  Complexity Pages
A non-technical introduction to the
new
science of Chaos and Complexity |
 Victor MacGill
  On this Site
Articles written by Victor
involving aspects of Chaos and Complexity Web
resources and links
A
Glossary of Terms
used
in Chaos and Complexity from http:// www.calresco.org Search this site |
Fractals
One of the unexpected patterns of order found in strange attractors is that they are self-similar. A self-similar system has the same basic pattern   repeated
at different levels. When you look at a picture of desert sands, you
cannot tell whether you are looking at the sand from a distance of five
meters or 500 meters. The picture could be a close up or a distant
shot. The basic shape of the sand dunes looks similar at all distances.
It is said to be "self-similar" or “scale free”. Its shape is similar
to itself at different levels of viewing and no matter from what scale
we look at the fractal it has a similar appearance. It is only when we
see something like the camels walking across the scene that we realise
the scale. You could equally imagine a tiny beetle walking across the
same desert sand.In 1975, mathematician Benoit Mandelbrot coined the term "fractal" to describe self-similar patterns. In 1967 he posed the puzzle, "How long is the coastline of Britain?" This is more difficult to calculate than it would seem at first glance. We might think to get a piece of string and put it on the outline of a map and then measure the length of the string. From that we could get a measure of the length. But, if we get a more accurate map, we will find many more small coves and bays, which did not show on the less accurate map and make the length much longer. If we were to pace out the real life coastline, we would find it even longer. So, if you think about it, the length depends on how we measure it, and the more accurately we do so, the longer the length gets. Ultimately the length of any stretch of coastline is actually infinite.  If you look at the map shown, you
cannot tell if it is one kilometer across or a thousand kilometers
across. The coastline is fractal. The basic convoluted shape remains
when you zoom in closer and closer. The lines could be dividing
off small farm blocks, or as they actually do, divide off whole
countries in South America.Trees are also self similar. A small twig from a tree  has the same
basic shape as the whole tree. If we take a basic 'Y' shape and repeat
it again on top of each branch of the Y, but at a smaller proportion
and then repeat this procedure again and again at the top of the
branches of the Y, we get the diagram shown. We see that it looks like
a tree. If instead of a perfectly even Y, we have angles and lengths
that vary somewhat in a random manner, the result is strikingly similar
to a tree we might see in nature. This fractal shape is also
found in a cabbage or broccoli. Our own bodies are filled with fractal
structures.The shape of the tubes in our lungs is based on an inverted 'Y' shape. Our kidneys, veins and arteries and even the neurons in our brain all use fractal formations. If we experiment with variations on the basic “Y” shape we find some other interesting patterns. Using the four simple lines as in the next diagram as a base pattern we find it which iterates to come up with the very life-like fern pictured in the diagram. The fractals found in nature are not perfect fractals like those created mathematically, which cover an infinite number of levels of magnification. In nature, we generally find up to about seven levels of pattern before the detail becomes too small to physical form any smaller levels. Fractals not only cover the natural world, but are just as evident in the 'man-made' world. Look at the shape of a city map. It has a self-similar look to it with main highways with major roads coming off them and then minor roads coming off them. So, why would nature use fractal patterns for so many structures? Natural selection tends to select out less effective structures, so that over time, only the very best, most efficient structures remain. Our lungs are an efficient means of drawing in a large amount of air and distributing that air over as large a surface area as possible, so the air can be effectively absorbed into the blood stream. By forming into a fractal pattern where the air tubes reduce thickness by the same proportion at each branching point, enables the most efficient diffusion of air into the blood. The fern leaf needs a large central branch, which is strong enough to support the whole fern and move nutrients efficiently into all the extremities of the fern. The fern shape allows the greatest absorption of sunlight, CO2, and water. The cells in our brain need to connect enormous numbers of other neurons. Large nerve tracts allow the fast transmission of electrical impulses, which then divide into smaller and smaller units so as to connect to other neurons all over the brain. A fractal distribution of nerves from large tracts to small neural fibres is most efficient. A fractal structure therefore allows the greatest number of agents to be linked in order to allow the greatest diffusion of energy or information through the entire system. While the lungs take energy in the form of air from a single source and distribute it over a large area, we can also use the same fractal structure to collect energy from a wide distributed area and bring it together to a single exit point. A river system collects water from a large catchment area collected in small streams which join to form ever increasingly large rivers that finally joins to one river entering the sea. A highway system works in both directions. One side of the road brings people into the city through large highways, which divide into smaller and smaller roads until we reach the driveways of individual houses. On the other side of the road, motorists from all over the city move into larger and large roads until they can efficiently and quickly leave on a large highway. As we have seen, fractals are very complex structures, which have simple rules creating them. Instead of the system having to retain information about the entire structure, it need only retain the information about the basic pattern and rules for repeating that pattern at new levels. Nature uses fractals so much because they create very efficient structures, which help improve the chances of survival in a highly competitive world. Fractals are found in so many places in the natural world that would often not suspect. Fractal and chaotic dynamics are found in cloud formation, the stock market, traffic flows, and the human heartbeat.  Ron
Eglash discovered that many African villages are actually built on
fractal lines. They are built intentionally and reflect the nested
hierarchies of the social structure. Depending on your position in the
social structure will determine where your house is in the village
fractal. He strangely found many different types of fractal villages,
some based on circles, some on rectangles etc. but only found them in
Africa and Southern India. He also found games and methods of
divination which employed different computational fractal devises.The psychological map used in the book, "When the Dragon Stirs" (MacGill, 1995), views our being as expressing four core qualities: courage, compassion, wisdom and authority. On an individual level, we can see personal courage, personal compassion, personal authority and personal wisdom. Equally we can look at a whole nation and talk of national compassion, national authority, national courage, and national wisdom. The same is true for other communities, or organizations such as business corporations. For an individual, authority is about the ability to make their own decisions to maintain a structured life, while for a nation it is about security and the law. The Police Force ensures that people take responsibility for their actions and the judicial system sends people to prison if they do not adequately take responsibility for their actions. |
© Victor
MacGill 2007, This
site is a part of the web site of Victor MacGill.
The
disclaimer
on that site applies equally to all pages on this site.