Chaos Theory and Projective Geometry
by Nick Thomas


An article from the 1994 Edition of the Golden Blade
An Anthroposophical Scientific Journal
https://www.waldorflibrary.org/images/stories/Journal_Articles/Golden_Blade__1994.pdf

Of God, Laplace said that he had no need of such a hypothesis; however in recent years the growth of the study of chaos seems to have turned the tables on him. Indeed we might say that rather than God being dead, Laplace is dead! The idea then current in science, and in one form or an other ever since, was that the future of the physical universe is in principle completely predictable in a mechanistic fashion. Laplace said that if he could know the positions and velocities of all the atoms in the universe then he could predict exactly what would happen in the future for all time. However, this assumes that the physical universe is a giant mechanism obeying the rather convenient laws of mechanics
that he assumed apply universally.

It is something of a travesty that the godless view of the cosmos arose because it was assumed that matter obeys ordered laws discoverable by the human mind, and in turn it was assumed by Sir Isaac Newton and others that those ordered laws operate just because the universe is divinely created! A good God would not build-in perversity merely to confound us. Yet the very concept that the physical universe
operates on the basis of order alone has been severely challenged by the scientific notion of ‘chaos.’

Chaos is often thought to be the state of the house after a wild party, or the mind of a madman, or the aftermath of a hurricane. It may also be seen in the light of spiritual science to have another possible significance, as a state in which new sensitive influences may play a formative role in contrast to a stifling order prohibiting the emergence of anything new. The scientific notion contrasts with the traditionally conceived behavior of a machine which behaves predictably. A clock depends on the regularity of the operation of its parts and the laws governing them, and a pendulum swinging in a small arc of about 10 or 20 degrees is completely predictable, as first discovered by Galileo when
observing a swinging lamp.

Yet it is quite easy to construct a slightly more complicated pendulum whose behavior is quite unpredictable. It is not just that we do not know enough to predict its future antics (literally!), but that no amount of knowledge would permit such a prediction. Laplace is dead! Why is this? Briefly stated, it is because the special pendulum has such a great sensitivity to the tiniest disturbances around it, such as slight air currents, that these continuously disturb its motion, having a much greater influence than the large scale laws of its form and construction. Not only this, it now seems that order and regularity are the exception rather than rule. Laplace is even deader! What order there is arises zis something enduring among the chaotic tendencies all around, something that can survive all this in a kind of darwinism for the lifeless world. Quite apart from the impossibility of knowing simultaneously the motions and velocities of all particles in the universe, even if we did it would not help, for the chaotic would not thereby be banished.

It is sometimes erroneously stated that chaos has only been known about since its ‘discovery’ in the 1950’s by a meteorologist named Edward Lorenz who found that his computer predictions of the state of the Earth’s atmosphere were not reproducible. However, the possibility was already known in the last century, and was certainly known to Einstein. Perhaps in earlier times it was hoped that it would not prove significant!

What about some real-world examples of chaos then? If you listen to a dripping tap, sometimes it will not drip regularly, but will depart from an even ‘drip — drip — drip’ to a ‘drip drip — drip drip — drip drip,’ or even become quite random sounding. The weather and the stock exchange hardly need emphasizing, although it is a significant advance to realize that a mechanistic model of the economy is now
not only difficult to achieve, but almost certainly unreachable.

The planet Jupiter hjis a large red spot that has intrigued astronomers ever since it was discovered by Cassini in 1665. It seems to be a stable feature of Jupiter’s atmosphere, and it was thought that when the Voyager spacecraft took closeup pictures of it all would be revealed. In fact nothing of the kind happened, and the pictures the author has on compact disc of all the NASA photographs taken by the Voyagers show
nothing but chaos on closer inspection. This is a paradigmatic example of chaos, where large scale order ‘rides on the top’ of chaos.

The weather, or more accurately the Earth’s atmosphere, is another such example, for there is (believe it or not) long range order there too. The human heart can stop beating and enter a state known as ‘fibrillation,’ typically caused by an electric or other shock, which in fact is a chaotic state in the technical sense, as is an epileptic fit. The Earth’s magnetic field has reversed itself a number of times in the past, so
that at times a compass would point south instead of north. A careful study of the magnetization of rocks in core samples has shown this to be true, but no regular rhythm was to be een. It seems that this is another example of a chaotic system, operating (fortunately) over a long time scale. Rudolf Steiner indicated from his research that the present mechanistic-like state of the cosmos (at least to outer appearances)
only lasts for quite a short time and will cease in the future.

So chaos is not merely an artificially contrived curiosity in the laboratory. It is a significant field of scientific research today, and in such circles ‘order’ is seen as boring while its synonym ‘counter-chaos’ is interesting! What is very interest ing is the idea that a machine can be such that it is more sensitive to minute disturbances than to its own form, as though it is not contained within its form but is really part of
the whole cosmos and only thus to be understood. The harnessing of cosmic forces such as Rudolf Steiner pointed to may be realizable by means of such a machine.

So chaos is not merely an artificially contrived curiosity in the laboratory. It is a significant field of scientific research today, and in such circles ‘order’ is seen as boring while its synonym ‘counter-chaos’ is interesting! What is very interesting is the idea that a machine can be such that it is more sensitive to minute disturbances than to its own form, as though it is not contained within its form but is really part of
the whole cosmos and only thus to be understood. The harnessing of cosmic forces such as Rudolf Steiner pointed to may be realizable by means of such a machine does. We find chaos on the edges, and a new field of research which studies such things is called complexity theory.
Here the greatest possible opportunity for variation and the greatest sensitivity to minute disturbances exists.

A most remarkable aspect of chaos theory is called the ‘chaos game.’ More complex variations on the Mandelbrot theme can model many realistic-seeming real world complexes such as coastlines, mountain scenery, weather patterns and so on. This is where the scientific interest arises, for this field holds out the hope of understanding complex matters quite beyond traditional non-chaotic approaches. Any engineer will recognize the Mandelbrot set as describing pictorially the stability characteristic of a machine that happens to be described by the corresponding equations. So-called ‘non linear’ systems were in the past a pain that had to be lived with, while now they are of great interest.

Is it possible to start with such a picture and then find the equations that produce it? In some cases the answer is ‘yes,’ and the technique is unbelievably simple in principle. A mathematical theorem called the ‘Collie Theorem’ validates the procedure, and the paradigm for this is the fern leaf. In this area of research, forms crop up which contain them selves as parts, and some of the specks of dust in the Mandel
brot set turn out to be themselves Mandelbrot sets when magnified. A fern leaf contains its own image in its parts, and the technique is to set up the outline of the leaf on a computer screen, and then ’tile’ that with smaller images of itself (see an example of such a ’tiling’ in Figure 2.1). The computer program then uses the results to construct point-by-point those areas that are ‘stable’ in the sense described above for the Mandelbrot set. The result is actually an in stance of a set called a ‘Julia set’ which is related to the Mandelbrot set. A developing sequence as the leaf appeared on the author’s computer screen is shown in Figure 2.2. It bears a striking resemblance to how a photograph gradually
develops when a scene is illuminated extremely faintly so that a very long exposure is necessary (Figure 2.3).

 

For long years Lawrence Edwards has done research in the application of projective geometry to living forms, as reported by him previously in this magazine, following the work of George Adams. Fundamental to this work are the so-called ‘path curves’ which arise in geometry in the most fundamental possible way. It is most interesting to find, then, that the Julia sets generated by the Collage Theorem can be seen as
interwoven spiral path curves, the stable points all being on intersections of such curves. At a stroke the question of how a number of path-curve systems may interact, and with what result, is revealed. This may have far reaching consequences for physics, for the low-light-level exposure in Figure 2.3 is deeply connected with important aspects of quantum physics.

Until now it has been thought that quantum physics suppresses chaos at the smallest levels of matter, leaving it in the realm of the macroscopic. That may not after all be the case, for the work of George Adams (and Louis Locher Ernst) in exploring geometrically Rudolf Steiner’s discovery of negative- or counter-space enables us to see that just at the submicroscopic level the cosmos works through such counterspaces. The question is how to relate many such spaces, and the Collage Theorem seems to be an important clue. As a vehicle for etheric forces we might expect counter-space to be related to systems containing their own images in their parts after the fashion of Goethe’s idea that in the living world the whole is contained in every part. This is just what chaos theory and so-called fractal geometry are all about.

The author has fiirther found that chaos appears unexpectedly (to him) deep within projective geometry itself when non-linear aspects of polarity are explored. The equation governing an aspect of the transformation, when applied in the same way as a much simpler one was to produce the Mandelbrot set, yields an equivalent picture (Plate 3, opposite page 33), and a magnified detail shown in Plate 4 (opposite page 33) clearly shows its ‘fractal’ or self-similar character.

 

This article is necessarily a very condensed account of a large and growing field of research which can fruitfully be allied with research into Rudolf Steiner’s indications.

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