"All things are numbers."

A Chaos Theory History

Eastern mysticism vs. Western science
Newton's universe was a linear, closed set of events governed by cause and effect and the conservation of matter and energy. Matter was considered to be composed of a finite number of indivisible particles. Energy was thought to be a characteristic of matter. If we only had enough facts, we could predict any event with absolute certainty. The universe was seen as a very large, complicated clock, which, when observed at an atomic scale, consisted of a very large number of billiard tables. Newton did find inconsistencies in his calculations, but these were attributed to divine intervention and seen as proof of the deity's activities.

In the 1940's, computers and statistical mechanics proved predicting most events theoretically impossible by the sheer number of variables. The universe was then considered unmeasurable, but still entirely linear and mechanical. It was also believed that the effect of small events was just so much unavoidable noise. Infinitesimal events were absorbed by the greater scheme of things and the universe-as-machine, ticked on.

The first chaos theorists operated with intuitive and interdisciplinary methodologies. Early models were ignored or derided as amateurish, by career mathematicians, for their lack of formal mathematical proofs. It was only with the availability of computers to formulate those proofs that chaos theory became an important factor in physics, meteorology, medicine, statistics, and materials engineering.


Edward Lorenz: modeling reality
In the 1960's, a new concept arose from attempts to computer model and predict the weather. Forecasters found that any change in a weather model, no matter how small, eventually caused a drastic change in the outcome of the weather prediction. This became known as the 'butterfly effect', in part because of graphic representations of these effects, but also because of an analogy made by meteorologist and computer scientist, Edward Lorenz: if a butterfly flaps its wings, weather all over the world will change as a consequence. A fluid system, such as the atmosphere, is completely interconnected, there is no dampening or interference which will remove the disturbance or cancel it to zero. For the next two decades, scientists in other fields began to see the butterfly effect appear in systems considered random and unpredictable. The formal term for the butterfly effect is sensitivity to initial conditions.

Sensitivity to initial conditions only holds true in an unfiltered system. A mathematically filtered system is one that uses only discreet finite variables, i.e. numbers and operations which consist of and only generate integers or numbers with a fixed number of decimal places. These operations are very rare in both mathematics and nature. Any math function which contains such operations as division or square roots or relies on approximate data does not necessarily generate numbers with a constant number of decimal place values and is therefore subject to sensitivity to initial conditions.

Let's return to the butterfly metaphor. What if the butterfly was captured in a jelly jar and flapped its wings? Would it still have an affect on the weather? We would have to say yes. The vibrations of the butterfly's wings would be dampened by a container, but never entirely. There is only one perfect filter in nature - a perfect vacuum.

The nervous system of a living being can also be a perfect filter. At some point, a nervous system must arrive at a yes or no conclusion, it must respond to information or ignore it. Ignored information has no reverberations, no effect on future activities of the system, no behavioral response. Because digital computers must also make yes/no decisions, round-off data, and ignore data, they too could be said to possess perfect filters.
The human filters have been subject to much debate among psychologists, information theorists, linguists and designers. At some point assumptions must be made as to the nature of the filters and their relative effects. The nervous system may perfect intellectual filters, but it is also possesses so many sensory inputs and filters interacting on non-continuous information that it must also be a chaotic, unpredictable system. This places most of the current research on perception, which assumes some perfect intellectual filter,operating in isolation, to be in question.

The cybernetic mind of the computer is more quantifiable, but recent large networks of computers have begun to exhibit chaotic properties and nervous breakdowns. If classically chaotic properties are evident in cybernetic systems composed of nothing but perfect filters (on-off switches are the physical basis for computing), then they must certainly be present in human sensory and cognition systems. This raises two questions for designers- what filters exist in human visual systems and what filters exist in cybernetic systems? Are these filters really perfect or are they subject to sensitivity to initial conditions?


Mandelbrot: a new geometry
Benoit Mandelbrot is the mathematician most responsible for codifying discoveries about structure within seemingly random systems into what is now known as chaos dynamics. Much of his career has been spent doing pure research at IBM's Thomas J. Watson Research Center. It is to IBM's credit that they supported Mandelbrot, when the academic community would not read his papers and no one in the scientific community would claim him as belonging to their discipline. Mandelbrot's work is focused on the visual representation of dynamic systems rather than the purely mathematical representation. Scientists and mathematicians working with turbulence or randomness had been approaching these systems from a strictly linear point of view which was not present in Mandelbrot's maps and diagrams. It is not surprising that most resisted his ideas, since they often discounted the basis for lifetime careers of research.

Perhaps Mandelbrot's first big breakthrough came with a problem in economics - commodity prices, to be exact. Commodity prices have never matched any normal curves of probability, no matter what method or level of economic analysis was used. It behaved much like the weather data, making unexplained and unexpected changes of state. It was unpredictable. Economists felt that more data or a new method of analysis would eventually enable them to model an economic system and make predictions. Lorenz's work with sensitivity to initial conditions show's that this is not possible, the data will never be exact enough, no matter what the method of analysis.

Mandelbrot was not just interested in predictions, however. He was looking for chaotic patterns which did not necessarily fit to any linear, predictable curve. What he did find were patterns within the patterns. The degree of statistical variation for a day's change in commodity prices was the same as that for a month or a year. One could not predict what cotton's price would be tomorrow, but one could predict the variation for the next month, based on the variation observed today. Mandelbrot used the computers at IBM to look at 60 years of cotton prices and found consistent variation across the entire time, when analyzed with his methodology. Using IBM's facilities, Mandelbrot discovered this quality of self-similarity to be present not only in commodity prices, but also in such data as the flood levels of the Nile, which have been recorded for the last two thousand years.
He used this new knowledge to help IBM engineers deal with random noise on transmission lines. Noise could not be eliminated, but if one could predict its variation more accurately, then it was possible to plan strategies for transmitting redundant information and then checking for errors. Mandelbrot's work does not so much provide specific problem-solving techniques as it provides frameworks for determining what is and what is not possible to control.

Mandelbrot's research led him more and more from the graphic representation of abstract data to the geometries inherent in natural phenomena. He became interested in cartography, specifically in the answer to the question: how long is a coastline? The answer is that the smaller the unit of measure, the longer the coastline. No matter what scale is used to measure, there are always irregularities below what is seen. Bays are always composed of sub-bays, the sub-bays are composed of smaller bays and so on. Units of linear measure are based on smooth Euclidian forms, such as straight lines or circles. There are always irregularities for which the measuring tool does not account. The length of a coastline is finite only at an atomic level, if then. For all practical purposes, a coastline is an infinite length contained within a finite boundary.

What interested Mandelbrot was that the various levels of sub-bays didn't look much different from one another. The titanic forces which created the bay were replicated in all of its subdivisions. Coastlines were not the only phenomena which evidenced self-similarity, in fact all dynamic systems showed a tendency to be self-replicating across different scales. Clouds and mountain structures look almost identical when seen from a great distance or a microscopic distance. Mandelbrot was fascinated by this continuity across scale. Since scale is infinitely large and infinitely small, what we perceive as a point, that is something with no dimension (dimension 0), must become two- and then three-dimensional as an observer approaches it. The question, then, is whether the transition between dimensions is sudden or are there fractional dimensions in between?

Mandelbrot began to define the irregular self-similar forms of nature in fractional dimensions. He used these fractional dimensions not to measure the infinite lengths of dynamic structures, but to describe their common patterns of irregularity. He named these geometries and their fractional dimensions fractals. The coastline of infinite length, contained within a finite Euclidian area could be thought of as more than a line (one-dimensional) and less than a plane (two-dimensional) and be assigned a fractal dimension such as 1.8457. A mountain with its infinite surface would have a dimension between that of a cone (three-dimensional) and that of a triangle(two-dimensional) which encloses it. Fractal dimensions have proven useful to engineers working with the interaction of irregular surfaces, such as tires and concrete.

Computer artists
Digital artists have begun to use fractal dimensions to create the illusion of such natural objects as rocks, water, vapors and hair. Other computer artists are exploring the creation of new images based on nothing but the pure mathematics of fractical geometry.

What is important to visual designers is not so much the ability to define objects or surfaces in fractal terms,or even to emulate the appearance of natural structures, but the ability to look at graphic structures in terms of their quality of self-replication across scale. I believe that just as weight and balance, area and direction are important parts of a system of visual aesthetic, so too are the concepts of fracticality, self-similarity and growth.

There are however, practical limitations which currently prevent us from utilizing this quality in an empirical manner. There is no convenient tool available to the designer for measuring fracticality. There is, however, a tool included in the desk-top paint program, PixelPaint, which allows one to create a line between two points with a given degree of fracticality. Graphic design is not just the ability to draw a line or a letterform, but the ability to measure it as well. I cannot see fracticality as a significant part of the designer's lexicon, until there is a tool available to measure that quality.

Mandelbrot, for the time being, should be better known to designers for his contributions towards a universal theory of chaos dynamics. Chaos dynamics is the fundamental theoretical body for describing the motion and interaction of the elements of any naturally occurring dynamic, biological, or chemical system. It is as significant a concept as the definition of a lever. Just as the concept of leverage is a prerequisite for the understanding of compositional balance, chaos dynamics is a prequisite for an understanding of randomness.


copyright 2006, M. Blair Ligon, all rights reserved worldwide.