GOLDEN ARCS

An Introduction to Nonequilibrium

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When we see a world class racer carve a smooth and powerful giant slalom turn, we cannot fail to be impressed. What we are witnessing is a skier in extreme "far from equilibrium" condition. In this condition, new organising principles take over and we are impressed by their power, accuracy and apparent total defiance of nature. The aim of this article is to introduce "nonequilibrium" and identify several of the associated organising principles.

The System

Skier, equipment and local environment combine together to form a system. The system is deemed "Open" because it takes energy from outside of itself (Gravity) and in turn loses energy through Friction. This loss of energy is called "Dissipation". Another feature of the system is "nonequilibrium" itself. This is basic in skiing as any change of speed or direction is an acceleration and constitutes "nonequilibrium". However, there is more to nonequilibrium than this as we shall see and we can start now to use the term "far from equilibrium", as with the world cup skier. Before we move on let's recap on what we have so far:

We have a system which is:

To see some of the other aspects of nonequilibrium we need an appropriate model. The best model appears to be the simple pendulum. There are two kinds of movement possible for a pendulum. One is "oscillation" and the other "rotation", shown as follows:

We must also make our pendulum model an Open system, which is done by keeping the pendulum moving by applying forces at the base (pivot point).

If the pendulum swings right round to the vertical 12 O'clock position, it finds itself on a boundary between the two different kinds of motion; oscillation and rotation. By applying forces at the base (pivot point) we can keep the pendulum near to this boundary position:

Some people will have already noticed that this model is also the basis of most ski simulators, using springs or elastic (and leg power) to supply the force at the base.

The force at the base is the fourth feature of our system and it is called "nonlinear feedback" which we will list as "D" along with our three other system requirements.

We now have a system which is:

The reason why the feedback is nonlinear (and not just a simple regular force) will soon become apparent. We now need to look closely at the "boundary" between oscillation and rotation because it has very special qualities. The boundary turns out to be "fuzzy" in the sense that it is not a sudden clear cut boundary. The following diagram represents forcing frequency plotted against pendulum position - don't let that worry you, just consider the white area as being oscillation and the black as rotation, with an obvious boundary in the middle. If we now take a small section of this diagram and zoom up on it we can see that it is constructed of fine irregular bands. We can now zoom up again and the image is identical :

We can repeat this zooming infinitely many times and will always see a similar but unique image - this is because we are looking at a "Fractal" which is an example of "Deterministic Chaos". This is the reason why our "feedback" is nonlinear (feedback (as with fractals) is about "encoding similarities"). It is also a common feature of boundaries in nature and is part of the reason why each snowflake we ski on is similar but uniquely complex. The following picture of an ice grain snowflake was generated on a home computer using a fractal algorithm:

Chaos however is not our aim, we only want to go as far as "The Edge of Chaos". Our nonequilibrium system has already revealed some surprising features beyond that of "accelerations", but none more surprising than that which we find on the edge of chaos. The edge of chaos is occupied mathematically by a number. This number is itself a fractal and has some very special properties. The number is 1.6180339887499... and is called "The Golden Mean".

The Golden Mean

The Golden Mean as a "ratio" has been known about for thousands of years, but only recently understood in terms of nonequilibrium dynamical systems. Before we look at what it does for our system we can first take a very brief look at The Golden Mean itself.

There are three ways to view The Golden Mean:

Bear with the following definitions A and B, because they help understand the more interesting definition C.

A : Pure Mathematics - The Golden Mean is represented by the Greek letter "phi" and computes exactly from (1+sqrt(5))/2. Phi possesses two qualities: Arithmetic proportion and Geometric proportion.

Arithmetic (linear) proportion is when a quantity is changed by ADDING some amount.

Geometric (nonlinear) proportion is when a quantity is changed by MULTIPLYING by some amount.

The special quality of phi is that it possesses both at the same time, making it simultaneously Linear and Nonlinear. This is mathematically represented as "Phi + 1 = Phi x Phi"

B : As a number series - If you make a list of numbers starting with "0,1" where each new number is the sum of the previous two you get the following series:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ......

If you divide any number by the one before it in the series (except the first pair) you get a ratio which tends towards phi - the further towards infinity the list goes the more accurately the fractal phi is defined. (21/13 differs from phi by only 0.003) This means that each new number in the series can be derived by multiplying by this ratio "phi" as well as by addition - it is both Linear and Nonlinear.

C : Geometry - The Golden Rectangle. This rectangle springs forth naturally from starting with a single square 1 x 1 in dimension. Swinging the rectangles forms a number series identical to the one above: 1 x 1, 2 x 1, 3 x 2, 5 x 3, 8 x 5, 13 x 8, 21 x 13, 34 x 21...

The Golden Arcs

OK. We now know that this ratio "phi" exists at the edge of chaos in our nonequilibrium system, but what, we ask, does that do for us? Well, the answer is: Just about everything ! Here are a few features which are relevant to us:

It is obvious in skiing that the whole body is used in the process. It is not like driving a car where the body is effectively a passenger. Due to this fact it would be natural to expect to see evidence of "phi" at this level, to visibly see how phi identifies our system trajectory. I suspect that we do, in the form of "Golden Arcs". By this I mean the tracks left in the snow by skiers executing good carved turns, especially on "parabolic" skis, which appear to be a major technological advancement. The following image is constructed using the ratio phi. The shapes should be very familiar.

Hold the page in front of your eyes so that it is at the angle of a 25 to 30 degree slope. The hook shape appears to smooth out - exactly as we see when we look back up a hill we have skied down. Two adjacent squares within the Golden Rectangle appear to form a complete "closed" turn. This is of considerable interest to anyone who has ever tried to set a race course and thought about where a turn starts ,ends and how wide to set the poles.

Equation of Skiing

It is instantly obvious that there is a clash here with the classical view that a turn has a continually varying radius. It would certainly be naive in the extreme to think that there were no other dynamics of "Flow" involved in determining a skiers trajectory. We will have a look at such dynamics a little later. Before that we need to look at the classical approach which is epitomised by John Howes Equation of Skiing. In this approach "feedback" is completely ignored and the turn is considered to be determined only by the ski - this being the only way an appropriate equation can be derived through classical mechanics. The equation allows you to calculate the radius of turn at any point in the turn. It uses a quadratic equation to solve for upper and lower quadrants of the turn, giving two radii in a similar manner to the Golden Rectangle. A random example of the calculation for traverse angles of 45 degrees in both upper and lower quadrants is given as follows:

The ratio of the two radii for the above example comes to 1.59, differing from phi by only 0.028, despite the curve being irregular and taking 45 degrees as only an approximation to the mid point of each of the curves inside each quadrant. If you compute with more points and average them, the result generally comes even closer to phi.

This points towards an oscillation around the value phi even when using classical mechanics, without the aid of feedback. Remember that by using feedback we can make constant adjustments which are not allowed by this equation. John Howes equation was formed over twenty years ago when he was designing skis for Head. It is interesting to note that Head has currently produced the most successful range of Parabolic skis. It would be interesting to see if Parabolic skis scribe an arc which is closer to Phi. There are more questions here than answers and on the whole it would be most surprising if overall ski dynamics did not contrive to deviate from phi.

Harmony

Our system operates as a Whole, within an "envelope". From our model system we can see that if we keep the pendulum close to the vertical, almost stationary with only small movements at the base, then the envelope is small. We can however push this envelope to the limits, which correspond to the incredible edge angles and inclination of world cup skiers - or Joe Bloggs on Parabolics.

If we can see the Golden Arcs of whole turns, we must realise that small, reflexive movements of the legs are feedback-led and fractals of the Whole. This creates a certain "Harmony". When people talk about being in "balance" they are referring to this feedback-led Harmony. Loss of "balance" is a loss of Harmony with a breakdown in feedback, leading to either oscillation or rotation straight out of the envelope - i.e. falling over.

Use of the term "balance" is first and foremost a result of complete ignorance of the above phenomena. Our "system" shows features and phenomena which operate at a level of organisation which cannot be reduced to or explained in the "equilibrium" terms of classical mechanics. It is equally false to try to accommodate the word "balance" just because it is in common use - it simply cannot express the above phenomena with any level of understanding. Nonequilibrium and Balance are opposites.

Synthesis

"Physical Harmony" works best when skiing is allowed to be fully dynamic, during full expression of the dynamic qualities of Flow, Form and Directing of Momentum. Ski instruction traditionally and currently deals mainly with the skiers' Form. This is inevitable when the main approach in teaching is "Analysis". The unfortunate side effect of this approach is that it suppresses many of the skiers' natural instincts, producing skiing which is not very dynamic. The skier is left worrying about minute details of Form (index finger pointed out etc.) yet remains almost clue-less about Flow and Directing of Momentum. Good skiers often reject the ski instruction world in favour of Dynamic Quality, understanding instinctively that the two are not compatible. There are things we can do to help make instruction compatible with Dynamic Quality. The first thing is to realise the limitations of "Analysis".

Skiing is mainly about "Synthesis"; whole system behaviour and dealing with external constraints. Synthesis is the only practical way to study Flow and Directing of Momentum. The required tools are directly linked to chaos theory. The specific tools required for understanding Directing of Momentum are relatively simple and come in the form of 6ft long red or blue plastic poles with springs and large screw thread bases - a direct "hands on" approach to chaos. In contrast the best approach I have found to study Flow goes back to mathematics and is called "Catastrophe theory", developed by French mathematician Rene Thom in the 1970s as a means of visualising the behaviour of complex systems in nature.

Just as our previous model system showed us that there was a boundary between oscillation and rotation - two different types of behaviour, we can use a different model to be able to see similar features of behaviour in more complex situations. Flow is complex because there are several factors to deal with at one time. The following model has two variables only (control factors), and so can be expressed in three dimensions - the third being the resultant flow. The "external constraints", such as snow condition, terrain gradient, ski design and stance are kept constant so as to more easily understand the relationship and effects of the two control factors, which are Alignment and Timing. I have found this model practical and informative, though obviously, models can be constructed with more control factors and higher dimensionality or with different combinations of a few control factors.

Features

The model is "qualitative" in nature. This means that there is no scale in terms of "quantity". If someone was to be backed by a generous research and development grant from BASI (?%$&!), then empirical measurements could be made in the field to establish quantitative values. However, the un-scaled axes represent the control factors of Alignment and Timing. The three dimensional shape is called a "cusp" catastrophe. It is a partially folded two dimensional equilibrium surface - on which we find represented all the possible characteristics of flow resulting from Alignment and Timing.

Timing

The Timing axis goes from "Late" to "Early". This corresponds to natural body mechanics, and the timing of our walking, running, jumping movements. Walking downhill with "the brakes on" is equivalent to "late" timing. It is also what is taught by most ski instruction systems!

Alignment

Alignment is a major management issue in its own right and cannot be fully addressed here. This model though gives a good starting point. "Flow" should be the main criterion for understanding Alignment. The Alignment axis goes from "under" edged to "over" edged. If you know someone who skis habitually with their feet wide apart, and is convinced that everyone else should do the same, then the odds are strongly in favour of this person being bow legged and under edged. However, never be lured into using Alignment to alter Form, as there are also many bow legged skiers who collapse into an "A Frame" stance in an attempt to find an edge to stand on. Most people are under edged and can benefit from widening their stance. The real solution though is to have correct alignment in the first place. You can change Form considerably without changing alignment, but this nearly always sacrifices Flow.

Warren Witherell's advice on "Knee" Alignment also appears to be based mainly on observations of Form. This becomes obvious when you study Alignment in terms of Flow and Timing. Without going into detail here - suffice to say - you cannot alter the knee alignment of a straight leg - without breaking it.

Model Interpretation

Much of the model is self explanatory if studied carefully.

Skid / Hook and Wobble : "Late" timing, unfortunately the domain of most ski instruction, is the Skid/Hook and knee Wobble area of the Flow surface.

Golden Region : The Golden Region is the set of values which allows carving to take place. You can see that the skier with optimal Alignment of body and equipment (Middle of axis), will have maximum potential to achieve carved turns. This skier will have the most sensitive feedback and can carve closer to the "cusp" - The Edge of Chaos. We see that Alignment is a kind of Tuning Mechanism for Feedback.

Butterfly Effect :The Butterfly Effect of "initial conditions" can be seen where a small change in Alignment either side of the optimum - combined with late timing, can create major changes in Flow, from skidding to hooking.

Many other aspects of commonly observed behaviour can be predicted from this model.

Summary

A : We ski on a moving fractal boundary, between two qualitatively different types of motion. This phenomena exhibits a different level of organisation from the equilibrium states of Statics, and cannot be reduced to these definitions or laws.

B : Nonequilibrium definitely rules, but sometimes Chaos wins !

Conclusion

"The Balance Myth"

Thor, Odin, Zeus, Apollo ... the mythical gods of ancient history all served as explanations of the natural world. Thunder and lightning were caused by Thor swinging his hammer. If Thor was "throwing a wobbler" with his hammer then everybody knew about it, everyone understood. I spoke to a BASI trainer a few days ago who said "When I talk about Dynamic Balance, everybody knows what I mean..."

If we look at the world cup skier and say, "Look at that Dynamic Balance!" then we are "defining away" the issue - not explaining it. When we accept that the skier is out of equilibrium, then we can ask questions.The alternative to Myth is curiosity. All explanations (however correct) are best regarded as only provisional and a step towards the next question. To quote once again : "In the beginner's mind there are many possibilities. In the expert's there are few."

Literature Cited

The same as in articles 1 & 2 but with the addition of the following:

Notes

Author: Ian Beveridge 1996.

The articles in this series dealing with "balance" were:

1. "The Emergent Fuzzy Nature of Skiing"

2. "Illusions"

3. "Golden Arcs"

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