Dilian Gurov

A great many real world phenomena have a repeating nature, or can be seen as the result of continuous or repetitive application of some force on material objects. The result of such physical transformations can vary considerably. Wind and rain cause stones to break into smaller pieces and eventually become sand, which is stable to these forces. This stable state of affairs can be seen as a fixpoint of the transformation: further application of the force does not change the state. Another common outcome is a different form of stabilization where the system oscillates between the same set of states, as for example a pendulum. And yet another possible outcome is chaos in systems in which small causes can have large effects, so that certain patterns of behaviour can be observed but not precisely predicted. A typical example of a chaotic system is weather, which is known to be principally unpredictable for more than, say, four days ahead.

There is a very simple mathematical function that illustrates fixpoints, oscillatory behaviour and chaotic behaviour in a visual and intuitive manner, called the logistic map. It is defined as f(x) = ax(1-x). By choosing different values for the coefficient a and the initial value of the parameter x we obtain different behaviours.

Here is a web link to an interactive Java applet, which allows the user to set a and x.

A rather unexpected place to find the foundational iterative construction of computing a fixpoint is the ancient Greek tale of Achilles and the tortoise by the Greek philosopher Zeno. The story goes roughly as follows.

Where is a fixpoint computation here, you may ask. Well, consider the mathematical transformation f over the set of points forming the path of the race, that maps every point x to a point y where the tortoise has been at the same time when Achilles has been at point x. You can surely see that if you start with the initial point of the race (that is, where Achilles started the race), then the infinite sequence of points x, f(x), f(f(x)), ... is exactly the one from the argument of the tortoise!

But where is the catch? Surely Achilles is going to overtake the tortoise at some point? Yes, indeed, and this point is the limit of the sequence, and at the same time the (earliest) fixpoint of the transformation. This iterative procedure of computing (in the limit) the (least) fixpoint of a (monotone) transformation is the principle technique used in mathematics and computer science.

An extremely beautiful example of the phenomenon of stabilization upon the repeating effect of a force on matter are the so-called Chladni patterns. They can be obtained by drawing a bow over a piece of metal whose surface is lightly covered with sand. The plate is bowed until it reaches resonance, when the vibration causes the sand to move and concentrate along the nodal lines where the surface is still. Depending on the frequency of vibration, different patterns are obtained.

Interestingly enough, the view on the meaning life that appeals most to me is the one expressed in W. Somerset Maugham's novel Of Human Bondage. Here is a web link to several excerpts from the book. Maugham's view draws obvious parallels with the idea of beautiful patterns emerging as the result of the evolution of matter. Maugham starts by observing that life has no meaning:

"The answer was obvious. Life had no meaning. On the earth, satellite of a star speeding through space, living things had arisen under the influence of conditions which were part of the planet's history; and as there had been a beginning of life upon it so, under the influence of other conditions, there would be an end: man, no more significant than other forms of life, had come not as the climax of creation but as a physical reaction to the environment. [...] There was no meaning in life, and man by living served no end. It was immaterial whether he was born or not born, whether he lived or ceased to live. Life was insignificant and death without consequence."

Still, humans can give purpose to their lives by pursuing a rather aesthetic goal:

"As the weaver elaborated his pattern for no end but the pleasure of his aesthetic sense, so might a man live his life, or if one was forced to believe that his actions were outside his choosing, so might a man look at his life, that it made a pattern. There was as little need to do this as there was use. It was merely something he did for his own pleasure. Out of the manifold events of his life, his deeds, his feelings, his thoughts, he might make a design, regular, elaborate, complicated, or beautiful; and though it might be no more than an illusion that he had the power of selection, though it might be no more than a fantastic legerdemain in which appearances were interwoven with moonbeams, that did not matter: it seemed, and so to him it was. In the vast warp of life (a river arising from no spring and flowing endlessly to no sea), with the background to his fancies that there was no meaning and that nothing was important, a man might get a personal satisfaction in selecting the various strands that worked out the pattern. There was one pattern, the most obvious, perfect, and beautiful, in which a man was born, grew to manhood, married, produced children, toiled for his bread, and died; but there were others, intricate and wonderful, in which happiness did not enter and in which success was not attempted; and in them might be discovered a more troubling grace."

Certain geametrical figures can be obtained by applying iteratively one and the same construction, which stabilizes in the limit. Such figures can thus be seen as the fixpoint of the iterative transformation. One well-known construction is the Sierpinski triangle.

Here is a web link to an animated version.

Fractals are typically self-similar patterns, where self-similar means they are the same from near as from far.

Fractal music is one approach to compose music algorithmically, by applying the fractal principle of self-similarity. Here is a web link to an article.