Chaos theory governs most aspects of human life and the order of most things in the universe but its significance was not understood by the scientific community until the last half century. Although it is dominant, its importance is not yet comprehended by the public at large.  Its name alone is a put off!  This paper, following on the heels of the CCCC Newsletter of January 2005 on Emergence, attempts, herewith, to provide information related to chaos concepts and to make them better known to people within the CCCC circle of influence.

Clearing the Bum Rap

Although there is a chaotic aspect to chaos theory, this misnomer gives it a bum rap because it is anything but chaotic.  In fact it creates order and predictability but from very simple, unsophisticated beginnings.  The ‘chaos’ portion falsely stems from two considerations:

(1) Although one can predict the general outcome, one cannot predict the specifics of the outcome when chaos theory is at work.

(2) The process itself defies our usual intellectual consideration of how things should be.  It behaves to most people, in a counter-intuitive fashion, and thus seems chaotic.

An example of consideration (1) is a snowflake.  It forms on the basis of two simple decisions being made about surface tension and amount of heat in the air; the crystal forms as it passes through each level of the atmosphere.  What we DO NOT know is the actual shape the snowflake will take as it forms.  What we DO know is that a perfectly formed snowflake will result. Consideration (2) occurs because the formation of snowflakes appears to be random, but it is not random.  It is influenced by the very tiny force of water surface tension and the varying temperature levels during the decent from the sky – a balance between the forces of stability and instability, respectively.  Since each combination of temperature and surface tension will never be identical, each snowflake will be unique.

 The Simple Beginning of the Chaos Process

 As stated above, the chaos process starts with very simple decisions being made in the formation of the snowflake.

The formation of the ant colony, as detailed in the January Newsletter [1], begins with the harvester ant’s foraging for food.  If a foraging ant notices that most of its neighbors are also foraging, it makes the decision to switch its role to nest building (and care of infant ants).  It picks up this information only from the neighboring ants.  If there is an equilibrium between the number of foragers and nest builders that it sees, the ant continues as is – a simple binary (0, 1) decision for the ant. “If there is equilibrium more or less, I continue as I am.  If there are too many ants doing what I am doing, I will switch my role to the other one.”  There is no examination of the global picture but simply a response to local conditions.

Everyday Examples

Beside snowflakes and ants, common phenomena uniquely defined by chaos theory are:

Turbulence of air currents

Water flow

Oscillations of the heart

Fluctuations of wildlife populations

Cigarette smoke breaking into wild swirls

A flag snapping in the wind

Dripping faucet changing pattern from steady to random

Disease cycles

Behavior of weather

Cars clustering on the freeway

Oil flowing in a pipeline

Airflow over an airplane wing

DNA, which is formed from an a-periodic crystal, rather than periodic (chaotic rather than orderly).

No matter what medium, the behavior obeys the newly discovered laws related to chaos.

Rules of the Chaos Game

Some keys that govern chaos can be stated in different ways:

  • Tiny differences in input will quickly become overwhelming differences in output.
  • Small perturbations in one’s daily trajectory can have large consequences – a person is one minute late for a regular daily train so cannot get to work for the day. His absence at work results in a million-dollar error by his replacement.
  • Locally unpredictable actions result in a globally stable result.
  • Life extracts order from a sea of disorder.
  • Patterns are born from formlessness.
  • A single set of rules embodies the final shape (or shapes).

This last point can be illustrated yourself by playing the chaos game, developing your own chaotic results.

Create your own Chaotic Results

Without a computer you may create your own example of achieving results the chaos way.

  1. Gather up a coin, a pencil and some squared paper.
  1. Make up your own rules arbitrarily that tell you how to take one point to another. Realize however, that those rules determine the final result.  For example the rules might be:

a) If the coin is heads: Move 5 cm. (or squares) to the southeast.

b) If the coin is tails: Move ¼ closer to the center of the sheet.

3.Start flipping the coin and marking the resulting points.

4.As an option, throw away the first 50 plots which will appear to be random.

5.The next 100 points, however, will produce a distinct shape, getting sharper and sharper as the game goes on. The first 50 points are the initial iterations and may be kept by a purist, but it is the latter points that produce the real result.  Imagine that nature does it, not 150 times, but millions of times to tie down the exact shape or results.

  1. Do it again with exactly the same rules and note the new image because of the randomness of the coin flipping – but not randomness in the overall process.  This will show you that any slight change in the variables will produce an obviously different appearance of the final result.
  2. Of course, it is better to play this game with a computer, a graphics screen and a random number generator.

The randomness is in the flip of the coin, but not in the process.  That randomness is to take things that occur naturally and to work with them – but within the overall structure or order – not unlike the harvester ant’s choice to forage or to nest.  What they choose to do is random from a mathematical perspective, although, for the ant, the ‘why’ is known.

As stated above, a single set of rules embodies the final shape.

The challenge then is: knowing the shape you want (such as a fern leaf), how do you choose the set of rules that define it?   These are now understood.

Second Order Equations

The first order equation, for example, X2 + Y2 = 1 describes a circle. However from the chaos perspective second order equations are used to iterate a shape rather than describe it.  Thus the equation becomes a process rather than a description – dynamic instead of static.  When a number goes into the equation a new number comes out; the new number goes in, and so on, the resulting points hopping from place to place (Xnext = px[1-X]).  A point is plotted not when it satisfies the equation but when it produces a certain kind of behavior such as steady state, convergence to a periodic repetition or out-of-control chaos.

Key Words in the Trade

Emergence: evolved from the observation of a network of self-organization of disparate agents that unwittingly create a higher-level order [2].  The results ‘emerge’ from self-organization that transcends local discipline to solve problems by comparing behavior in one area to behavior in another.  That is, the behavior of the group grows smarter over time, responding to the changing needs of the environment.  The common features of emergent systems are that they solve problems by drawing on masses of relatively simple elements rather than being organized by any hierarchy.  Emergence therefore is defined as the movement from low-level rules to higher-level sophistication.  In a word, it is bottom-up behavior that organizes the final desirable result vs. top-down thinking – decentralized systems vs. centralized systems – “the eerie invisible hand of self-organization”.  Within it, a neighbor is influenced by neighbor, which in turn influences back – such as the harvester ants.  All emergent systems are built on the two-way feedback, the two-way connection that ultimately fosters higher learning.

Attractor: a fine structure hidden within a disorderly stream of data – an artificial pair of points (or wings) around which plots of the data circulate, never crossing one another but forming a beautiful image [3].  First developed by Edward Lorenz in 1963, the attractor provides a visual image of the mathematics of chaos built around two spiral wings. A steady equilibrium is reached of points that attract all others; no matter the starting population, it will bounce steadily towards the attractor.  Then, with the first attractor period-doubling, the attractor splits in two, like a dividing cell.  At another period-doubling, each point of the attractor divides again at the same moment to rest in one of the two wings.

Fractals: the name applied for irregular shapes such as those of shorelines, trees, leafs or body parts.  It offers a means to measure and specify the degree of roughness, brokenness, irregularity, jaggedness or fragmentation of an object.  A fractal description, of bronchial-tube branching, provided answers where none existed before. (Also, the urinary collecting system, the biliary duct in the liver and the network of fibers in the heart that carry pulses of electric current to the contracting muscles proved fractal.) How did nature manage to evolve such complicated architecture?  Not so: the complications exist in conventional mathematics, as fractals branching structures can be described in transparent simplicity, with just a few bits of information.  Fractal’s new geometry was nature’s own!

Mandelbrot Sets:  these are the mathematics of fractals (see above) that allow drilling down into finer and finer detail without limit (that, as a consequence visually display more and more beautiful patterns).  The mathematics involves removing segments of the numbers, level after level until you are left with a ‘set’ or ‘dust’ of points arranged in clusters, infinite in quantity yet infinitely sparse.  Imagine a set that displays the shore of Britain, then a piece of the coast, then a specific beach, then a yard of sea-edge, then a grain of sand, then portions of the grain, etc.

Bifurcation: the chaos-based pattern moves from extinction to a simple steady state and then to a doubling state and re-doubling and then to chaos.  It is the doubling and re-doubling state that is termed bifurcation.  An example close to home is cell division of the embryo.

Key Names in the Trade

 Edward Lorenz, a meteorologist, postulated the butterfly effect. Discovered when he cut off the last two decimal places in some numbers in an experiment he was repeating, he found that he had a totally different weather pattern result.  A small numerical order was like a puff of wind interjected at the start that creates a totally different pattern – the butterfly flapping its wings in the Amazonian jungle creates a storm in Texas two weeks later.

James Yorke, a mathematician, followed Lorenz’s work and impressed upon other mathematicians the importance of understanding that first there is disorder – not order.  Disorder is not something to be waved off or described as ‘noise’ in the experiment.  He also proved that if ever a regular cycle of three appears (in a one-dimensional system) the same system will also display regular cycles of every other length and hence chaos.  He showed that chaos is ubiquitous (everywhere), it is stable and it is structured.

Robert May, a biologist, created a complete graphic from dormant to chaos for a second order equation.  He argued that chaos should be taught to science students because the science being taught gave the wrong impression.  No matter how elaborate linear mathematics could get – Fourier transforms, orthogonal functions, regression techniques, it was a very small piece of the world dominated by second order mathematics.  Thus the young scientists were misled about their linear world in an overwhelming nonlinear world.  May showed that the student at the time was ill-equipped to confront bizarre behavior exhibited by even the simplest systems.

Benoit Mandelbrot, a mathematician in economics within IBM, while looking for predictable variations showed that cotton pricing remained constant over a 62-year period despite two World Wars and a depression.  Within a disorderly realm of pricing lived an unexpected kind of order.  He asked:  Why should any law hold at all?  And why should it apply equally well to the arbitrariness of personal income, cotton prices and noise in telephone lines?  Solving the noise problem with out-of-the-box thinking, Mandelbrot identified a consistent geometric relationship between the burst of errors (noise) and the spaces of clean transmission using a mathematical tool he called the Mandelbrot Set.  Bursts of errors had always sent the engineers looking for a man sticking a screwdriver somewhere.  But Mandelbrot’s work suggested that noise would never be explained on the basis of specific local events.  He showed that the relationship of errors to clean transmissions remained constant.  He applied the same methodology to help the Egyptians forecast the behavior of the flooding Nile River.

Mitchell Feigenbaum, a physicist at Los Alamos nuclear laboratories, discovered a constant for all situations – thus, that all functional equations used to describe the mechanics were irrelevant.  When order emerged, it suddenly seemed to have forgotten what the original equation was.  The regularity was the same for each – parabola, sine wave – it didn’t matter.  He discovered that he could predict the precise values of each point on the attractor (see above) and predict the actual populations reached in the year-to-year (for example) oscillations.  The numbers obeyed a law of scaling; functions could be scaled to match each other.  He created a universal theory which meant that different systems would behave identically. The theory expressed a natural law about systems at the point of transition between order and turbulence.  Now the different turbulent frequencies could be seen to come sequentially.  It could be measured and predicted, spanning the disciplines of mathematicians, physicists, geologist, biologists, etc.

The Conflict

Simple shapes such as skyscrapers of New York are unnatural, despite the architects’ efforts to dress them up.  The buildings fail to resonate with the way nature organizes itself.  In fact they fail to resonate with the human perception: humans find beauty in the silhouette of a storm-bent leafless tree against an evening sky in winter but rarely in skyscrapers – an example of the orderly situation conflicting viscerally with the chaotic approach.

 Practical Applications of Chaos

(a) Commercial Problem solving:  The process starts by accumulating information, from a group of people seriously affected by the problem, in a random and thus, disorderly fashion – any idea, any thought, important or trivial, hurts and joys, etc. [4].  Participants accustomed to orderly processes are disturbed by the approach that seems to be leading nowhere and even try to thwart it by developing top-down suggestions instead of going along with this bottom-up management strategy.  Of course, it is exacerbated for them because the posting of so many thoughts stimulates the creation of even more thoughts.  These resisting people may attempt to organize the data before the accumulation has been completed.  We tell them to hold on till we’re done.  With the build up of a hundred or more thoughts now before the group, we then move onto an organizing of the information, not by a planned route but by allowing the pieces of information themselves to tell us how to form the structure.  For example: “Many of these thoughts are about time frames, so let’s group all the time-related ideas together.  Quite a few of the concerns have a financial connotation, so let’s put these financial ones into the same column.  If a thought spans both orders, then let’s place it in both.”   The first order creates a path to a further order in exactly the same way.  It also stimulates more thoughts and concerns and each such new idea is explored and is added to the accumulations for sorting as well.  Then the order converges and converges until a solution appears – a beautiful solution because everyone in the room sees its elegant practicality.  That solution is unique to the situation based on all the preliminary information – which is, at the end, tested against it.  There is unanimous consent – a solution without compromise, the epitome of the 100% success rate – thanks to a chaotic start.  No one, not even an experienced facilitator, could have seen the solution in advance.

Simple choices are made from local information leading to a global solution.  One sub-result leads iteratively to another result and so the final solution is achieved.

(b) Commercial Job Finding: Searching for a career position travels the route of the unseen or hidden job market where about 75% of new jobs lie [5].  Since they’re hidden, the challenge is to find them.  There is no predictable path, only a predictable method.  We begin by peeling back the curtain that exposes the hidden job area by having the candidate talk with personal acquaintances.  Understand that many of these candidates are new to the city or the country.  “I have no friends or contacts here; I don’t know where to start.”  We start chaotically, i.e. with a simple unplanned path for Tom – his sister, his insurance agent and his neighbor.  These lead us to a new level of contacts and by the third iteration we are seeing signs of jobs in the candidate’s area of expertise.  By the sixth iteration we hone in on one or more ideal job choices – ones that provide a thrill, not a struggle for candidates.  (We have developed statistical histograms that show medians and distributions for contacts.)  The point is that we don’t know where we are going or exactly how it will end up with our ‘local-first’ approach but it leads to a global result – the finding of a job in the specialist’s desired field.  We coach many professionals – engineers, PhD’s, scientists and software developers – in bad times and in good, usually with remarkable results.

The arbitrary aspect is that the information interview (not a job interview) either leads to another interview or it halts the process for that stream, in which case a new stream must be initiated.


Order is merely a small subset of chaos.  It is important to appreciate chaos, and to allow ourselves to sometimes be led by it.  Chaos is the much more prevalent aspect of nature, not only because it affects us daily but also because it leads to elegant solutions every time – as elegant as a snowflake or a fern leaf.