Seeing Past The Edge*
An Original Work of Non-fiction
By: David G. Yurth Copyright 1997
All Rights Reserved
Looking For Simple, Elegant Solutions
The Fractal Universe
Information Without Context
The camera is mounted vertically so that its lens points straight down towards the ground during level flight. As the plane flies over the surface, the images it produces are stunning. Repetitive patterns of shimmering golden light ripple and writhe before us in stark contrast to a snaking counterpoint of darkening shadows.
At first, because our view is restricted to a narrowly defined frame, the patterns of light and dark have no meaning. They have no context or scale. While the flow of images is certainly beautiful and pleasing to observe, at this primary level the camera provides a montage whose meaning we can only speculate about. As beautiful as the images are, we feel distinctly uncomfortable after awhile because it is impossible for us sort out what we are looking at.
As the camera view ascends to a greater height, we eventually realize that the undulating patterns flowing across the screen are the peaks of an endless sea of sand dunes, defined with astonishing clarity by the shadows of sunset which separate them. Suddenly, we see the shadow of a Gypsy Moth biplane moving laterally below us across the surface of the desert – it dawns on us in a flash of insight that we are seeing what the pilot sees as he flies above the searing desert.151 How we come to recognize what we have been viewing is the topic of a later chapter – it has to do with the way the human equipment is architected and imprinted with pattern recognition programs, structures and operating routines which give meaning to the things we apprehend around us.152
The scenes which open the motion picture The English Patient provide a stunning example of how Nature works. It is no surprise that the beauty and magic captured by the images are made of sand, a seemingly endless sea of it, displaying all the fractal attributes of a self-organizing system, operating in the interference fringe, at the point of criticality. The patterns emerging from the sea of sand constitute a perfect example of fractal behavior and hold the key, I believe, to understanding one of the greatest of all the mysteries of creation.
Let us make a distinction here about the difference between the remains left by chaotic decay and the fractal appearance which represents the evolution of a self-organizing system. Benoit Mandelbrot of IBM’s T.J. Watson Research Center in New York, coined the word “fractal” as descriptive of the geometrical structures which display self-referent features of all length scales.153 In plain English, this means that wherever a naturally occurring geometrical feature is found which holds the record of evolution of a self-organizing system, with common features, attributes or structures occurring at each scale or level of measurement, then the image displayed by the structures is said to be fractal.
Features which represent fractal displays are all around us – the structure of mountain ranges,154 the surface features which define a shoreline (which distinguish the points of interface between the land and water),155 the surface features of the human brain156; the undulating landscape of the tropical rainforest157 or the delta of any major riverine system when viewed from a satellite;158 the structure of a coral atoll or the ancient Hindu pattern known as the paisley.159 All of these patterns contain the record of evolution of complex, open, self-organizing systems.160
In contrast to the attributes of the fractal patterns arising from the behaviors of complex systems, we immediately recognize the results of chaotic, catastrophic events occurring in closed or insufficiently organized systems.161 Impact sites where asteroids have struck the surface of the planet162, the ejecta patterns defining the eruption of Mount Saint Helens163, the dispersion patterns which characterize the explosion of a star in the heavens164 – all these patterns
demonstrate the effects of chaos, the catastrophic, irreversible evidence which chronicles events occurring in systems which are either closed or which have not yet become sufficiently sophisticated to operate at the point of criticality. Chaos operates in terms of events precipitated by external causes – self- organizing behavior operates spontaneously without regard to external causes.
Nature is Fractal
On March 1, 1980, Benoit Mandelbrot made what can only be described as one of the most important discoveries in all of science. In addition to announcing the discovery of fractal geometry, he was the first to make the astounding, revolutionary observation that Nature is by definition fractal.165 We can appreciate the importance of his discovery by recognizing, for example, that the coast of Norway, which appears to be based on a hierarchical structure comprised of fjords, is instead comprised of fjords within fjords within fjords within fjords, ad infinitum. The question, ” How long is a typical fjord?” has no answer – the phenomenon exhibits a set of attributes which are said to be “scale free,” that is, an examination of the fjord attribute leads to the discovery of fjords within every level of fjords, regardless of our standard of measurement.166 If you see a picture of part of a fjord, or part of the coastline, you cannot make any reasonable guess about its length or size until the picture also provides a ruler to give you a sense of context or scale. Also, the length to be measured depends on the resolution of the measuring device. A very large ruler designed to measure features only on the scale of kilometers will yield much smaller estimates of overall length than if a fine ruler, which can follow details on the scale of meters or centimeters, is used. This attribute of geology, the creation of a fractal image which chronicles the history of a system’s development and evolution, is common to all self-organizing systems.167 This includes graphical representations of the behavior of traffic168, the long-term behaviors of the stock markets169, the extinction and emergence of new species in biology170, the history of warfare among humans, the structure of DNA and all other biological systems. Manfred Schroeder’s extraordinary book Fractals, Chaos, Power Laws: Minutes From An Infinite Paradise171, provides a splendid, breathtaking view of the beauty of fractals, illustrated by their general occurrence at all levels in Nature.
University libraries are filled with an ever-growing number of volumes which attempt to characterize the geometrical properties associated with fractals. However, the fundamental question, what Bak refers to as “the dynamical origins” of fractals, has yet to be satisfactorily answered. Where do fractals come from? What underlying mechanism defines and produces them? Are they common to all self-organizing systems? Do they operate at every scale of the Cosmos?
The importance of fractals and Mandelbrot’s insight into their functions is as important a discovery as the work of Copernicus172, who first observed that the planets in our solar system orbit around the Sun. Isaac Newton developed a set of mathematical formulations to describe the dynamics of planetary motion173 and Albert Einstein provided the theoretical bridge which allows us to rise above the outdated notion that Nature is just a clockwork mechanism174. The astonishing thing about fractals is that absolutely nothing arising from the evolution of physics since Galileo even hinted at their emergence. In the language of the science of complexity, this discovery was truly an “avalanche” event.
The significance of the discovery of the fractal nature of the Cosmos on our search for meaning and primal causes cannot be overstated. Taken by itself, our recognition of the role of fractals as an attribute of natural occurrence has already exerted a significant impact on the way we live. Satellite photographs illustrate the eerie majesty of surface features of the planet which are caught in the act of self-organization one frame at a time. We see the unmistakable hand of self-organizing criticality at work every time we watch the satellite photographs used by television weather forecasters. In the local weather behavior evidenced by tornadoes, hurricanes and floods, we witness a daily sequence of catastrophic events of varying magnitudes, all driven by the same set of forces. When we analyze weather as a global phenomenon, we see the record of evolution displayed in fractal patterns.175
Recognition among scientists and engineers of the fundamental nature of fractal geometry has led to the development of a whole new variety of terrain recognition programs176, pattern recognition algorithms177, image and data compression engines178, military hardware, law enforcement search engines and the sophisticated software routines which drive the Hubble Space Telescope. Law enforcement agencies now routinely use fractal recognition programs to analyze satellite photography of the mountainous regions of Columbia, Bolivia and Peru. Wherever the dimensioned corner of a manmade structure interrupts the fractal patterns of the rain forest, it is possible that a drug processing laboratory has been built there in concealment.
Economists have yet to come to terms with the fractal nature of the self- organizing behaviors of the capital markets179. CitiCorp recently paid the Santa Fe Institute eight million dollars to develop a tool which would enable the corporation to forecast capital and economic trends just twenty-four months into the future180. In spite of the sophisticated mathematics provided by a team of Nobel laureates, the model quickly proved to be inadequate and unreliable. So long as economists attempt to deal with the ebbs and flows of our capital economies in terms of empirical statistical predictors, rather than recognizing the invariant fractal nature of self-organizing systems, there is no likelihood that the Federal Reserve will demonstrate any higher degree of accuracy in its economic forecasts than the state of California has in its failed attempts to predict the location, timing and magnitude of earthquakes or energy supplies.181
Nevertheless, the role of fractals in our consideration of how Nature works
has been seriously understated. Consider this: after nearly 100 years of intensive investigation by the best minds in the human community, three things [among others] have become abundantly clear. First, the Universe we live in is a quantum universe182. The universality of this insight could not be more important, for if it were not so, we would have no significant ability to understand or predict the workings of the material world beyond its grossest attributes.
Second, the Universe we live in is holographic.183 While Descartes and Newton were creating their model of the material world in terms of “physical stuff,” they completely and deliberately disregarded something which has now become self-evident. It is no longer possible to conduct valid scientific inquiry without incorporating “spirit stuff” as an intrinsic and indispensable source of information. What this means, why we know it is so and how it works has a direct, inextricable relationship with the phenomenon Mandelbrot called fractals. Third, the Universe is self-referent, “conscious” by its very nature.184
The Fractal Equation Z D Z2 + C
Taken together in a single context, the self-organizing attributes of the Universe can all be described in terms of a set of simple, elegant rules which are universally applicable to all of Nature. The nature of those rules, their attributes and functions, are largely functions of mathematical expressions. It is an axiom in publishing, at least with regards to books which are slated for general distribution in North America, that the addition of a single mathematical formula can be expected to result in a decrease of book sales by 50%. If this year’s SAT scores are any indication, this may very well be true. In any event, nearly everyone who speaks the English language is familiar with at least one mathematical formula, Einstein’s famous equation, E=MC2. Perhaps we can take the liberty of bringing another disarmingly simple formula to your attention. While there is now considerable question about the validity of the first, there is little disagreement about the validity and significance of the second.
In the case of fractals, the formula is equally simple: Z D Z2 + C.185 There are two key differences between this equation and Einstein’s. First, you will note that there is no “equals” sign (=) in the equation. It has been supplanted, instead, by a symbol which means “goes to and returns from” (D). This means that the simple set of numerical values which are specified for the equation (this is the second difference) modify each other in an infinite, endless sequence of self-referencing computations, a feedback loop, until one side or the other either becomes “free” (reaches infinite values) or goes to zero.
This is the essence of the fractal function – a series of simple calculations based on the simplest of rules refers back and forth on each side’s results until a numerical freedom is attained, in which case the value becomes represented by a series of infinitely complex patterns. The geometric patterns which arise in fractals are self-similar and infinitely variable at the same time. Again, we find unmistakable evidence of the primary principle of complementary at work.
In the alternative, the computational results occasionally produce a null [zero] value which is totally deterministic. This means that when the computational result of a cycle becomes zero, the map coordinates represented by those values become captured, bound or defined by the “free” values on the other side of the equation.
This reciprocal function of the fractal variables produces both bounded states, which appear in the Mandelbrot Set as black, darkened areas (which are the “seed” areas from which other patterns emerge and differentiate), and “free” states, which manifest themselves in individuated patterns which are both similar and mutually distinct from each other in infinite variations of shape and color.186
It is in precisely this way that DNA, for example, is able to contain the encoding necessary to both define biological structures and create infinite similarity and variability in all the tissues, systems, organs and structures of living things.187 This is the same set of baseline functions which operate in concert with Bak’s rules of self-organizing criticality, to define species whose members are clearly similar but distinctly individuated. These functions operate in concert to produce the Nature we are expressions of.
The Cosmos operates in the same manner at all scales. While Mandelbrot and his colleagues may have intuited the extension of their discoveries to include the Cosmos at large, it was not until the Hubble space telescope was repaired and put into full operation in the Spring of 1993 that astronomers were able to actually verify the extraordinary extent of fractal functions operating in the most distant reaches of the Cosmos188. High-resolution color images of the Eagle and Crab Nebulae captured by Hubble and transmitted to astronomers on Earth, for example, stunningly demonstrate the operation of fractal geometry in the vast cosmic nurseries in which stars and star systems are at this moment being born. Indeed, every attribute of the heavens subscribes to the functions of fractal geometry, at every observable scale in the macrocosm.189
In his wonderful book, The Artful Universe,190 physicist John Barrow explains why we find inexpressible beauty and majesty in this recognition. Because the Cosmos is fractal, because we are fractal, because our architecture is fractal, because our essence is inextricably interwoven into the fabric of the Master Fractal, humankind has always recognized an ineffable connection with the extraordinary beauty we witness every night, as the vision of the heavens emerges from the darkness of night.
Whether fractal geometry functions as a defining force at an infinite scale of smallness throughout the microcosm is a question for which we do not as yet have a definitive answer. Stephen Hawking’s belief is that the physical universe as we know it, in our four-dimensional continuum, does indeed appear to have a primary lower limit.191 This dimension, cited as 10-33 cm (a diameter defined as a period followed by 33 zeroes), was first defined by Max Planck, one of the fathers of Quantum Theory and one of the intellectual giants of the 20th century.192 This dimension, known as the “Planck Length,” constitutes a cornerstone of current mathematical estimations of the smallest possible size in the Universe. It is at a diameter approaching this point, for example, that sub- quarks demonstrate their dash–space–dash–space behavior. And it is at this point that super string [M] theory, theories of super-symmetry and other Western calculations of the origins of the material world, all find a common point of resolution.
Ian Stewart’s Insight
Dr. Ian Stewart, Chairman of the Mathematics Institute of Warwick University, first intuited the value of fractal geometry as a tool for general use by scientists and engineers.193 Instead of applying linear-sequential standards of measurement conventionally employed to derive meaning from large volumes of data, he intuited that fractal formulations could be employed to allow the data to manifest itself in natural ways which cannot be intuited by conventional means of measurement. As a result, he and his colleagues at the Georgia Institute of Technology developed a revolutionary new method for compressing and decompressing digital images.
Dr. Michael Barnsley of the Georgia Institute of Technology, one of the leading lights of research into the nature and uses of fractal geometry, had an epiphany one night as he slept.194 After working for months to find a way to apply fractal geometry to practical considerations, he experienced a repeat performance of a dream he had had repeatedly for more than 20 years. He describes the image in his dream state
.of an infinite array of switch terminals connected by an equally infinite array of connecting lines, so that every junction was connected to every other junction throughout infinity.
The jumble and clutter of the image as it grew in his imaginings would eventually become so disturbing and incomprehensible that it would wake him up. But in the last of these dreams, he suddenly realized that by using simple fractal geometrical formulas, he could both understand and recreate the images of the infinite interconnectivity he saw in his dream.
When he awoke, Dr. Barnsley drove immediately to his laboratory and began crafting computer codes which provided his software routines with a way to reduce an aggregation of complex data into a single set of variables, described by a simple fractal formula. Eventually, after much work and refinement, Dr. Barnsley was able to create software routines which recognize images of natural objects, derive the fractal formula which defines their appearance and then, using the formulas alone, reproduce the objects themselves in stunning variety and detail. Today, the power of this brilliant insight is being harnessed for a number of commercial applications which hold the promise of enormous productivity for the future.
Since that time, other scientists, mathematicians and engineers have developed variations on fractal geometries which now make it possible for artists, engineers and scientists to create images of clouds, mountains, riverine deltas, geographical coastlines, thousands of varieties of plant and animal forms and so on.195
What this means for our considerations is simply this: the universe we live in is fractal to its most primary functions. This applies universally because in the final analysis, what fractals do is provide a template which defines the forms and functions for everything which exists, by applying a simple, elegant set of rules to the self-organizing behavior manifest by the most basic set of data. Fractals organize information to produce the manifestations of physicality and, as we have shown, everything, absolutely everything, is information.196
In order to understand why this is so and how it works, we are compelled to see if we can discover any primary relationship between fractal geometry and the laser-produced light effect known as the hologram. Unraveling the particulars which comprise this fundamental relationship constitutes the ragged Edge of our understanding. It is also key to answering many fundamental questions not yet answered about fractals, such as, where do they come from and what role do they play in the evolution of the Cosmos?
There is only one other phenomenon in the known universe which manifests the localized functional attributes found in fractal geometry on a non-local basis. We have come to know it as the hologram. The importance of the relationship between fractal geometry and the attributes of holograms has not been explicitly stated before. I am convinced that the relationship between them is significant and constitutes one of the foundational premises supporting the new model we are creating.197
Fractals Are Not Just Ideas
As we pass through this hole in the looking glass, remember one important thing: the fractal forms produced by the Mandelbrot Set are not just imaginary. They are as real as the air you breathe and the ground you walk on. They manifest an infinite variety of similarly individuated forms, all of which contain in each local scale of detail all the information contained in the whole set, even if the full expression reaches virtually infinite physical proportions. Regardless of where you begin examining or displaying the pictorial images produced by the Mandelbrot Set, eventually you will find that you have obtained access to all the information contained in the entire formulation.
Dr. Barnsley estimates that a fully expressed Mandelbrot Set would have a radius reaching from the Earth to the star system known as Sirius, the Dog Star, located 3.7 light years away. The importance of this attribute of fractals is stunningly illustrated in the film produced by Arthur C. Clarke entitled, “Colors of Infinity198,” which is available in most video rental and sales outlets. I highly recommend it. Nothing in our experience reflects more graphically and immediately how breathtakingly beautiful this notion is.
Holograms and Fractals
Similarly, when any part of the photographic film plate used to produce a holographic image is introduced into the image-production environment, regardless of how physically small the segment may be, the entire original image can still be produced. What this means, among other things, is that within the information set which defines the structure of each phenomenon found in Nature, there is embedded a fractal record containing sufficient information, at any level of resolution, to enable us to reproduce a complete three-dimensional image of the entire object. Remember: when we create a holographic image with laser light, we are relying on nothing more than the interference patterns produced by coherent light, passed through a two- dimensional image (a photographic plate) of the object, to recreate a three dimensional image of the object in real space.199
What does this mean with respect to our quest for understanding how
Nature works? For one thing, it means that if it can be established that the Universe is holographic as well as fractal, we will have discovered a solid link between the simple, elegant set of rules which define the appearance, form, behaviors and attributes of all things found in Nature, at all scales, with the fundamental template implicit to the implicate structure of the Cosmos. Is Nature holographic? I believe we can show without question that it most certainly is.