Journey to Infinity

and what you might find along the way

 

3 Chaos and Order

As you might expect, there are some strange contradictions in the world of fractals. One one hand, we're talking about order, repetition and progression. On the other hand, there is more than an element of chaos. Compare these two fractals.

The first one, above, is called Bifurcation. That word means simply 'dividing into two branches'. As you can see, it starts on the left with a single line, which divides into two lines. However, those two lines are not exactly the same, are they? The directions of the curves are different. When they, in turn, break into two, the new curves go in different directions. After the fourth division into two, the whole thing seems to become chaotic. This graphic does not show that the divisions continue in what looks like a mess of dots.

The second fractal is appropriately called Plasma. In medical science, plasma is a colourless watery fluid which itself has no cells, but other cells are suspended in it. This version of the fractal is in green, blue and red. It doesn't matter which colours are used the overall appearance is still of a shapeless, perhaps chaotic, mess. It seems to have no shape, progression, mutations or order.

Remember the fractal you studied earlier, which looked like a butterfly's wings or perhaps a pair of staring eyes? Although it was irregular in shape, and not symmetrical (not the same on both sides) it looked orderly. However, this is what you see if you zoom deep into its sweeping curves and look between them.

What happened to the apparent orderly pattern? Why does it seem to be so chaotic inside?

The butterfly storm

The discovery of fractals and how they work led to the development of a new branch of mathematics: the geometry of chaos. In some areas of science, such as meteorology, it is difficult to predict results on the known facts. The weather is unpredictable; something could disturb what seems at first to be a nicely organised sequence of events.

In weather forecasting, there is a half serious, half humorous, idea called the Butterfly Effect a butterfly flapping its wings today in Hong Kong can transform storm systems next month in London. Yes, this is an exaggeration, but it conveys the idea that even the smallest event can change a very much large pattern. And that is exactly what is happening in these fractals a small change of direction in the lines in Bifurcation produces an effect which is repeated over and over again. The result looks like chaos, but if you zoom into the fractal, you see more and more magnifications of the same pattern amidst what can only appear in the original graphic print as a random mess of dots. Deep inside the fractal itself, of course, there is not a random 'mess' at all.

That leaf again

In the first picture, above, it is clear that soon as the two curves branch out, they go in slightly different directions. When the next two curves branch out from the first pair, they also go in different directions. The are similar, but not the same. Something is happening to cause this. In the case of a fractal, the cause is the way it has been programmed in mathematical algorithm.

You saw the same sort of thing happening in the scan of the leaf you looked at earlier. If you want to have another look, click here and then use your browser's Back button to come back. Notice how the left and right hand sides are similar, but not the same. Track the way each vein curves upwards, divides, and continues. The leaf, of course, is not infinite, so the birfurcated veins don't go any further than the edge.

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