The same patterns crop up everywhere, in the shape of a cloud, in the eddies of a stream, in the movement of the stock market, even down to the frequency of division, of a persons arteries from the heart, down to the capillaries, that feed every cell in our bodies and most important of all our history. Mathematics is a way of modelling aspects of reality. Chaos Theory is a revolutionary way of mathematically modelling living systems, coastlines, population dynamics, the weather and the history of our species, it has now become possible to describe life in a new way.

The way this can be understood is with a new kind of geometry, called fractal geometry. The marvellous thing about this geometry is that it can model natural order with an equation. The way this is done, is to take an equation, feed some figures into it, then take the answer resulting from that equation, display that answer graphically, then feed the answer back into that equation. For as often as required. This process except for a few examples, was never practical before the coming of computers Some equations are not graphically relevant till this has been done for many thousands of times (not an interesting process with pencil and paper). In other words by feeding a few numbers into a simple equation taking the answer and feeding it back into the equation and keep repeating the process. You can produce an infinitely complex picture which changes continually, never repeating itself. In this sense it mirrors life which is both complex and dynamic ever developing and changing.

When we analyse reality, we strive for simplicity. When this `simplicity` is really looked at closely, it is usually found not to be quite that simple! Conversely when any aspects of reality are analysed and found too complex for understanding, the mind gives up trying. Chaos theory gives us a way out of this paradox, if the concepts of simplicity and complexity are discarded and replaced by the idea of a dynamic pattern across scale. It then does not matter how complex a system is, once the dynamic pattern has been identified the system is understood. The revolution with Chaos theory comes because for the first time in science non linear dynamics (Chaos) can be qualified. Before chaos theory we had anthropomorphic ideas of order and chaos, or put another way, it is to think that before humanity evolved upon the earth, order did not exist. This leads to our species having the collective ego trip of thinking that God is somehow human. The fossil record tells us that life has been on the earth for about four billion years. Some estimates put the modern human species existing for about 200,000 years.
Is not the idea of a human God rather arrogant?

I believe it wrong to think that our creation was a random event, or that we are an accident. The order that is observable in our bodies is also observable in the universe. One of the breakthroughs with chaos theory was the discovery by an American physicist called Mitchell Feigenbaum of a fundamental constant, this constant is a ratio which is 4.669. This constant can be demonstrated by the diagram invented by Feigenbaum.

Each bifurcation happens with a shorter distance before the next division, this distance change makes a statistical ratio, this ratio is 4.669 it is a constant that is observable everywhere in the universe. When the blood circulation system is analysed, it is in the shape of a tree. The trunk of the tree being the main artery from the heart called the aorta, which splits or bifurcates into two arteries which divides into four arteries then eight arteries on and on, down and down, until the veins become small enough to be able to deliver the requisite amount of blood required by one cell  When the big red spot on Jupiter is analysed, it is found to be a giant circular storm or eddy, the point is, that there is one big eddy, surrounded by smaller eddies. The ratio between the size of the big eddy and the smaller eddies, is 4.669 the same as the arteries bifurcating coming from our hearts. This observation should tell us, that the order observable in the universe, is also observable in our bodies. This ratio is also visible in the home; Take a glass, then put in it some washing up liquid and cold water, cover the top and shake vigorously. If you look closely at the bubbles, you will see the sizes of the bubbles fall into classes. The statistical ratio between the size of the classes of bubbles is 4.669. Just the same as our arteries.

A good example for non linear dynamics (chaos) is to take an island and measure its coastline, the greater the accuracy of measurement, the longer the coastline, this sounds counter intuitive, but if you physically get a tape measure and walk around the island measuring each inlet and bay, as each smaller inlet on every smaller inlet is measured the figure for the circumference increases. This principle can be mathematically shown with a geometric construction called the Koch curve, this can show the seemingly impossible idea of putting an infinitely long line, in a finite space. It is very simple, draw a circle and in the circle draw an equalatrical triangle with each vertex touching the circle, in the middle third of each side of the triangle, draw another equalateral triangle. It is possible to keep on drawing smaller and smaller triangles on the sides of the larger triangles. There is always enough space to draw another triangle. The point is every time another triangle is drawn the line in the circle gets longer. In classical mathematics using calculus to work out the gradient of a curve, is done by magnifying it until it approximates a straight line. No matter how much the Koch curve is magnified, it never becomes a straight line, the same pattern is found no matter how much the magnification, it is a fractal and has self similarity.

I have discovered a way to describe the meaning of our existence in terms of self similarity, the thing that is true of life and true of us, is that it is generative. The pattern that exists common to all life is procreation, it exits on a cellular level, in vegetation everywhere, all life wants to grow into more space, nature abhors a vacuum, this urge is true in a business sense and a human sense. ( Earwigs want to turn the universe into a universe of earwigs! ) We have evolved the ability greatly to increase the space for all life to grow into. Many of our problems are caused by the fact that, we are running out of space to be dynamically generative in.

A common metaphor for chaos theory is; 'The flap of the butterflies wing', this phenomena was first identified by the meteorologist; Edward Lorenz in the early 1960's. He was trying to model the weather, on a computer with three non-linear differential equations. Like many other breakthroughs, this breakthrough happened by accident. He started his program with three values:

X=8.1646812788: Y=8.9676166531: Z=25.500286726

The computer program he wrote took these X, Y and Z start values, then passed them through the equations, he had to written to model the weather, the answers that were then obtained, were displayed graphically '@'. Then his program took these new values and passed them through the same equations, then displaying them graphically. ( return to the '@'). So basically he ended up with a graph, in the form of a wavy line, to represent what the weather was doing. His beautiful accident happened (like it does with most accidents), because he was in a hurry. Next time he ran his program. Instead of inputting his start constants in full, he rounded them off:

X=8.165: Y=8.968: Z=25.50

Entered them in and ran his program.

His first thought when he looked at the resulting graph, was there was either; Something wrong with his program, or something wrong with his computer. When he compared both graphical results although both graphs started off the same, they gradually diverged and went into completely different patterns. The difference between the starting values, each time he ran his program, was only a thousandth. The reason he thought there was something wrong, is because he thought; "How could such a small change in start conditions, lead to such a big change later on?" Then the penny dropped. He realised that non-linear phenomena, like the weather, is inherently unpredictable, because of the infinite sensitivity of start conditions. Because irrational decimal fractions, like Pi can be ever determined, more and more accurately, the further one calculates to the right of the decimal point. This means it is impossible, to accurately define start conditions. Before science discovered this unsettling information, the idea was; Given enough sensors over the surface of our planet and a powerful enough computer, we would be able to predict the weather months into the future. They have found that no matter how powerful the computer, or how many sensors we have, the weather is only partially predictable 3 to 5 days. The sensitivity of these affairs, is encapsulated by the metaphor 'The flap of the butterflies wing'. Thus we have the idea, of a butterfly flapping its wing in South America, this flap starts a vortex, and because the surrounding conditions are exactly correct, this butterfly started vortex, begins to get bigger and bigger, and because of the prevailing westerly winds across the Atlantic, by the time it reaches England. One tiny vortex has turned into a hurricane.

The standard model ascribed to by much of academia, is one of fundamental non determinism. Which has been followed by most people since Darwin's breakthrough, with chaos theory it is possible to see a new kind of determinism arising. Not with a local predictability, but global predictability. For change to happen the right conditions must prevail globally. To understand this concept, imagine a forest in a drought, the longer the drought lasts, the more chance there is of fire. The unpredictability is understood when it is realised that one does not know where, or when the fire will start. Nevertheless, one can state with some certainty that given no rain, eventually it will catch fire. Understanding the new determinism, leads to the perception of the causality of reality. Thus we have the concept, that the future has been written by the past.

Two very important concepts in chaos theory are: 1) Phase space. 2) The strange attractor.

The way it works is thus: -

1) Phase space is the term, covering the parameters of unpredictability in a dynamic system.
2) The strange attractor is the pivot around which the dynamic system moves.



Another definition for a strange attractor could be stated is that it behaves like a dynamo. If a piece of string with a weight on the end, is tied to a finger and whirled around the head, the orbit of the weight would not be exactly circular, it would in fact vary slightly, so a very simple example for a chaotic system could be created. The way this chaotic system can then be understood, is to term the variation in the orbit of the weight as the phase space and the strange attractor as the finger. When the world is looked at in terms of evolution, human history, religion and the weather, they are all chaotic. Science has never explained before now, how they all relate to each other, however by using chaos theory: If the Earth is stated as the dynamic system, the variation in evolution, human history, the weather and religion as the phase space. Then the Sun is the strange attractor.

The point is that the Sun is varying in terms of Sun spot activity and has also been found to vibrate! Who knows what subtle effects these activities have on our phase space (environment) in terms of cause and effect.
(Our Father in heaven the progenitor of change. The engine of causality
(The Sun makes the butterfly flap its wings)).

Chaos theory is a way of understanding of how small patterns connect to create a larger pattern. Chaos theory is about non linear dynamics, science before the chaos revolution ignored anything non linear, because non linear equations are usually unsolvable. ( One of the foundations of science is mathematics.)

Usually non linear system dynamics are not accurately predictable or predictably controllable, all that can be done in terms of the prediction of any particular non linear dynamic, is the study of the underlying pattern.

One of the most beautiful aspects of chaos theory, is that it resolves the dichotomy between simplicity and complexity in terms of the definitions of natural system dynamics, because simplicity is no longer seen in terms of small scale and complexity in terms of large scale. It is the idea of having the same elemental system, at a different dynamical stage of development, at a seemingly different degree of complexity.

One of the most astounding examples for this is the Giant Sequoia tree it starts as a simple seed and grows into one of the biggest living things on earth. This extreme degree of dynamical generative change, is shown by the fact that the simple seed can weigh 4.7 milligrams and ends up as a massively complicated tree weighing 2145 tonnes, this represents a weight gain of approx. 250,000 million fold (courtesy Guiness Book of Records).

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