I believe that scientific knowledge has fractal properties, that no matter how much we learn,
whatever is left, however small it may seem, is just as infinitely complex, as the whole was to
start with. That, I think, is the secret of the Universe.
- Isaac Asimov¹

The term fractal was invented by Benoit Mandelbrot, who created the whole field of fractal
mathematics. One of many definitions of fractal is that a fractal is a complex geometric figure whose
small-scale and large-scale structures resemble one another.

The most famous traditional attempt to portray this quality is the Taijitu, the classic Yin Yang symbol: This symbol represents the unity of yin and yang,
where each contains the other. The small dots inside each half portray this containment. But, by implication, there is no end to this process: the yin within the yang in turn contains both yin and yang, and the process continues endlessly.

Mathematics had many such unusual geometric figures, which predated Mandelbrot’s discovery of fractals. For example, in a 1904 paper, the Swedish mathematician Helge von Koch presented what has come to be known as the Koch snowflake. Take an equilateral triangle (i.e., each side is the same size. Then divide each side into thirds and on the middle third build another equilateral triangle. Continue this process with those triangles, and with all subsequent triangles. The figure on the left below shows this process progressing through four stages. The figure on the right shows the result after eight stages.

A closely related figure is the Sierpinski triangle, described by mathematician Waclaw Sierpinski in 1916. In this case, take another equilateral triangle, but this time, divide it into 4 equilateral triangles and remove the middle one. Continue this process with all the triangles that emerges. Or instead of a triangle, take a square, divide it into smaller squares and remove the middle ones. Again the process continues. The process can be used just as readily on 3-dimensional objects or even higher dimensional objects, though of course we can no longer see what those look like. The most famous of the 3- dimensional objects is the Sierpinski cube. Here are pictures of the Sierpinski triangle, square, and cube.

Until Mandelbrot, however, these strange figures were considered anomalous, or even “pathological.” Mandelbrot broke the ice with a three page paper that somehow slipped into the illustrious (and usually staid) pages of Science in 1967, called “How long is the coast of Britain?”² This paper has become known not only to mathematicians, but to the general public as well, usually for the wrong reason. Reference books often give figures for the length of the coast line of various countries. But it’s self-evident that the length depends on the measuring device. If a ruler is used instead of a yardstick, the coast line will be much longer, because the yard stick can measure smaller differences in its shape. If we let the unit of measure get smaller and smaller, ultimately the length of any coast would
be infinite. That’s what most people think Mandelbrot’s paper was about.

Actually he had discovered something quite different. He had come across a 1961 paper by Lewis Fry Richardson, who Mandelbrot later referred to as “a great scientist whose originality mixed with eccentricity.”³ Richardson had found that the approximate length of a variety of actual coast lines could be calculated by a simple formula into which you plugged the length of measurement you were using. The formula had a constant “D” that varied by the coast line.⁴ So for the coast line of Spain, there
was one “D”, and another for “Britain,” and so forth. This was exactly what Mandelbrot meant by fractal (though he hadn’t yet come up with the word yet). Mandelbrot’s breakthrough was to recognize that this constant “D” was actually a dimension, even though it wasn’t a whole number. In other words, in addition to the 1-dimension of a straight line, the 2-dimensions of drawings on a plane, the 3-dimensions of solid objects, etc., he was proposing that there could be fractional dimensions. For example, in the paper, heproposes that the dimension of the west coast of Britain is approximately 1.25.

We needn’t be concerned with the math here. What is significant is that the math produced figures which were self-similar no matter what level you looked at them. By 1973, Mandelbrot was working for IBM as a fellow at the Thomas J. Watson research center, and had access to high-speed computers. This was like a Disneyland for Mandelbrot. He worked out formulas which, when fed back into themselves over and over (as with everything else in chaos theory), produced coast lines that actually looked like real coast lines. Mandelbrot went on to generate clouds and trees and rivers and almost anything else imaginable in nature. The implication was that, since he could generate something that looked remarkably like a natural phenomena by building it up out of little self-similar pieces, then nature itself might use the same technique. Nature might be fractal. Or, as the Emerald Tablet had it: “as above, so below.”

The most famous of all fractal creations is the Mandelbrot set, which is the most complex mathematical object in existence. As with all the earlier fractals, it’s generated from a very simple formula⁵ with the results fed back over and over again. It’s normally generated using colors to show distinctions, but a beautiful black and white version follows, created by mathematician Ed Pegg, Jr. The shadings in color between black and white were generated by the average differences between iterations.

Again imagine that you had an enormously powerful microscope which you could use to look at
any part of this figure. If you would do so, you’d find that what you saw would look remarkably like the
whole picture you’re seeing. What makes this more interesting than the fractal objects you’ve seen earlier
in this chapter is that you would find creative differences between the original figure and its copies at
each deeper level. These approximations to the original look very similar, but each has variations of its
own, as if some celestial artist kept working with it at every level. The new insight that chaos theory and
fractals give on the ancient idea of “as above, so below” is that there may be beautiful variations on
theme that emerge without any way of predicting what they will be.

Notes:
1. Asimov, Isaac, I, Asimov: A Memoir, New York, Bantam, 1995. Quoted in Clifford A.
Pickover, A Passion for Mathematics. (Hoboken, New Jersey: John Wiley & Sons, 2005), p. 90.
2. Mandelbrot, Benoit, “How long is the coast of Britain?”. Science 156, pp. 636-638.
3. Mandelbrot, Benoit. The Fractal Geometry of Nature. (New York: W. H. Freeman & Co.,
1977), p. 29.
4. The actual formula was L(€) ˜ F€^1-D.
5. Z_n+1=Z_n+c, where c is a complex number; that is, a number with both a real and an imaginary
component.

Introduction:

How is it that transformation comes about? Let us look at two models of that process, both of which ostensibly look at outer transformation, while unknowingly also speaking about inner transformation. One is an ancient model - Western alchemy - which came into existence in the Western World during the 1st through the 3rd centuries, reached its peak in the 15th and 16th centuries, and still existed in some form during the 17th and 18th centuries. It was only in the twentieth century, however, that the idea first occurred that alchemy was actually a model for psychological transformation. Psychologist C. G. Jung argued that the alchemical opus (i.e., the stages of the process leading to the production of the Philosopher’s Stone), was equally a portrait of the stages of the individuation process of the alchemists, which they were unconsciously projecting out onto the physical world. Religious historian and philosopher Mircea Eliade placed the emphasis slightly differently: he said that the processes that underlie alchemy lie deep in both the psyche and the world, and must then inevitably occur in any transformative process, whether in chemical beakers, or inside the human soul.

The other model - chaos theory - is much more recent. Deterministic chaos was first recognized explicitly in the early 1960’s, with meteorologist Edward Lorenz’s early computer models of weather. His work led to what has been termed “sensitive dependence on initial conditions.” Or more popularly - the butterfly effect - the idea that the flutter of a butterfly’s wings in China might affect the weather in Los Angeles. This is because very tiny initial changes can have very large final effects, when the changes are fed-back into a system and amplified, over and over.

Anyone who is called to the difficult inner journey of self-transformation ineluctably awakens deep structures in the psyche that can be seen in both the alchemical opus and, properly translated, in chaos theory. These two parallel models share five critical insights (as well as many smaller ones) which provide a template for transformation. No matter what particular spiritual path we have chosen (or which has chosen us), these insights can help enrich our understanding of the process of transformation, whether outside in the world, or within our own lives.

1. The self-referential nature of reality¹
In alchemy, this is represented in the core belief of “as above, so below,” first presented in the fabled ‘Emerald Tablet.’ In Chaos theory, this occurs in several different presentations, including the fact that global (i.e., as above), and local (so below) are inextricably mixed. The chaos theory example which is equivalent in renown to the Emerald Tablet might, however, be Benoit Mandelbrot’s mathematical concept of fractals.

2. Feedback
Here I’m using the more modern term. All of chaos theory is based on feeding back information from one stage of a process to the next stage of the process. The computer has allowed scientists, for the first time, to model the way nature continually feeds back actions into themselves. But alchemy tried earlier to model such behavior by building up-and-down movements into their alchemical stages such as sublimatio (sublimation, aeration, rising, spirituality), and mortificatio (mortifying, falling into matter), or more explicitly in the circulatio (continual cycles of rising and falling). But the most ancient alchemical symbol of feedback is the image of the uroboros, the snake that swallows its own tail.

3. Transformation through cycles of taking apart and putting back together
In alchemy, this is accomplished through stages such as solutio (i.e.,dissolve), and separatio (break into parts), followed by stages such as coagulatio (coagulate, come together) or coniunctio (conjunction, joining). This process is implicit in all of chaos theory, especially in ‘strange attractors.’ The most famous example is the baker transformation, in which a baker kneads dough over and over, separating parts of the dough that are close together, and bringing together other parts that were widely apart.

5. Chaos/Emergence
In alchemy there was an explicit stage of darkness and confusion: the nigredo. There was an understanding that because this chaos was beyond all definition, it implicitly contained all possibilities within itself. Eventually, points of light (scintillae of light) would begin to emerge within the darkness of the alchemical experiment. There was no possibility of predicting which spark would suddenly grow into the light of new birth, but new birth there would be.

And, of course, as one can tell by the name chaos theory, chaos is explicit part of the modern model. As we will see in more detail, order gradually bifurcates, splitting into first two possibilities, then four, and so on. It is impossible in advance to predict which fork will be taken. Then at some further point, the bifurcations change into chaos. But what is a new insight in chaos theory, is that even this chaos has structure at a global level, that out of it emerges a new order.

6. Inseparability of Experiment and Experimenter
There are innumerable references in alchemical texts to the idea that in order to fully carry out the opus, the alchemist must first purify himself in order to become worthy of the stone. Though less explicit in chaos theory, it is nevertheless found there. For example, neurobiologist Walter J. Freeman, a pioneer in the application of chaos theory within neuroscience, discovered that: “Instead of minds shaping themselves to their sensory inputs from the world, minds shape sense impressions according to their innate categories.” Freeman demonstrated this conclusively through his study of dynamic attractors in the brain for odor and other senses.

This last topic - the inseparability of experiment and experimenter is, of course, why alchemy and chaos theory can teach each of us something about our own process of self-transformation. This conjoined relationship between the world and the psyche was perhaps best expressed in the words of Alain de Lille, 12th century theologian: “God is an intelligible sphere whose centre is everywhere and whose circumference is nowhere.”² Each of us through our own process of growth and transformation in turn affects everyone and everything.

{pagebreak}

Lesson 1a from Alchemy: As Above, So Below

Tis true without lying, certain & most true. That which is below is like that which is above & that
which is above is like that which is below to do the miracles of one only thing. And as all things
have been & arose from one by the meditation of one: so all things have their birth from this one
thing by adaptation.
- From Isaac Newton’s translation of the Emerald Tablet.

The idea that the macrocosm (the universe or God) and the microcosm (the physical world, a
human being) are inherently connected is the first crucial element of alchemy that we will study. The
equivalent understanding in chaos theory is that reality is self-referential at every level. We will deal with
alchemy in this lesson, then its equivalence in chaos theory in the next.

Perhaps the single most important core belief of alchemy is contained in the phrase “as above, so
below,” which first appeared in the fabled Emerald Tablet, also known as the Smaragdine Table or the
Tabula Smaragdin. Though the earliest documented source for the Emerald Tablet itself is an 8th century
pseudo-Aristotelian Arabic text, which was translated into Latin in both the 12th and 13th century, it is
likely, however, that it evolved from earlier documents and may well have its origin in more primitive
versions that go back to the period just before the start of the Christian era “when the thought models of
the Greek philosophy met with the experimental practices of the Egyptian traditions.”³

Paracelsus

The mysteries of the Great and the Little World are distinguished only by the form in which they
manifest themselves; for they are only one thing, one being.
- Paracelsus.

We are now going to jump forward to the 16th century, when
alchemy reached its peak. One of its most influential expositors was physician Philippus Aureolus Bombast von Hohenheim, more commonly known as Theophrastus Paracelsus, or simply Paracelsus [1493–1541]. Paracelsus was born the illegitimate son of a wandering doctor who settled in the small Swiss town of Einsiedeln. As a young man, he was hardly physically prepossessing, with “a stature of a mere five feet, an unhealthy appearance, an upper lip that was too short and did not quite cover his teeth . . . and, so it seems, a pelvis that struck everybody by its femininity.”⁵ Despite that appearance, or perhaps because of it, he strutted around life with an arrogance that earned him many enemies, especially among his fellow physicians, whose methods he reviled.

He was a physician, an alchemist, an astrologer, an occultist, and one of the Renaissance’s most prolific authors. While still in his teens, he studied medicine at the University of Basel. On the move as always, he then went first to Vienna and later gained his doctorate degree from the University of Ferrara in Italy. Unsatisfied by the methods he learned at the Western universities, he continued his traveling ways, through Egypt, Arabia, the Holy Land, Constantinople, Lisbon, Spain, and England, always looking for newer methods, methods that fit in with his holistic view of reality.

The Renaissance was a time when thinkers turned their eyes onto the physical world around them and began to observe and measure it carefully in a way that few had done earlier. They looked both above and below: outwards toward the heavens and down onto the earth they lived on. Paracelsus was interested less in one or the other of the two worlds than in the Hermetic doctrine that the two were necessarily connected. In Paracelsus’ words: “Heaven is man, and man is heaven, and all men together are one heaven, and heaven is nothing but one man.⁶ This lovely phrase means not only “as above, so below,” but also “as below, so above,” which we will see fits closely with chaos theory. He believed that each of us was composed of an earthly part and a heavenly part, or as he said “Man is the child of two fathers - one father is the earth, the other is heaven . . . . From the earth he receives the material body, from heaven his character.” Thus our characters were, as all believers in astrology argue, determined at the moment of birth by the total form of the heavens. That also meant, in his view, that each of us carried a particular heaven with us at all times. And further, that the heavens above were composed of all these individual heavens. Unlike most astrologers, Paracelsus insisted, however, that: “It is not true that the firmament exerts a compelling action upon man; on the contrary man himself acts upon the world more than he is influenced by it.”⁷

When Paracelsus returned to Europe, his medical treatments went completely against the grain of most of his colleagues. Just as he believed that the heavens determine our character and the earth our physical being, he also believed that there is a wholeness to our body and character that is reflected at all levels. The central concept of his treatments was that “like cures like.” This breakthrough concept led physicians for the first time to use chemical treatments for diseases. As such it is the foundation for all modern medicine. In his treatments, he used arsenic, mercury, sulfur, iron, copper sulfate and other metals. His idea of like curing like is also the basis for the more controversial modern method of “cell therapy,” where cells are taken from an organ of the body, then injected into the body to cure the damaged organ.

But Paracelsus felt that these methods, powerful as they were, could only be effective it the physician approached the patients with compassion and even love, because it was love that could bridge the seemingly inseparable distances between human beings. “First of all it is very necessary to tell of the compassion that must be innate in a physician.” “Where there is no love, there is no art.” “The practice of this art lies in the heart: if your heart is false, the physician within you will be false.” For again, within each of us lay a piece of heaven, and through that heaven, each could be joined to the other. In our next lesson, we will look at the equivalent insights from chaos theory.

Notes:
1. These five processes will be individually explored in later articles in this series.
2. This idea was thought to have originated in the Corpus Hermeticum of the 3rd Century. It later found many variant expressions in authors as famous as Pascal and Emerson.
3.. von Franz, Marie-Louise. Alchemy: an Introduction to the Symbolism and the Psychology (Toronto: Inner City books, 1980), p. 11.
4. quotation from Jacobi,Jolande. Paracelsus: Selected Writings (Princeton: Princeton University Press, Bollingen Series XXVIII), p. 19.
5. Jung, C. G. “Paracelsus,” in Collected Works, Vol 15: the Spirit in Man, Art, and Literature (Princeton: Bollingen Series, Princeton University Press, 1942), par. 6.
6. quotation in Jacobi, Paracelsus: Selected Writings, p. 39.
7. quotation in Jacobi, Paracelsus: Selected Writings, p. 153.