter is different from linear and non linear reading discussed in a previous editorial (Mollenhauer 1987c). Linear functions are the mathematical equivalent of 'cookbook' procedures in the clinical disciplines. That is, linear is the 'if A, then B' approach. This has the advantage of being shnple to teach but unfortunately, it implies a predictable or guaranteed conclusion. Short courses and introductory courses tend to be restricted to this approach. The word technique also has this connotation and should rightly be considered an anachronism in orthodontics. Whilst training has to start somewhere and obviously with the basics, there has been an unfortunate tendency to keep it at that level. The 'cookbook' approach also is concise for publication purposes. Our literature can be said to suffer from the K.I.S.S. curse, but elaboration (to use Leshan and Margenau's term) requires very long articles--see the 22 page clinical article (Mollenhauer 1987a) where Expert System rules are used to give copious clinical details. Non-linear equations, as would be expected, are the recognition that, after a period of time, order and predictability can break down even without outside influences. Gleick's book refers on a number of occasions to the irregular dripping of a tap. It is probably the irregularity which disturbs sleep. This is an example experienced by most people and thus lends itself well to lecturing to an audience of several disciplines. It has been the habit of mathematicians to circumlocute complex non-linear approaches and instead obtain approximation with linear equations. Another necessary differentiation is that between randomness and chaos. We recognise that the term 'random' has a specific connotation in science compared to the lay term where it is often used in place of haphazard. Chaos is an even more unfortunate term because of its lay usage. Naturally there was some dispute originally among mathematicians

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about the term 'Chaos' as it is used in this editorial. The ahernative terms for Chaos are cumbersome, yet the list is worth reproducing from Gleick because it conveys something of the concepts of Chaos Theory: 'The complicated, aperiodic. Attracting orbits of certain dynamical systems. A kind of order without periodicity. Apparently random recurrent behaviour in a simple deterministic (clockwork-like) system. The irregular, unpredictable behavior of deterministic, nonlinear dynamical systems. Dynamics with positive, but finite, metric entropy to . . . . ' Compared to randomness in the scientific sense, mathematical chaotic systems have an underlyhzg pattern. This is more precise than Plato's Theory of bbrms since it can be quantified. In the early development of Chaos Theory, the concepts were visualized by means of topological theory and transformations. Then the rigorous approaches of mathematics began to confirm the patterns seen on computer screens. These patterns, such as from the Julia and Mandelbrot sets, are now ubiquitous on the covers of physics textbooks, software catalogues, etc. As stated above, there are underlyingpatterns, so attempts have been made to apply chaos theory to all manner of things, including decisions. Obviously experienced practitioners subconsciously develop such patterns. The evidence for this is when brilliant and well trained neophytes rush into problems 'where angels (read experienced practitioners) fear to tread.' An orthodontic example is a severe skeletal Class II case where the patient is unlikely to be compliant--lower extractions are not prescribed lightly by an experienced clinician . . . . well, this rule applies without competency with Jasper Jumpers. A recent feature of studies in Chaos Theory which fascinates mathematicians, and to a lesser extent physicists, but certainly not biologists, is reorganisation. This is because reorganisation contradicts the Second Law of Thermodynamics, whereas biologists are quite familiar with reorganisation and repair rather than continuously evolving randomness. One branch of mathematics, statistics, is oriented towards the taking of a minimum number of examples as a sample to make valid conclusions about the population being studied. This obviously has important financial implications. But to provide the full picture by establishing the underlying chaotic pattern, far more data is required. Questioning the basis of statistics for applied biology but not for manufacturing was raised in previous editorials on Observability Theory (Mollenhauer 1987d, e). Indirectly, this supports yet another plea in previous editorials for the need for practitioners within the field of study to be consulted for research design, especially for students' research. That is, experienced practitioners have had the large number of observations required to determine the most rational approach for formal studies. Furthermore, the concept of average is central to statistics. But in Gleick's book the question of just what is the average climate over a period of time, is discussed. Dr Mitchell Feigenbaum, another of the eminent developers suggested 'chaos exists out there, and the brain seems to have more flexibility than "classical physics in finding the order to it.' Orthodontists have no problem with a multidimensional

Am. J. Orthod. Dentofac. Orthop. August 1992

view. In fact, one of the significant features of orthodontic clinicians is the need to take the fourth dimension of time fully into account. It frequently distinguishes orthodontists from other dental specialists who cannot understand why treatment is often delayed. But orthodontists would usually be unfamiliar with the concept of fractional dimensions which gave rise to the term 'fractal' by Dr. Benoit Mandelbrot. We are used to whole dimensions, e.g. one, two, three etc. as previously discussed under Diagnosis and treatment planning in 9-dhnensions (Mollenhauer 1985). As mentioned earlier, fractal geometry allows for similarities, or more specifically, self similarity. As interesting and relevant the concepts of Chaos are, time may prove that the real benefits to orthodontics may come more from fractal geometry. For example, it will be possible to discard the dental interincisal angle which is not patient-oriented. Self similarity may show that, whilst the upper and lower incisor sagittal angulation may appear similar in relation to the occlusal plane, different factors determine their separate optimum angulations in terms of the labiofacial surfaces which are gleaned from study models and photographs as much as cephalograms taken without metal brackets. The various factors determining the different labiofacial surfaces of upper and lower incisors are discussed elsewhere (Mollenhauer 1991). Undoubtedly the upper and lower lips will also be studied in the future using fractal geometry. As discussed in another part of that publication's editorial, both areas must be considered in relation to the changes five to ten years post-treatment. A reasonable question is; why is it worthwhile considering Chaos and Fractals together? Jurgens et al. (1990) provide that answer: 'The correspondence between fractals and chaos is no accident. Rather it is a symptom of a deep-rooted relation: fractal geometry is the geometry of chaos.' Possibly of more interest to the biological disciplines is the Lindenmayer system, or L-system, since it was originally a mathematical theory for plant development. It was an exercise in understanding how the natural processes of growth can be specified, modelled, controlled and predicted. Orthodontists would certainly like that. Finally, there are a number of other concepts which are important to understand Chaos theory. But this author does not have the qualifications nor the confidence to explain them. Hopefully, this will be rectified in a future issue of the journal since our local University has experts in Chaos Theory, and one may be prevailed upon to contribute an article. These other concepts are phase space, period-doubling, fractal basin boundaries, bifurcations (mathematical in topological transformations, not tooth roots) strange attractors and imaginary numbers. Most biological people are not yet used to working with such terms.

INTERESTING COROLLARIES One of the exciting prospects for the health sciences is the ability of fractals to store images in less computer memory. Images such as cephalometric and occlusal tracings take up considerable memory. Reduced code would expedite communication, thus allowing pooling of knowledge (as per University of the World Newsletter--For the Health professions,

Votume 102 Number 2

including dental research projects p20, Steele 1990), raw data (Mollcnhauer 1988a), categorical data (see Lotion and Rethman 1990) and techniques for research design (Mollenhauer 1988c). Dr. T.M. Graber, in a recent keynote address to a meeting of orthodontic teachers at Dallas, Texas, stressed the need for sharing and pooling teaching material and methods in North America. This could be extended beyond that continent, if the prediction of Jurgens et al. (1990) comes to fruition. They said 'The time, complexity and cost of transmitting satellite images to the earth could be drastically reduced by converting them into codes using fractal algorithms.' To make this point, it has been claimed that Bamsley's technology of using 'iterated function systems' has achieved compression ratios of 500: 1, and in some cases as much as 10,000:1 (Valdes 1991). To facilitate communication, a basic format must be established e.g. the ISDN one. Teaching material can certainly be shared with Computer Based Training software (McSherry and McArthur 1991). This author has personal copies of the demonstration software of Author referred to in the article and it is certainly impressive, especially if random numbers can be used to generate new problems. But like all didactic material, it is better for imparting knowledge, rather than imparting judgment which only comes from the experience of dealing with large sample sizes--as required by consultants and Chaos theory. Nevertheless, CBT appears to have motivational advantages for imparting facts, because its efficacy for teaching basic physics has been instantiated on two continents (Kokot and Ellis 1988, Ellis 1991). A significant fact is that fractal geometry can handle irregular shapes. To quote Jurgens et al. (1990) again 'Once one has a command of fractal language, one can describe the shape of a cloud as precisely and simply as an architect might describe a house with blueprints that use the language of traditional geometry.' Gleick frequently stresses the need to start with graphic representations and then move onto formal mathematics. By this approach, Lorenz was able to use Chaos methods to model the weather eventually with just three equations. As mentioned in a previous editorial, it is sometimes necessary to struggle through a storm of complexity before the calm of a simple approach can be reached. The disadvantage of a K.I.S.S. approach is that people try to start at this end point. Still on the subject of graphic representations, the original purpose of the Bolton Growth Study was to produce templates for each age group. Dr Reed tloldaway and then Dr Robert Ricketts via VTOs, Dr Alex Jacobson via the Proportionate Template (1985) and to some degree, Dr Lysle Johnston, returned to graphic representations to visualise growth and treatment planning. In fact it is probably necessary to alternate periodically between qualitative and quantitative methods-according to the concept of spiral thinking (Mollenhauer 1990b). An important feature of graphic representations is th,qt they are often more easily comprehended than discipline specific language and figures. For example, the flow charts of Professor Alexandre Petrovic's cybernetic models (1982) for proposed research can be understood by clinicians, with a

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little effort. The question of whether growth and treatment response prediction is possible is a common one discussed by orthodontists. Gleick records, in relation to the modelling of whether 'The European Centre's assessments suggested that the world saved billions of dollars each year from predictions that were statistically better than nothing.' Could it also be said that sensible routine Visualised Treatment Objectives (VTOs) save considerable chairside and treatment time? It has been my experience that well chosen extractions can save at least 4-6 months of orthodontic treatment time; and one proper session of quiet treatment planning before starting is more efficient than making complex decisions every time the patient visits. Achieving better and more consistent treatment is also more likely because professional pride will drive the average clinician to attempt to achieve the goal as often as possible. Chaos theory has had the benefit of opening a Pandora's box of honest observations about physical mechanisms which were previously assumed to be completely understood and were not, such as the common pendulum. Previously data points outside the linear range were considered 'noise' and therefore disregarded as are outliers in statistics. 'Noise' can be significant--one has only to record the overjet in centric relation on several occasions, especially when they are more settled and relaxed, for different recordings to confirm this point. In orthodontics, every now and then, one gets new insights into the benefits of traditional study models. As stated before, the facial angles of the labial surfaces of incisors, should take priority in VTOs over the interincisal angle which is of little interest to the patient. Occlusal stops are far more accurate on study models than cephalograms. Similarly, with restrictions due to radiation hygiene, facial photographs including smiling photographs, are the best means of establishing lip and smile lines. In other words, the new concepts of this editorial allows us to take a fresh and valuable look at basic things. Gleick quotes several cryptic statements which are of interest to biologists, such as one attributed to Dr. Joseph Ford: 'Evolution is chaos with feedback.' But of greater significance to orthodontists are several quotes from Dr. Arnold Mandell. These really take some pondering but are worth the effort. One is ' . . . when you reach an equilibrium in biology you're dead.' This, of course, flies in the face of traditional thinking in orthodontics about the stability of archform. Mandell is a psychiatrist whose main interest is the dynamic physiology of the brain. Another example shows this: 'If I ask you whether your brain is an equilibrium system, all I have to do is ask you not to think of elephants for a few minutes, and you know it isn't an equilibrium system!' He also posed much more radical questions 'Is it possible that mathematical pathology, i.e. chaos, is health? and that mathematical health, which is the predictability and differentiability of this kind of structure, is disease?' when referring to the pathognomic erratic eye movements of schizophrenics as they watch a pendulum. Gleick says 'Traditionally, because the surface tension effects are so small, researchers assumed that for practical purposes they could disregard them. N o t s o . . . ' and earlier

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'Where heat diffusion tends to create instability, surface tension creates stability. The pull of surface tension makes a substance prefer smooth boundaries like the wall of a soap bubble. It costs energy to make surfaces that are rough.' Does the latter apply to root resorption? Non-lineaxity also confirms, as is well recognised in quantum mechanics, that the very act of observing can influence the findings. The famous Rat maze psychometric experiments are a classic on research into research itself (Rosenthal 1966). In these, whichever group of rats were randomly labelled 'clever' was found to produce the best intelligence scores by the researchers. The orthodontic equivalent is that treatment alters the oral system and therefore treatment response must be expected to be different from normal growth and development. Somewhat less recognized is that it could also apply to comparative clinical research e.g. one clinician may be better at motivation and attention to detail than another. Chaos Theory's reorganisation concept means recovery to the original pattern. In orthodontics, this was called 'a return to the morphogenetic pattern' and was often used as a euphemism for relapse. The Chaos software suggests that Chaos tends to refute the Epigenetic hypothesis more than support it. The term reductionism is of interest to all biologists. It implies breaking down a procedure or phenomenon into its smallest parts; studying them individually; and then adding the effects--along the lines of finite element analysis. This latter term has appeared recently in the orthodontic literature as F.E.M. (Finite Element Analysis Method). It's origins are in engineering for, for example, studying the suspension ot" automobiles. Reductionism is not at all favoured by the proponents of Chaos who try to look at the phenomenon as a whole. Dental ankylosis has been classified in a previous editorial (Mollenhauer 1990b) on clinical grounds for this very reason. That is, the histological picture does not explain temporary and variable ankylosis which one sees occasionally. An interesting treatise on reductionism is found in 'The Search for Scientific Truth' section of Leshan and Margenau's book (1982). Topology may be called the geometry of rubber sheets. When topology became linked to dynamical systems by some of the early workers in Chaos, pattern recognition became significant. Thus, pattern recognition, where humans still lead computers (Mollenhauer 1989), is starting to receive the attention it deserves from abstract mathematics. Currently, lip profile is still better evaluated by pattem recognition. This should be done by clinicians who fine tune their judgment with constant observations, rather than a few simplistic measurements. Observations here include taking note of fashion changes. For example, as shown by Dr. Edward Crawford at the Combined Conference in Canberra in March 1990, many currently successful teenage photographic models have strong chin buttons. Fortunately, augmentation genioplasties have improved with the use of angulated sliding techniques. These have both cosmetic and functiona.l be~aefits (ttookey and Gooday 1989). It will be interesting to see five year or longer term follow-ups of the new genioplasties, since 5 years seems to be necessary to allow recovery of the lips

Am. J. Orthod. Dentt~lc. Orthop. August 1992

from conventional orthodontic treatment (Holdaway 1983). Gleick stated of Mandclbrot, whose mother was a dentist, by the way, that 'unlike most nmthematicians, he confronted problems by depending on his intuition about patterns and shapes. He mistrusted analysis.' This confirms the Caveat Communique about digitizers (Mollenhauer 1990c). But the real problem is the over-simplistic use of cephalometric analyses by reductionism, rather than synthesis to mentally look at the face as a whole, or globally, as the mathematicians would say. Not only had traditional mathematicians generally disregarded or discarded non-linear representations, orthodontics had too. For example, the Steiner 'S' line was lost along the way because it was too difficult to simplify its usage quantitatively, yet, in hindsight as shown by the 'tt' line of Holdaway, it could have been one of the more significant parts of the Steiner analysis. The circulatory system is a fractal structure by having a large surface area to serve the cells but it has a limited volume in comparison to the body. The lining of the digestive tract and the lungs have a similar structure. As more and more non-linear systems are studied it becomes obvious that each is unique. Does this explain the common observation that each professional person seems to take twenty continuous years of practice to achieve some degree of mastery of his work? One of the fascinating stories related by Gleick concerns the different approaches to the study of light by two past geniuses, namely, Newton and Goethe. We all learned about Newton's prism experiment at school but Feigenbaum ferreted out Goethe's study. This was with some difficulty because it had been outshone by Newton's more precisely mathematical model. Feigenbaum established the value of Goethe's work in the light of Chaos. It proves that 'honest' and diligent research may some day be of value. Most of my generation were influenced by the wholesale retractive orthodontics of the 1960s. But we had to re-evaluate our thinking when colleagues were able to show non-extraction cases which were as stable, and sometimes more stable 5 years post-treatment. Interestingly, a comparison of Sadowski and Sakols' (1982) and Little et al.'s (1988) long term post-treatment studies showed that good selection of extraction or non-extraction treatment by experienced clinicians in Sadowski and Sakols' study yielded much better stability of mandibular incisor alignment than students' cases. A Chaos term of interest to orthodontists is intransitive. This implies that an observer can see one type of behaviour over a long time, yet another stable arrangement may also be natural. Only an outside influence can change it from one state to another. This relates to the fact that systems are invariably 'more complicated than the quadratic (equation) one.' Could this explain the problem of late adolescent lower incisor crowding and the often presumed implication of the third molar? The italics for the word 'complicated' are mine to draw attention to the possibility than mathematics used in such studies have been linear, or even over-simplistic. Gleick puts it this way for other disciplines, but it probably applies to dental crowding 'scientists studying fractal basin boundaries showed that the border between calm and

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catastrophe could be far more complex than anyone had dreamed.' As he points out at the top of p21 of the Software manual: ' . . . a person trying to make a decision between two important, attractive, but incompatible alternatives might imagine a sort of boundary between t h e m . . , such bounda r i e s . . , may be extremely complicated, so that some slight change in circumstances throws one into an altogether different region.'

PERSPECTIVE Despite the desirability of larger numbers for samples than that required by statistical techniques, one thing disturbing to this author is the instruction of Chaos theory to discard the first fifty data points. This, unfortunately, is reminiscent of the early Fibonacci series. It is disturbing because some phenomena are very rare and so it is sometimes neeessary to draw conclusions from small numbers. For example, in 26 years of practice, only one case of haemophilia has been treated by this author with full fixed appliances. It was 24 years ago, when full banding rather than bonding was done, and strangely, bleeding was not a problem during band seating and cementation--apparently due to haemostatic agents in the gingival fluids. Only two blind patients have been.treated with full fixed appliances in that time. Both sufferea unusual root resorption. Is there an association as there is considered an association between root resorption and turned up noses, or was it coincidence? Only two possible reports of anabolic steroids producing complications with orthodontic appliances are personally known. (Mollenhauer 1988b, Cadell 1989). To overcome the problem of small samples, when more are required for more sophisticated analysis such as for Chaos, previous editorials have suggested that all raw data should be published. Con'espondence to overseas journals in other disciplines to implement this has not been successful. Some journals supply extra tables on request, and of these, some are on microfilm or microfiche. This joumal has experimented on two occasions with small and condensed printing such as with Brown and Travan's article (1988) and for an editorial on Exploratory Data Analysis (Mollenhauer 1988a). The interesting finding was that, by reducing the size of the data to one quarter size which is still readable, eight pages can be reproduced on one page. In the field of information theory, further interesting observations have been made about 'noise': 'The more random a data stream, the more information would be conveyed by each new bit.' Think of that in terms of listening for extraterrestrial messages. One simplistic orthodontic example may be with digitisers. For example, if exactly the same printout of cephalometrie measurements came from two successive patients, even if they were twins, then one would naturally expect that the wrong computer file had been printed on the second occasion. For the Mershon lecture at the AAO meeting in Washington, Dr. Lysle Johnston spelled out the difference between cynicism and Skepticism. For new things, he advised us to be skeptical by looking for real evidence. We could extend

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his advice to all models and simulations which are not of the real world. A theoretical perspective is stated by one physicist quoted by Gleick 'Relativity eliminated the Newtonian illusion of absolute space and time; quantum theory eliminated the Newtonian dream of a controllable measurement process; and Chaos eliminates the Laplacian fantasy of deterministic predictability.' But did Newton and Laplace state these concepts so simplistically, or is it what subsequent authors thought they s a i d - - a s is so often the case? See the comments on Tweed in the second column of Viazis's article (1991). But, to quote folklore, we must guard against 'throwing out the baby with the bath water.' At two recent meetings of Australian orthodontists, the suggestion was made that we should have part of a meeting devoted to 'Things which seemed a good idea at the time . . . . ' This suggestion is being implemented as a segment of a meeting in Canberra so it will be interesting to see what eventuates. Surely we could all make a list for such a segment: the original plastic brackets, carboxylate and EBA cements, fine coil springs as intermaxillary elastics, combination braking arches, .011" Supreme aligning archwires, intermittently worn vertical elastics. On the other hand, some apparent mistakes (overuse) lead on to better things. For example, certain procedures are most appropriate if used selectively, such as functional appliances which are generally thought to be suitable in 15-17 percent of cases, ttowever, initial overuse can lead to mastery of a procedure which can then be used selectively, but effectively, later.

OBJECTIVES One purpose of this editorial is to convey some of the concepts and language of Chaos Theory and Fractals so that it may be followed up more formally. Interdisciplinary groups for this should comprise mathematicians with expertise in chaos theory, and experienced practitioners. A researcher and recent postgraduate student familiar with the recent literature would also be valuable to the team. By working with such mathematicians, it should be possible to produce a lexicon of terms used by the disciplines involved, to enhance future communication. Furthermore, these terms and their various connotations should be documented in the literature. There are several prerequisites for such groups. As stated in a previous editorial on Cognitive Science (Mollenhauer 1989) each discipline needs to ask the correct questions for its clinical and research problems. When the correct questions have been asked, only then can the correct requirements be passed on to mathematicians. Experience has shown that more is achieved if we speak openly in lay terms initially rather than provide what we thhzk mathematicians want. We should not feel that we will make fools of ourselves--that will happen more likely later if we do not get our thoughts across clearly. It takes time to establish understanding and rapport between disciplines due to different terms, different connotations of terms, etc. In like manner, practitioners ought to report honestly their

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observations rather than contribute stylised statements 1o the literature. For example, in the editorial on Cognitive Science, it was noted that friction is affected with changing surface texture by friction being greatest when the surface textures are extremely smooth or extremely r o u g h - - a n d less in between the two extremes. As a result, the orthodontic/mathematical lexicon would yield the practical idea of 'not too much or not too little' as corresponding with the mathematical term 'parabolic.' Until the lexicon is established, it does not matter that the clinical observer does not know the precise mathematical t e r m - - a s long as the observation is described precisely and honestly in lay terms. To repeat the point above, too often people start interdisciplinary dialogues by using terms which they think the second person wants to hear, but all too frequently the full connotation and ramifications of the term is not understood by the first person and therefore it is misused.

CONCLUSION An earlier point will be briefly expanded. If Chaos Theory, along with expert systems (Mollenhauer 1987b), upgrade the status of the detailed working knowledge of a profession, as they should, then it will have a significant effect on the literature. Firstly, the proportion of clinicians as referees may need to be increased. Secondly, the day-to-day expert knowledge of practitioners should be utilised by more contributions to the literature and by being consultants for research design because of their familiarity with the parameters of the subject related to Observability Theory (Mollenhauer 1987d, e). This would ultimately mean that a greater percentage of a profession would be involved in the literature which would be of benefit from all perspectives--producing, reviewing and studying. Chaos adds yet another method to the suggestions made in the editorial on Cognitive Science--that studies need a range of methodologies. Chaos Theory adds a method using sample numbers well above the large number of observations seen even during many years of practical experience. That is, data should 1re pooled by many practitioners to give the type of patterns seen with Chaos methods. Conversely, below even the statistical sample sizes, many decisions must be made from meagre information, due the rarity of some conditions and responses. This qualitative area must utilise insight. The practitioner's definition is 'hzsight

is the art of drawing tenable conclusions from i{zsufficient data.' Naturally, these tenable conclusions need to be confirmed in the long term with more information, but the advantage of tenable hypotheses is that the correct path for formal exploratory analysis is chosen. Furthermore, because large numbers are seen during years of experience, insight can also be applied 1o subgroup trends seen by a practitioner. Therefore, to reiterate an earlier suggestion, clinical and practical observations ought to be recorded in the literature to guide non-practicing researchers and mathematicians in research design. Finally, it is easy to predict that there will be an irruption of Chaos based articles in the literature in the years ahead. Unfortunately many of the sophisticated advances in statistics and cost benefit analysis in the late 1980s have not yet found

Am. J. Orthod. Dentofac. Orthop. August 1992 their way into the orthodontic literature, and there is a danger that they will be swamped by the attractive concepts of Chaos. As so many of these editorials conclude--the correct balance must be achieved between all research methodologies.

Acknowledgments Recognition is given to the trouble taken by Mr. James Prato, of the Queensland University of Technol~y, for supplying reprints and computer software regarding Computer Based Training; and to Dr. William Weekes, a Sydney orthodontist, for supplying a copy of the Newsletter of The University of the World. In fact, Dr. Weekes' father is the Australian representative of that organisation. REFERENCES 1. Brown T, Travan GR. The Garn pattern profile analysis on personal computers. Aust Orthod J 1988;1 i(2):165-70. 2. Cadell WB. Steroids and orthodontic treatment. Aust Orthod J 1989;11(2):130. 3. Crilly AD, Eamshaw RA, Jones H. Proceedings of conference on fractals and chaos. London: Springer-Verlag, 1991. 4. Davies P. Introduction. In: Davies P, ed. The new physics. Cambridge University Press, 1990. 5. Davies TM, Shaw WC, Worthington HV, Addy M, Drummer P, Kingdon A. The effect of orthodontic treatment on plaque and gingivitis. AM J OR'roODDErcrOrAr ORTHOP 1991;99:155-61. 6. Ellis HD. Experiences of computer based education at Queensland University of Technology. CTISS File 1991;11:38-41. 7. Ford J. What is chaos that we should be mindful of it? In: Davies P, ed. The new physics. Cambridge University Press, 1990. 8. Freeman WJ. The physiology of perception. Sci Am 1991;264(2):34-41. 9. Freer TJ. Interview. Aust Orthod J 1989;11(1):49-53. 10. Gleick J. Chaos: making a new science. London: Cardinal by Sphere Books Ltd, 1990. It. Goldberger AL, Rigney DR, West BJ. Chaos and fractals in human physiology. Sci Am 1990;262(2):34-41. 12. Holdaway RA. A soft tissue cephalometric analysis and its use in orthodontic treatment planning. AM J OR'mOO 1983;84:1-28. 13. Hookey SR, Gooday RG. New developments in genioplasty. Aust Orthod J 1989;11(I):3-6. 14. Jacobs SG, Wright J. Troubled orthodontic patients: the role for the social v,'orker. Aust Orthod J 1990;1 l(4)[in press]. 15. Jacobson A, Caulfield PW. Introduction to radiographic cephaIometry. Philadelphia: Lea & Febiger, 1985. 16. Jurgens It, Peitgen H, Saupe D. The language of fractals. Sci Am 1990;263(2):40-7. 17. Kokot S, Ellis D. A useful working CBE in chemistry program based on the AUTIIOR system. Proceeding ASCILITE 1988;88:i75-9. 18. Leshan J, Margenau H. Einstein's space and Van Gogh's sky. New York: Macmillan, 1982. 19. Little RM, Wallen TR, Reidel RA. Stability and relapse of mandibular anterior alignment--first premolar extraction cases treated with traditional edgewide orthodontics. AM J ORTHOD 1981;84(4):350-1. 20. Little R, Reidel RA, Artun J. An evaluation of changes in mandibular anterior alignment from 10 to 20 )'ears posttreatment. AM J OwruoD DE."rroFAcORrHOP 1988;93:423-8. 21. Lorton J, Rethman MI'. Statistics: curse of the writing class. Am J Endo 1990;16(I):13-8. 22. McSherry A, McArthur E. Computer based training with 'Author.' Your Computer 1991;Jan:94-5.

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23. Mollenhauer B. Comparison of qualitative and quantitative research designs. Aust Orthod J 1984;8(3):104. 24. Mollenhauer B. Diagnosis and treatment planning in 9-dimensions. Aust Orthod J 1985;9(2):251-9. 25. Mollenhauer B. New approaches to the Begg technique. Aust Orthod J 1987a;10(2):67-89. 26. Mollenhauer B. Expert system rules. Aust Orthod J 1987b;10(2):128-32. 27. Mollenhauer B. Non-linear reading. Aust Orthod J 1987c;10(2):126. 28. Mollenhauer B. Observability theory--Part I. Aust Orthod J 1987d; 10(2):I 26-8. 29. Mollenhauer B. Observability theory--Part II. Aust Orthod J 1987e; ! 0(2): 186- 8. 30. Mollenhauer B. 'Exploratory data analysis' display. Aust Orthod J 1988a;10(4):352-4. 31. Mollenhauer B. Warning about anabolic steroids during orthodontic treatment. Aust Orthod J 1988b;10(4):264. 32. Mollenhauer B. Expert system database forresearchdesign. Aust Orthod J 1988c;10(3):190. 33. Mollenhauer B. Applicationofcognitive science to orthodontics. Aust Orthod J 1989;I I(I):61-8. 34. Mollenhauer B. Classification of dental ankylosis. Aust Orthod J 1990a;[in press]. 35. Mollenhauer B. Spiral thinking. Aust Orthod J 1990b;[in press]. 36. Mollenhauer B. Digitisers (In: Caveat communiques). Aust Orthod J 1990c;[in press]. 37. Mollenhauer B. The need for a science of philosophy for orthodontics. Aust Orthod 1990d;[in press]. 38. Mollenhauer B. (In: Editorial) Non-related labial surface angles for optimum aesthetics and stability. Aust Begg Orthod Newsletter 1991;1 I-2. 39. Parker AG. Interview. Aust Orthod J 1988;10(4):252-8. 40. Petrovic A, Stutzmann J. The concept of the mandibular tissuelevel growth potential and responsiveness to a functional appliance. In: Graber LW, ed. Orthodontics: State of the art. Essence of the science. St Louis: CV Mosby, 1986:59-74. 41. Rosenthal R. Experimenter effects in behavioral research. New York: Appleton-Century-Crofts, 1966. 42. Sadowski C, Sakols El. Long-term assessment of orthodontic relapse. AM J OR'nIOD 1982;82(6):456-63. 43. Steele WW. Survey of information for health professions education: Conducted as part of the Clearinghouse Project. California: University of the World Newsletter, 1990:1-29. 44. Valdes R. What is biocomputing? Aust Personal Computer 1991; ! 2(4):77-83. 45. Viazis AD. A cephalometric analysis based on natural head position. J Clin Orthod 1991;25(3):172-81. 46. Vig PS, Weintraub JA, Brown C, Kowalski CJ. The duration of orthodontic treatment with and without extraction. AM J ORmoo DEt,rroI:ac ORnioP 1990;97:45-5 !.

Example study: ROOT RESORPTION FROM EXTRUSIVE MOVEMENTS Several students in the Melbourne orthodontic department have sought to find literature references to root resorption due to extrusive traction, since some older practitioners have consistently stated that they were taught its importance decades ago and have noticed it themselves over the years. There are two problems associated with getting such information into the literature. One is that the research design is difficult, as discussed later, and the other is that few are willing to show such a condition in their own treated cases,

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in case it comes back to haunt them or their technique--later. Despite the idealised Visualised Treatment Objectives supplied by some commercial companies, patients with the same morphology do not respond the same way to the same appliance with the same activations. What is more, even the same patient can respond differently at different times. For example, the clinician may be struggling to reduce an excessive overbite for many months or a year or more, when suddenly the patient presents with clear signs that it has reduced within two appointments. Questioning the patient will frequently show that the patient has had a cold in the intervening period. The same may apply to patients of a mild dolichofacial type who have limited overbite at the start of treatment, and produce annoyingly frank open bite during treatment. In other words, it is not possible to predict with great certainty and accuracy which patients will suffer backward mandibular rotation and when. Thus interceptive measures may not be indicated or routinely applied. Therefore, as well as open bites before treatment, it is not rare to see the need to close down an anterior open bite which has occurred during treatment. If it occurs, it is far more convenient to instruct the patient to place anterior vertical elastics than change to new light archwires to reduce the open bite. Light vertical elastics seem to be associated with a higher incidence and degree of root resorption than other movements. Practitioners experienced in the use of the so-called 'Squeeze Technique' where many heavy vertical elastics are employed state that less resorption is noted than conventional vertical elastic traction. This may be due to the rapidity of closure of the open bite with less jiggling. Other practitioners insist that elastics should not be used for vertical extrusion at all, for two reasons. The main one is that intermittent wear, either due to non-compliance, or removal for eating, causes the jiggling. Another reason suggested for resorption is that some patients develop a habit of opening their mouths to exaggerate the effects of the elastics, but again, intermittently. Such clinicians usually prefer archwires to close open bites. Some of these clinicians state they prefer heavy arches with step up bends. Others prefer lighter archwires, and some ultralight. The important point of this paragraph is that while all this information is anecdotal, it could be closer to the truth than stating that vertical traction is not associated with a higher incidence of root resorption than other movements. The problem with a research design to study this is that patients respond differently to tooth movement. Most of the literature indicates that females suffer more resorption than males. Around puberty the problem seems to be accentuated in females. Therefore, one requires larger numbers for observation than the usual statistical samples to glean trends as discussed in the body of Chaos Theory and Fractals. Another sampling problem is that normal treatment with modern mainstream appliances does not produce a high incidence of gross root resorption, hence there is a need for pooling observations by a survey of many experienced practitioners. Since no institutional ethics committee would condone prospective studies on root resorption, apossible protocol would be: 1. Do a phone survey as did Vig et al. (1990) of orthodontists

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to seek (a) those who had had more than ten years experience, and (b) who routinely take radiographs near the end of, or after treatment. Such radiographs could be periapicals, OPGs, or 'hard' cephalometrie films which show the dental features better than films taken with less penetration. 2. Ask those Who meet the above criteria if they would be prepared to answer a written survey anonymously--so as to encourage honesty. 3. The survey sent to these practitioners should include such questions as: (a) What do you think are the major causes of root resorption in descending order of significance? (b) Have you noticed a higher incidence of root resorption where vertical movements are involved? If so, extrusion or intrusion? Furthermore, heavy or light forces? and continuous or intermittent? (c) How haveyou tackled the various factors you have considered important over the years, and what have been your more successful experiences? (d) Have you done any studies, either formal or informal? (e) How much resorption do you feel is acceptable? (f) How do you rate the following factors? 1. Facial type

Ant. J. Orthod. Dentofac. Orthop. August 1992

2. 3. 4. 5. 6. 7. 8. 9.

Age Sex Hormonal (eg around puberty) Microrhinodysplasia Turned up noses Racial and ethnic groups Orthodontic force values Other: (f) Any further comments? The critical part of the questionnaire would be the interpretation of the answers as Davies et al. (1991) had to do in their paper. Those familiar with the 'Rat maze experiments' (Rosenthal 1966) will see a parallel: it will not be so much the aetiology which is being determined but the relative incidence of root resorption related to vertical traction. In the Rat maze experiments, the researchers' ability to do objective research was really being studied rather than the intelligence of the rats. As with studies in quantum physics where the observer affects the results as mentioned before, unfortunately these experiments showed that observers' preconceptions influence the results of studies. Question 3(e) may be significant since it may show the priority the clinician gives to the problem. The clinician who agonises over eliminating all resorption, like those who wish to eliminate all loose brackets, may try the hardest to ascertain and remove the various factors.

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