"All things are numbers." -Pythagoras
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.
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 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?
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. 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.
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. |