For want of a nail, a horse was lost;(Mother Goose)
For want of a horse, a rider was lost;
For want of a rider, an army was lost;
For want of an army, a battle was lost;
For want of a battle, a war was lost;
For want of a war, the kingdom was lost.
Several years ago at a lecture I presented the following analogue to effectively introduce the notion that fractal geometry (and "sensitive dependence on initial conditions / strange attractors") has meaningful relevance in our everyday life.
The scene: a normal, crowded, buzzing confluence of people somewhere in midtown New York City. All around, yellow taxicabs blare their horns, smelly street peddlers shout, smoke belching busses hiss their brakes, belly-bouncing workers operate their clangorous jackhammers, a thousand and one scents vie for dominance, and a particularly attractive woman unconsciously blocks out 99% of this maddening assault on her senses. Most especially, she blocks out the hardened scowls pasted on so many faces. Subconsciously, she is very aware of every one of these chaotic sensory inputs but rarely does any of it percolate onto the conscious architecture of her mind. She'd literally go mad otherwise.
Her mind is focussed on getting to work, in her downtown medical laboratory at NYU. Somewhere, her mind is wondering how she'll ever find the solution to the virus challenge she has been working on for years. Some way to introduce, perhaps by a simple vector, a small change in the destructive virus' genetic coding.
Then it happens.
A radiating, simple smile from a child in the crowd draws her attention. The child's unadulterated mannerism instantly penetrates all her defenses, and a wild pang of joyous, contagious energy surges through her. And in that moment, the answer to her problem, which had been lurking there all along behind all those defenses, bursts through in a wild epiphany.
Years later, when AIDS is relegated to the same historical status as perhaps polio, and she is awarded a Nobel prize for her efforts, she will wonder who that simple child was.
Wildly, you open both eyes and look about yourself, and listen. New World monkeys, macaws, toucans, thousands of songbirds, a panther somewhere, hundreds of thousands of insects, the wind rustling the upper flora econiches, the constant dripping of condensation from leaf to leaf to the seething, bacteria-rich soil, and your own apprehensive breathing. These are the sounds. But there is no air conditioning; only the not so subtle stench of moisture-laden decomposition. Your skin itches horribly. But it's what you see which is most startling of all; what causes you to think only in terms of utter chaos. In every direction that you dare look there is a strangling, seemingly unpatterned knot of schizophrenic undergrowth and overgrowth, illuminated by a diffused green haze from above and darkened by the opaque hues from below. All the order to which you are accustomed has given way to utter confusion.
What the heck happened? You went to bed after watching Miami Vice within your well ordered life, in your safe little cocoon of a bed, occupying an economic niche within the stratified, seemingly coherent, pattern of Western society.
But then a different sound captures your attention. It has a curiously familiar pattern. It resonates evenly. It has seeming order within apparent chaos. Ah. . . it is the sound of bulldozer's diesel engine. --And there is the well structured sound of a human voice giving that ever familiar shout in the Amazon jungle, "--Timber!!!"
In the not-so-fictional story above, the subject perceives his or her environment in terms of what is familiar. A more precise way of looking at the term "familiar" is to see it as "events and states that can be related to." Naturally, I've hinted at even deeper levels in the allegory, but to stick with the thesis of this paper, I will refer to Stephen Hawking, who currently holds the world's most prestigious position in physics--Newton's Chair at Cambridge University: "We see the universe the way it is because if it were different, we would not be here to observe it" (Hawking, 183). This is based on fundamental principles of thermodynamics; energy must be transformed (as calories expended by the brain)--I.E., entropy--in order for observations to occur. So, that means that for observation to occur, the observer must live in an entropic, or "chaotic" universe.
Culturally, these "observations" had very rudimentary beginnings; namely, that environment specific to the subject's survival. A mathematical way of stating this type of subject-limiting environment is system, or perhaps, a fractal state. As the entropic transformer of observation(the brain) evolves, more and more of the environment can be described (Hawking 137). Another way of looking at this is to think of the universe in terms of the total set of all the subsets of probabilities as well as all the subsets of knowns. Each subset is merely a relative way of seeing a system, of using a system, of analyzing a system, or otherwise having a mathematical relation to understand or describe some aspect, in an abstract way, of the universe.
Traditionally, science has had a tendency of compartmentalizing the universe into neat, little packages based on the repeatability index of these systems' mechanics. Physics would look at the Amazon jungle in terms of mechanical vectors of force, I.E., gravity, resistances, thermal gradients, time, etc. A chemist would look at the various molecular dynamics of energy transformation. An ecologist would look at the floral and faunal interactions and climaxes, predator-prey equilibriums, and macroscopic biomass input-output values through the defining of a system's boundaries. An anthropologist might look at the Amazon in terms of how to perceive present day conditions of a rainforest and extend analogous relationships into man's primordial past. In each case, depending on the scientist's disciplinary background, the same chunk of the universe is viewed differently. It is often the boundaries of various scientific disciplines which touch upon areas for which science is not yet equipped to describe.
What we observe of the universe is channeled into our brain through unique reflection points, reflection lines, and reflection planes (depending on the dimensions and senses that we are dealing with--certain animals do not perceive the world in 3-dimensions, for example.) These focal points--in an inverse sense--will become clearer later on as I describe the various components to what some theorists are considering a revolution in scientific thought.
This is new nomenclature for traditional science. As such, there is a great deal of ambiguity regarding the terminology one theorist may use over another. Generally, the arguments between scientists are only in terms of semantics. The several books and numerous journal articles discussed here, and work I conducted, left me initially flustered because of these language problems. It requires one to manage the jargon barrier pertaining to mathematics.
Another difficulty will lie with the un-initiated reader. Since most of the research in this field stems from theoretical physics (Quantum Mechanics and the two theories of relativity) the reader must bear with me as I draw a picture of a multi-dimensional (n-dimensional) universe whose endless, interactive and interweaving systems (often visually conceptualized by fractal geometry) are the self repeating or transforming influences that shaped the evolution of all matter, energy, and ultimately--life. Naturally, I will draw considerably upon the works of those who have already researched these relationships. Incredibly, it was at the turn of this century when D'Arcy Wentworth Thompson of Cambridge University first archived these relationships in his "On Growth and Form." He, like Mendel, was misunderstood or otherwise ignored until others at science's cutting edge could understand what he was saying by using the language of nonlinear mathematics.
A simple way of understanding this would be to accept the old adage, "as above, so below." This may seem too complex (actually, too simple) to conceptualize quantitatively. If the whole universe is a system, what does elementary particles physics have to do with human evolution? To best realize it, one needs to be quite familiar with the differential equations in calculus, the very ones Einstein used to develop his principles of general and special relativity, only to then utterly disregard them. You see, you must understand the why-nots in order to understand the why-fors.
The reasons all this relates to anthropology are numerous. In the past decade most major universities or research institutes studying molecular genetics and population dynamics have been using the mathematical models of fractal geometry, or scaling properties, in order to understand how life evolves in terms of an underlying pattern.
I will borrow Stephen Hawking's words to capsule the thesis for this paper. In 1988 he wrote, "If everything in the universe depends on everything else in a fundamental way, it might be impossible to get close to a full solution by investigating parts of the problem in isolation" (Hawking 11). What I hope to do in the proceeding pages is to explore the background and the progress being made to universalize our understanding of the world, eventually focussing on its applications in anthropology.
At this point it becomes necessary for me to lengthily diverge into the realm of basic theory and nomenclature.
The first thing we need to do is differentiate linearity from nonlinearity, as these are key concepts in the study of chaos. The "Dictionary of Scientific and Technical Terms" defines linear to be anything which has "an output that varies in direct proportion to the input. . . relating to a line" (McGraw 917). A linear system is one "where all the interrelationships among the quantities involved are expressed by linear equations which may be algebraic, differential, or integral" (McGraw 918). I use the expression linearity quite often in this paper; it defines "the property whereby a mechanical system is well behaved with regard to addition and scalar multiplication. . . a quantity which produces directly proportional change in another" (McGraw 917). Linearity, therefore, is the keystone to all the mathematics taught in high school and college, and is based on the number line. It's also based on the idea that for every action there is an equivalent, quantifiable reaction.
A nonlinear system, on the other hand, describes "a system in which the interrelationships among the quantities involved are expressed by equations, some of which are not linear" (McGraw 1091). That doesn't seem to be saying a whole lot! But the dictionary defines nonlinearity to be a state which responds in ways "other than directly or inversely proportional to a given variable" (McGraw 1090). So, a system which is described through nonlinear terms would not be bounded by the action-reaction phenomena of Newtonian physical sciences. Nonlinear mathematics generally deal with peculiar scaling properties, which may or may not relate to the number line so fundamental to conventional mathematics.
What is a system ? According to Professor Joe Rosen of Tel Aviv University, "a system is any object of interest with regard to its symmetry properties" (Rosen, 7). "A system could be a geometric figure, like a square, or a physical entity such as an elementary particle, an atom, a molecule, a crystal, a plant, an animal, the earth, the solar system, our galaxy, or the whole universe. It could be a process taking place in time: the scattering of elementary particles by each other, a chemical reaction, the fall of a stone, a beam of light, biological growth, a piece of music, the flight of men to the moon, the evolution of the solar system, or the development of the universe. The system might even be abstract: the laws of physics, an idea or concept, a mathematical relationship, a feeling. There is no system to which the concept of symmetry is inapplicable" (Rosen, 5-6).
So, then, what is symmetry? According to WEBSTER'S NINTH NEW COLLEGIATE DICTIONARY, symmetry is "(1): Balanced proportions; beauty of form arising from balanced proportions. (2): Correspondence in size, shape, and relative position of parts on opposite sides of a dividing line or median plane or about a center or axis -- compare bilateral symmetry, radial symmetry. (3): A rigid motion of a geometric figure that determines a one-to-one mapping onto itself. (4): The property of remaining invariant under certain changes (as of orientation in space, of the sign of the electric charge, of parity, or of the direction of time flow) --used of physical phenomena and of equations describing them" (Webster's, 1196). All of these definitions pertain directly to this "new" science.
So, where does the term chaos fit in? Well, here is a semantic problem. (According to Dr. James Gleick, author of "Chaos: The Making of a New Science", nonlinear dynamics, underlying symmetry within complex systems, and even "catastrophe theory" are part and parcel to the same scientific concept.) Basically, "chaos breaks across the lines that separate scientific disciplines. Because it is a science of the global nature of systems. It makes strong claims about the universal behavior of complexity. It applies to the universe we see and touch, to objects at a human scale" (Gleick, 6).
I began this paper with a story which allegorically emphasizes the anthropocentricity which traditional science has been all about. Newtonian mathematics were controversial because they peered at nature and the cosmos in ways not centered on human activity or the perfect symmetries of Euclidian geometry. Newton, like Galileo, was challenged by the dogmas of literal Biblical interpretation. However, if we trace mathematics further back, to the time of Plato's Greece, we see that the basic comprehension of seeing the universe through geometry was already evolving. "One of the themes dominating the scientific revolution of the 17th century was the Platonic-Pythagorean tradition, which looks on nature in geometric terms, convinced that the cosmos was constructed according to the principles of mathematical order" (Westfall, 1).
We should therefore forgive the narrow view the Greeks gave us of the universe's geometry. To anyone of that Hellenic era trying to solve simple dimensional problems, such as constructing a house, these simple shapes and linear formulations (the number line) were a light at the end of a dark tunnel, not unlike Socrates's "Allegory of the Cave".
Somehow, many of the softer sciences, including anthropology, to this date have been adverse to the intrinsic mathematical cultivation found to be necessary in chemistry and physics. But even within the realm of the hard sciences, symmetry is not taught. Mathematics in these sciences is used to describe separate activities without much reference to the symmetry patterns which cross all boundaries of all scientific disciplines. Don't forget that science is a human invention, and what science studies and proposes theories about is not concerned with these abstractions.
Joe Rosen, professor of elementary particles physics at Tel Aviv University, eloquently states his disdain at this mediocre acceptance of science when he wrote, "The concepts and principles of application of symmetry. . . was not taught . . . except at the graduate or post-doctoral level. Only in children's books or the heights of Weyl's 'Symmetry'"(Rosen VII).
I worked with George Sukahara, who is a professor of nonlinear science, with a Ph.D. in Biology, at U.C.S.D.'s Scripts Institute. He said that "because of the traditionally boring aspects of mathematics at virtually all levels of education, I [G. Sukahara] dropped out of a Ph.D.. in biology and went back for two years to study undergraduate math. . . until kindling and nurturing a love of math"(Sukahara).
In my opening story, the subject perceived only chaos. There was no apparent, underlying symmetry which bound him or her to the jungle environment until the familiar noise of a bulldozer came into the story. The Science of Chaos tries to explain:
"How life begins? In a universe ruled by entropy, drawing inexorably toward greater and greater disorder, how does order arise? Traditionally, when scientists saw complex results, they looked for complex causes. When they saw random relationships between what goes into a system and what comes out, they assumed that they would have to build randomness into any reasonable theory. The modern study of chaos began with the creeping realization in the 1960s that quite simple mathematical equations could model systems every bit as violent as a waterfall. Tiny differences in input could quickly become overwhelming differences in output--a phenomenon given the name 'sensitive dependence on initial conditions.' In weather, for example, this translates into what is only half-jokingly known as the Butterfly Effect-- the notion that a butterfly stirring the air today in Peking can transform storm systems next month in New York"(Gleick 7-8).
Basically, science has for countless centuries been constrained by the Greek sciences of a structured universe; structured by symmetrical squares, circles and triangles. "Sunspots were at first hard to accept because of the Hellenic belief that the sun had to be a perfect body. The notion that the planets could only move in circles hindered the acceptance of the Copernican-Keplerian model of the solar system" (Thomsen 184).
All of science is now undergoing a major metamorphosis. Take the most fundamental theoretical mathematics: calculus. "Differential equations [calculus] represent reality as a continuum, changing smoothly from place to place and from time to time, not broken in discrete grid points or time steps. So by the time a scientist masters this way of thinking about nature, becoming comfortable with the theory and the hard hard practice, he is likely to have lost sight of one fact. Most differential equations cannot be solved at all [emphasis mine]. 'If you could write down the solution to a differential equation,' Yorke [a mathematician who helped Einstein] said, 'then necessarily it's not chaotic, because to write it down you must find regular invariants, things that are conserved, like angular momentum. You find enough of these things and that lets you write down a solution'" (Gleick 67, 68) Unfortunately, in every tiny and large facet of the universe this sort of predictability irrefutably does not
occur. Why?
Gleick explains: "Since the time of the Greeks, there have been two ways of looking at things: that of Hericlitus (everything moves). And that of Paramenides (everything is at rest). This is the problem of pattern and process. It is also a problem of two kinds of time: Newtonian and Bergsonian. The 1st is the time of classical physics, a time that is in no way different from the other physical dimensions and participates equally in the pattern of the Great Being. Bergsonian time is irreversible and is linked in some fashion to increasing entropy and the accumulation of information; it is inextricably involved in the process of the Great Becoming.(Gleick 11)."This problem of pattern or process, the problem of the two times occurs at all levels of biology. For example, consider an enzyme molecule. It is the result of the evolutionary process s s .It is a pattern of chemistry and may be derived in principle from the axioms of quantum mechanics. And it is a participant in the process of cellular metabolism. Biology is thus an amalgamation of pattern and process. Great scientific success may be had by concentrating on either. Darwin achieved fame by rejecting the religious pattern of creation and the fixity of Linnaeus's species and introduced the process of organic evolution. Since the 1930s, mathematical population geneticists have collectively achieved equal fame by successfully exposing the logical patterns of this process"
Due to recent technological advances, the need for greater accuracy in predicting nature has advanced proportionately. Strangely, laws and theories once considered absolute are crumbling. However, since these breakdowns generally are not so shattering that cars will stop running, or the lights won't come on when the switch is hit, they are not a concern with everyday life; except when catastrophic accidents occur due to tiny and intrinsic flaws in statistical analysis or other seemingly tiny variations in mathematical logic. This could be the linear calculations involved with factoring the stress of an aircraft wing or the containment of a nuclear reaction, both of which have proven to be devastating. But even more importantly, linear mathematics seldom ever considers the far-reaching effects of science--I.E., environmental impact. Thus, humanity is acting like the stork hiding its head in the sand to avoid calamity by ignoring the fundamentals of chaos.
"Implicit in science is the belief that experiments are generally repeatable. . . . It is never possible to reproduce exactly the conditions under which an experiment was performed [or an observation made]. The quantity of one reagent may have been altered by 0.001%, the temperature may have increased by 0.0002 K, and the distance from the lab to the moon will probably be different as well [this is crucial since Einstein's fundamental concern with relativity]"(Saunders 17). In truth, then, we really can't expect to repeat any experiment or observation under precisely the same conditions. By repeating experiments or making observations under approximately the same conditions we will obtain only approximately the same results; within certain limits, let's not forget about the Butterfly Effect. "The Lorenzian quality of 'sensitive dependence on initial conditions' lurks everywhere. A man leaves the house in the morning, thirty seconds late, a flower pot misses his head by a few millimeters, and then he is run over by a truck. Or less dramatically, he misses a bus that runs every ten minutes--his connection to a train that runs every hour. Small perturbations in one's daily trajectory can have large consequences" (Gleick 67).
In science, "small nonlinearities were easy to disregard. People who conduct experiments learn quickly that they live in an imperfect world. In the centuries since Galileo and Newton, the search for regularity in experiments has been fundamental. Any experimentalist looks for quantities that remain the same or quantities that are '0'. But that means disregarding bits of messiness that interfere with a neat picture. If a chemist finds two substances in constant proportion of 2.001 one day, and 2.003 the next day, and 1.998 the day after, he would be a fool not to look for a theory that would explain a perfect two-to-one ratio" (Gleick 41).
Isn't this what Mendel did when he ignored the tiny variations observed in his phenotypical ratios?
Galileo challenged Hellenic ideals. Newton challenged some of Galileo's ideals. Einstein challenged Newton's ideas on gravity, time and basically, the universe. Now, due to rapid information dissemination and advances in technology, the cutting edge of every science is challenging the entire frame of our historically acquired parameters.
The underlying basis to most natural processes has been randomness. It is the characteristic which Darwin used to explain adaptation and selection. Later, Mendel was to use it to help explain adaptation's genetic basis. More recently, population and molecular geneticists have also supported their arguments using the principles of "random behavior" as the driving force. Although randomness is not being yanked out of the evolving view of natural processes, it is being challenged, and with considerable force. Gleick states that "the understanding of nature's complexity awaited a suspicion that the complexity was not just random, not just an accident" (Gleick 94). That "lurking behind every apparently balanced, dynamical system was chaos" (Gleick 42). The idea is that there are patterns through which "randomness" pass in order to make the processes visible.
So how does one describe such systems in a viable and useful way? Feigenbaum, a leading chaos theorist, responded that "one has to look for scaling structures--how do big details relate to little details. The process doesn't care where it is, and moreover it doesn't care how long it's been going. The only things that can ever be universal, in a sense, are scaling things" (Gleick 186). The renowned professor of mathematics at Queen Elizabeth College of the University of London, Dr. Leonard M. Sander further elaborated by stating that "an increasing amount of evidence suggests that nature's love of fractal shapes is a deep one. Random patterns are almost certainly fractals as well"(Sander 94).
"The first chaos theorists had an eye for pattern,
especially pattern that appeared on different scales at
the same time. They had a taste for randomness and
complexity, about evolution-- the analysis of systems in
terms of their constituent parts: quarks, chromosomes or
neurons [etc.]. They believe that they are looking for
the whole"
(Gleick 5).
Traditional sciences teach about the equilibriums in the systems they describe. "Chaos may undermine ecology's most enduring assumptions. What passes for fundamental concepts in ecology, is as the mist before the fury of the storm--in this case, a full, nonlinear storm" (Gleick 315).
So, traditional approaches in science deal with finding order; and the paradox is that the more science strives to search for order, the more it finds chaos. "The simplest systems are now seen to create extraordinary difficult problems of predictability. Yet order arises spontaneously in those systems--chaos and order together. Only a new kind of science could begin to cross the great gulf between knowledge of what one thing does--one water molecule, one cell, one neuron [one individual in a gene pool]--and what millions of them do" (Gleick 8). In effect, what traditional science has done has been to isolate and then create theories about the seemingly ordered phenomena that arise out of the seething chaos of the universe.
What the science of chaos is attempting to do is to represent this activity with metaphors which are as recognizable by traditional scientists as possible. Thus: fractal geometry, which is a graphic method of seeing scaling properties. It is also the type of geometry nature herself uses in her various forms. Gleick says that fractal geometry is "the only way to make sense of such numbers [probability breakdown, etc.] and preserve one's eyesight" (Gleick 63).
Recently, due to these graphs' often mesmerizing features, many publications have published numerous articles on the subject. These articles often include visually stunning photographic representation of chaos. A 1989 SCIENTIFIC AMERICAN article stated that "many complex phenomena, such as crystal formations and changes in animal populations, can be represented with fractals" (Unmasking 61). That's a very conservative statement, according to the growing proponents of chaos.
It is becoming apparent that "the tradition of looking at systems locally--isolating the mechanisms and then adding them together--[is] breaking down" (Gleick 44). The shapes of classical geometry, which is at the root of linear mathematics, are lines and planes, circles and spheres, triangles and cones. They represent a powerful abstraction of reality, and they inspired a powerful philosophy of Platonic harmony; in the arena of an emerging human intellect they were revolutionary. This was a geometry which Euclid made last more than two thousand years and it's still the only geometry most people ever learn. Artists, scientist and spiritualists alike have found an ideal beauty in them. But for understanding complexity, which is being found now in everything, no matter how grand or small, they turn out to be the wrong kind of abstraction.
We need now to go on to a new, more powerful abstraction of the universe if we are to succeed in making progress in understanding it. Let's briefly journey into abstraction: I. Begin the sequence with the simplest abstract possible: a point. A one-dimensional singularity--a point--[a black hole, for example; or the initial condition in the universe prior to the big bang]. II. This point, without any equation other than to tell it to go into a 2nd dimension, transforms into a line.
A point, therefore, is actually the cross-section to the 2nd dimension. All the information you need to know about a line is contained in the point. The point is a line's pattern. A point with a formulation could be the x, y coordinates on a Cartesian grid. A line is the cross section of a plane, which can also be plotted, either straight, angular or curved through Cartesian geometry. III. A plane is the cross section, or pattern if you will, of depth. . . of space. . . of volume. . . of matter. IV. Some mathematicians consider 3-dimensional space to be the cross section of time.
The phase transition of a point is a line. The phase transition of a line is a plane. The phase transition of a plane is depth. Any additional formulations in any of these phases are called determinants or phase transformers. A given point, with a formula, can describe any 3-dimensional object. That point is called the transformation point. What you are actually doing with the formulation is telling the point to draw vector lines into a three dimensional grid (x, y and z coordinates).
The fourth dimension--time--is difficult to reckon within a linear fashion. I'll get into its mechanics later, but to stay simple, time does not exist in a point. You can have two dimensional time when a point is allowed to go into infinity, or specific sections thereof, through an equation. But it's not relevant; the thermodynamic laws of the universe require at least a 3-dimensional space to operate. Time only becomes important when dealing with expansion or compression of mass: the processes of being (or, becoming). Metaphorically, time is represented in a two dimensional way, as is all of geometry, by the predictable motion of radii around a circle.
Scientifically, time began when the initial point of the universe exploded into the 3-dimensional patterns we can now observe in retrospect by peering into distant space. Therefore, time is relative to acceleration, speed and gravity; three nearly analogous components of the mechanical universe. The furthest galaxies out there--called radio galaxies, since they are not substanced in the expected way of our local galaxies--are the ancient imprints of what the universe looked billions of years ago; and are thus closer to the patterns to be found soon after the Big Bang. They are also the fastest things in the universe. Thus, a process began then, for which patterns are awaiting to be uncovered; some of these are temporal, others are spatial, and we'll probably one day find out they are both. Gravity, resistances, thermal gradients, time, etc. A chemist would look at the various molecular dynamics of energy transformation. An ecologist would look at the floral and faunal interaction
Gleick said that "those who recognized chaos in the early days agonized over how to shape their thoughts and findings into publishable form. . . [It seemed] too abstract for physicists yet too experimental for mathematicians"(Gleick 38). As with any formalized science, chaos needs to define its boundaries; in this case, I mean the boundaries of description and analysis. Or better yet, how the science of chaos is integrated.
In looking at a system, which I've hopefully quite well defined, a chaos scientist invariably must scale the system's many activities. Dr. Richard S. Gruber writes:
"It is common to describe nature in terms of its
strata structure. The crudest classification of this
kind is given by the series of cosmological objects,
everyday objects [animals, mountains etc.], molecules,
atoms etc. One then hastily points out that the
everyday objects consist of molecules, which consist of
atoms, and so on, and that phenomena belonging to
different strata are usually described by different
theories. A particular distinction is made between the
level of everyday objects and that of molecules. The
objects which belong to the level of molecules or
smaller are usually called microscopic, and other
objects are called macroscopic.
"However, it is obvious that nature is too complex to be grasped by such a 'linear' viewpoint. To point this out, let us recall the example of dislocations in a crystal. Dislocations are classically behaving macrocosmic objects which are created in a crystal and interact with various microcosmic objects such as phonons [a quantum of vibrational energy]. In one word, crystals represent examples of systems in which microcosmic and macrocosmic objects coexist and interact with one another. Even when one speaks of an object of cosmological scale, one faces the question of asking how the cosmological objects have been created out of very fundamental microcosmic objects such as quarks [sub- sub- atomic] or something more fundamental. One significant aspect of this question is to ask how the macrocosmic objects come out of microcosmic systems" (Gruber 412).
The rationale here is that according to quantum mechanics, nature comes closer, without ignoring lateral influences, to the time of origin the more micro-cosmic one looks.
Conversely, this scheme can easily be self-contradictory. When scientists look closely at fluid dynamics, a great deal of unexplainable and uncalculated phenomena arise. In a river, a fluid scientist would look at the forces creating the various eddy currents. Tributary volume contribution, gravity-induced inertia (slope), bank and bottom topography, effluent suspension, temperature gradients, and animate motion all contribute to these interacting eddy currents. To a point, the eddies can be predicted. "Eddies form, and smaller eddies within them, each dissipating the fluid's energy and each producing a characteristic of rhythm. This energy cascades through smaller and smaller scales until finally a [microcosmic] limit is reached, when the eddies become so tiny that the relatively larger effects of viscosity takeover" (Gleick 123). Turbulence is a steady accumulation of conflicting rhythms in a moving fluid. It's at this point that classic science and mathematics fail and when chaos science can he lp predict nature.
Fluid dynamics is a good example of studying nature for several reasons. It is easily considered, it creates its own graph (wave motions, etc.) and it is also where life evolved--which is not an arbitrary statement; many of the early life morphologies in the ocean reflected the mechanical activities of turbulent water. The surface of water, after having been impacted by a drop of rain, resembles a polyp. These are observations which I'll discuss in greater detail later on. But the crux of the preceding three paragraphs has been that nature interacts through patterns which are nonlinear; in other words, it's not a simple progression but one of overwhelming interaction.
Now we look at another aspect of nature where chaos is apparent: phase transition, such as the change of gasses to liquids to solids; or even, the change of atoms into the structured molecules, cells and organs that constitute life.
"Like so much of chaos itself, phase transition
involves a kind of macrocosmic behavior that seems hard
to predict by looking at the microcosmic details. When
a solid is heated, its molecules vibrate with added
energy. They push outward against their bonds and force
the substance to expand. The more heat, the more
expansion. Yet, at a certain temperature, the change
becomes sudden and discontinuous. A rope has been
stretching; now it breaks. [In evolution, a species
changes with gradual, predictable equilibrium; then
suddenly, there is radical change.] Crystalline form
dissolves, and the molecules slide away from one
another. They obey fluid laws that could not have been
inferred from any aspect of the solid. The average
atomic energy has barely changed, but the material--now
a liquid, or a magnet, or a superconductor--has entered
a new realm"
(Gleick 127).
A realm where one classic and differential theory must suddenly acquiesce to another, utterly different one. That's the new metaphors science is considering.
Chaos scientist have a word for what happens when one symmetry suddenly gives way to another; they say that a strange attractor has emerged and visibly influenced the system in the underlying pattern. Disorder was channeled into patterns with some underlying theme. The recognition of strange attractors gave mathematicians a clear program to carry out research. "One finds strange attractors wherever nature seems to be behaving randomly" (Gleick 152). One way of comprehending strange attractors is to remember that systems have a documented and sensitive dependence on initial conditions. Generally, these initial conditions, since they are so relatively insignificant, become buried in a researcher's complex analysis. Further, "in an apparently unruly system, scaling [influences passing from one phase to another] meant that some quality was being preserved while everything else changed. Some regularity lay beneath the turbulent surface" (Gleick 166).
In other words, when life evolved out of the simple atoms and molecules in the antediluvian oceans, patterns ascended through this process, being continuously redefined at the broadened scales, all the way to Homo sapien sapien. The randomness which anthropologists have been so adept at postulating to be the driving force is the chaos. By using fractal geometry to scale this chaos, the underlying patterns can be conceptualized. There are, of course, various sorts of patterns; it all depends on what part of a system you want to look at and what sort of information you want. And depending on the scale at which you are studying a system, the pattern can look different. But what's unique is that you are using the same metaphors of analysis regardless of whether you are a psychologist, physicist, anthropologist, chemist or mathematician. That's what is currently the focal point of the research being done at Los Alamos National Laboratory.
San Diego radio station KPBS in 1987 joked about how Los Alamos was the place where ultimate death, in the form of the atom bomb, was created; but that now, it represents the place where life is being grasped.
To continue with the track I developed regarding strange attractors and underlying patterns, let's look at another word: self referential. In essence, this describes what happens when one type of behavior is guided by the behavior of another hidden inside it. Fiegenbaum, a mathematician working at Los Alamos recently spent three solid weeks in isolation, communicating only with a supercomputer and consuming only coffee; he was trying to develop a theory which is nearly universal. Initially, he looked at the smallest sub-atomic particles to date documented: quarks and meons. A medical doctor at the institute eventually halted Fiegenbaum's work because he had become so absorbed that he was compromising his health.
When looking at complex behavior, which anything random is, scientists have traditionally been analyzing the wrong behaviors. Because of our Hellenic tradition, scientists look at things with the expectation of finding increasing complexity which must be unraveled. According to the chaos way of looking at things, "any point in phase space can stand for a possible behavior of the dynamical system" (Gleick 138). Since phase space signifies the spatial/temporal environment of an event or system, the chaos scientist abstractly cross-sections the graphed representation of his or her observations to comprehensively describe what is taking place.
(Gleick 139)."In the long term the only possible behaviors are the attractors themselves. Other kinds of motion are transient. By definition, attractors had the important property of stability--in a real system, where moving parts are subject to bumps and ziggles from real world noise [a random attribute], motion tends to return to the attractor. A bump may shove a trajectory away for a brief time, but the resulting transient motion dies out.
"In phase space the complete state of knowledge about a dynamical system at a single instant in time collapses to a point. That point Is the dynamical system--at that instant. At the next instant, though, the system will have changed, ever so slightly, and so the point moves. How can all this information about a complicated system be stored in a point? If the system has only two variables the answer is simple. It is straight from the Cartesian geometry taught in high school--one variable on the horizontal axis, the other on the vertical. If the system is a swinging, frictionless pendulum, one variable is position and the other velocity, and they change continuously, making a line of points that traces a loop, repeating itself forever, around and around. The same system with a higher energy level--swinging faster and further--forms a loop in phase space similar to the first, but larger. When friction variables are added, every orbit must eventually end up at the same place, the center: position '0', velocity '0'. This central point 'attracts' the orbits. Instead of looping around forever, they spiral inward. The friction dissipates the system's energy, and in phase space the dissipation shows itself as a pull toward the center, from the outer regions of high energy to the inner regions of low energy. The attractor--the simplest kind possible--is like a pinpoint"
So this is how chaos would look at the rudimentary components of mechanical motion. To view much more complex and interactive systems in the same way one cannot use simple points, but must use multi fractional scaling. "Multi fractional scaling provides quantitative description of a broad range of heterogenous phenomena. The key to this progress has been the recognition that many objects with random structure possess a scale symmetry. Scale symmetry implies that objects look the same on many different scales of observation" (Stanley 405).
"The vast majority of complicated natural phenomena have virtually defied understanding" (Sander 94). So "mathematicians have to accept the fact that systems with infinitely many degrees of freedom--untrammeled nature expressing itself in a turbulent waterfall or an unpredictable brain--requires a phase space of infinite dimensions" (Gleick 137) That's why fractal geometry is so ideally suited; it allows the scientist to pinpoint and develop a quantitative analysis of monster systems. In fact, nature itself emulates this abstraction: just look at the sprawling, seemingly chaotic construction of a tree; or the zig-zags of lightning; or the human venous system; or taken to abstraction, evolutionary history as outlined by paleontological records. There is a great deal of multi fractional scaling observed here. "Growth in nature can produce the sprawling, tenuous patterns called fractals"(Sander 94). These fractal patterns imply that there are attractors present in the underlying pattern. "Nature forms patterns. Some are orderly in space but disorderly in time, others orderly in time but disorderly in space. Some patterns are fractal, exhibiting structures self-similar in scale. Others give rise to steady states or oscillating one"(Gleick 309).
It becomes necessary to grasp a little physics in order to better understand anthropology. Not just for dating purposes, but for the reasons I've outlined above: the underlying themes and patterns of nature are not bounded by scientific disciplines.
Quantum physics has probably made the greatest contribution to technology in the shortest period of human history. In just over fifty years, we've gone from the crude idea of electricity as an ill-defined river of electrons to a nearly unified theory of energy and matter which allows for supercomputers to be built and for black holes to theoretically be tapped as an energy source. The theorists necessarily had to constantly change their fixed perceptions about nature in order to advance. In fact, Einstein himself presented a major stumbling block to the development of quantum physics, even though he was awarded a Nobel prize for his work on the subject.
In a nutshell, since this is not a paper in elementary particles physics, let me quote Dr. Hawking on the subject of quantum mechanics. Based on Newton's assumption regarding gravity, until 100 years ago scientists believe that a "hot object, or body, such as a star, must radiate energy at an infinite rate: a 'hot body' ought to give off electromagnetic waves (such as radio waves, visible light or X rays) equally at all frequencies. . . . Since the number of waves a second is unlimited, this would mean that the total energy radiated would be infinite. [And this violates the most basic rule in science.] The German scientist, Max Planck suggested in 1900 that light, X rays, and other waves could not be emitted at an arbitrary rate but only in certain packets that he called quanta" (Hawking 54). This allowed for a finite view of a radiating body's energy potential. These quanta of energy will disturb particles in space and/or mass and change their velocities and/or energy relations in certain ways, much like waves of water will disturb floating objects.
Let's look at what Newton and Einstein thought. To predict the nature of things Newton felt that mass had an attractive force around it, which he called gravity. Einstein proved him wrong. What Einstein did was create a four dimensional coordinate system--three of space and one of time. Inherent in this coordinate system is his theories of relativity which make the assumption that lightspeed [or any such radiation] is exactly the same wherever one is and no matter what one's velocity is. The relationship between mass, relative speed and velocity is very important for several reasons. The faster an object is pushed, due to inertial resistance, the more energy that object accumulates. The relationship such speeding objects have with their more restful environment is intriguing. The higher the velocity of an object, the slower time becomes and the more outside objects become condensed and closer. However, the light being emitted by a relatively static object and the moving object does not fluctuate in relation to one another.
Simply put, if two spaceships are speeding toward a distant object at two different speeds and at the moment one overtakes the other they both emit light, both signals would arrive at the distant object simultaneously. The faster ship did not add its velocity to the beam of light. Another feature to the theory is that if two ships were speeding toward each other at 95% the speed of the light at a given distance, it might be reasonable to assume they would cross that distance such that their combined speed equated to 190%the speed of light. Not so! The two ships, no matter how fast, will span that distance such that they will rendezvous with a combined velocity of 99% the speed of light. However, due to the principle of the conservation of energy, time and mass relationships relative to the ships and outside observers will be markedly different.
Einstein called it the "twin paradox": If a person flies to a distant star at near the speed of light, the ship's clock appears to be going slower to Earth observers. When the astronaut returns to Earth, his friends might be long dead. This is explained by the coordinate system that Einstein developed. The greater the acceleration--which is exactly the same as gravity--, the more "bent" space becomes. His Special Theory of Relativity deals with the event horizons between objects in relative stasis with objects in motion without the phenomenon of acceleration.
On planets with greater gravity than earth's, clocks will go slower and people will age slower as well. The inverse is true of people living on the moon. Why can't the speed of light be broken? Because the speed of light represents the entire energy capacity of the universe; it would require the total amount of energy in the universe to go a fraction beyond the speed of light, at which point the universe would cease to exist in time or space. Incidentally, space is curved due to it's total mass.
Quantum mechanics presents a curious paradox as well; the smaller something is the less likely it is to obey the fundamental laws of nature, I.E. , the laws of thermodynamics. These laws, especially the second one which deals with entropy, structure the way atoms explode, water boils, animals move, evolution occurs, the stars shine and how the universe is expanding after the "Big Bang". But at the elementary level, these laws don't hold quite as well and for a good reason.
In the 1920s, physicists stumbled upon an essentially correct description of the world around them because "it tells you how you can take dirt and make computers from it. It's the way we've learned to manipulate our universe. It's an extraordinarily good theory--except at some level it doesn't make good sense. You can't think of a particle moving as though it has trajectory. You are not allowed to visualize it that way [through quantum mechanics]. Upon asking more subtle questions--what does this theory tell you the world looks like? In the end it's so far out of your normal way of picturing things that you run into all sorts of conflicts. There is a fundamental presumption in physics that the way you understand the world is that you keep isolating its ingredients until you understand the stuff that you think is truly fundamental. In a way, that's a waste of effort, because what really happens has nothing to do with a fluid or particle [etc.] equation. It's a general description of what happens i n a large variety of systems when things work on themselves again and again. It requires a different way of thinking about a problem" (Hawking 56).
It would be far too complicated in the context of this paper to intimately explain the increasingly microcosmic activity at work within the sub- sub- subatomic properties which provide the framework for a reasonably stable universe composed of matter and energy. However, I will state that at these subatomic levels scientists are finding particles (within very brief time relations) that can only be metaphorically described by giving them a so-called "spin factor"; spin factors of 0, �, 1, or 2. I say metaphorically because scientists are looking at things so microscopic that they can never ever be viewed directly due to the constraints of our bodily senses. At an elementary level, these spin factors also describe the four fundamental energies present in the universe: electromagnetism (electrons), gravity (gravitons), the force that binds protons, and the force which deals with even smaller particles called quarks.
When dealing with a great deal of energy, which is the same as slowing down time, which can be the same as saying that we're looking at the universe in it's infancy, scientist are discovering extended objects. "When the size of the extended object is much larger than the quantum fluctuation, the extended object behaves as a classical object [as defined by the laws of thermodynamics]. As soon as the creation of an extended object breaks the translation and rotational symmetries, there appears a quantum coordinate which recovers the translational invariance" (Gruber 421). This translates that there are infinite variations to the types of particles and energies possible; that the way the universe is currently shaped is due to the patterns first adhered to right after the Big Bang. By looking at very elementary particles, scientists are discovering that the universe could easily have developed very differently and with very different laws.
Some of these spin factors describe a geometry which seems to intersect more than the three dimensions of space and one of time Einstein laid out for us. Hence, in a universe filled with evolving creatures, these dimensions must play a patterning role. But can conventional science help us find these relations?
Gleick explains that "the whole tradition of physics has been to isolate the mechanisms and hopefully all the rest flows. That's completely falling apart. To know the right equation [of classic physics] is not helpful. You add up the microscopic pieces and you find that you cannot extend them to the long term" (Gleick 175).
Ever since atoms were first mapped, writers have alluded to the curious similarity between the subatomic orbits of electrons around protons with planets around a sun. The fractal relationship between the universe and its subatomic components is 3.6 to 100 billion. Staggering!
Moving away from the microscopic, let's look at our everyday world of life and form. If two forms look alike, must we look for like causes? What about "the multi pronged falling droplets of liquid, hanging in sinuous tendrils, displayed next to astonishingly similar jellyfish?" (Gleick 199). Dr. Thompsen, at the turn of this century, stated it most eloquently when he wrote, "We may ultimately find a certain analogy between the slow, reluctant extension of physical laws to vital phenomena and the slow, triumphant demonstration by Tycho Brake Copernicus, Galileo, and Newton(all in opposition to the Aristolean cosmology), that the heavens are formed of like substances with the earth. And that the movements of both are subject to the selfsame laws"(Thompsen 11).
Thompsen did not have computers to generate the intricate graphs which nonlinear scientists use to quantify their work. He looked at nature and used the simple geometry of his time to make simple correlations, and published his findings in a twelve-hundred page book. One of the guiding themes or, as nonlinear scientists would say, attractors, in evolution is that "things in nature will fall apart unless proportionate changes are induced, as with the hollow bones in birds" (Thompsen 27). There's not much to argue with here, except in terms of the cause behind evolution. Thompsen did not limit himself to the accepted notions of Mendelian or Darwinistic randomness. Thompsen's premise is that all organisms are "but a portion of that wider science of form which deals with the forms assumed by matter under all aspects and conditions, and in a still wider sense, with forms which are theoretically imaginable" (Thompsen 1026). Moreover, "every natural phenomenon, however simple, is really composite, and eve ry visible action and effect is a summation of countless subordinate actions" (Thompsen 1028). This paraphrases the notion coined sixty years later: the sensitive dependence on initial conditions.
What Thompsen did was show us the similitudes between the incredibly complex organisms that have evolved on earth with the very simple shapes which physical science prefers. "There is no essential difference between the phenomena of organic form and those which are manifested in portions of inanimate matter. The mathematician knows better than we do the value of approximate results [emphasis mine]. The child's skipping-rope is but an approximation of Touygen's catenary curve--but in the catenary curve lies the whole gist of the matter. We [observers] may be dismayed too easily by contingencies which are nothing short of irrelevant compared to the main issue" (Thompsen 1029).
The principles of similitudes are obvious all through nature. Thompsen argued that the larger ratio of surface to mass in a small animal would lead to excessive transpiration, were the skin as porous as our own. This accounts for the hardened, or chitinous, skin of insects and other arthropods. Large and fat life forms must be suspended or lie on the ground, as with melons and whales. In quadrupeds, a large head must be supported on a neck that's either thick like a bull's or short like an elephant.
These are generalities. Thompsen was very specific and used scaling symmetries to illustrate his observations. He pointed out that "the effect of scale depended not on the thing in itself, but in its relation to its whole environment. A common effect of scale is due to the fact that, of the physical forces, some act either directly at the surface of a body, or otherwise in proportion to its surface or area; while others, and above all gravity, act on all particles, internal and external alike, and exert a force which is proportional to the mass, and so usually to the volume of the body" (Thompsen 25).
In mechanical physics the student is taught vectors of influence and orders of magnitude. Thompsen applied these principles to nature and evolution. "A fish, in doubling its length, multiplies in weight by no less than eight times; and it doubles its weight in growing from four inches to five inches long" (Thompsen 23). The formulation is quite simple:
W = (k) (L[cubed]).
Two generations after Thompsen, Dr. Saunders in his "Introduction to Catastrophe Theory", explained in contemporary terms: "What D'Arcy Thompsen discovered was that by drawing one species of fish on a rectangular grid, and then performing what we would call a simple diffeomorphism, he would obtain remarkably close likenesses of three different, though related, species. Similar transformations were found for a number of other examples, including the skulls of primates and other mammals, the pelvises of fossil birds and the carapaces of various crabs. This is strong evidence indeed for the claim that the topology is basic to the plan [evolution], with the geometry being filled in later and consequently being more susceptible to change during evolution" (Saunders 116).
So it is becoming increasingly apparent that there are patterns at work underlying what causes nature to produce its shapes and sizes. Beside topological morphology, Thompsen also dealt with size-limiting factors, caloric consumption and organ shape ratios. Smaller animals produce more heat per unit of mass than larger ones in order to keep up with surface-loss; extra heat production means more energy spent, more food consumed and more work done. Thompsen showed that the larger an animal is the less calories per kilo of bodyweight it could consume.
A guinea pig weighs at most seven kilograms but it consumes 223 calories per kilo in a day. That's not very efficient, but it's absolutely necessary due to its volume to surface area ratio. Of all porous skinned, and hence, warm-blooded creatures, the humming bird--being the smallest--has the highest metabolic rate. It will easily go into protein toxicity and has a heart rate of 600 beats per minute. An average elephant weighs 4000 kilos but needs only 13 calories per day per kilo. A whale can weigh as much as 150,000 kilograms but can support itself on as little as 1 calory per day per kilo. Man's average weight is 70 kilograms and he or she needs 33 calories per kilo per day. The smaller an animal is, the more it has to be very near its food source. The larger it is, the more it can wander.
But beyond a certain size, a mammal needs such thick skin and such a bulging shape due to gravitational and thermodynamic considerations that it would not survive. If it were more cold-blooded, greater size could be possible. This interesting fact reaffirms the older theory that dinosaurs were fairly cold blooded. The thermodynamic advantage toward increased size that they had was due to their use of direct solar radiation for warmth rather than food.
Man consumes 1/50th of his weight daily. A mouse eats half its weight in a day. "A warm blooded animal much smaller than a mouse becomes an impossibility; it could neither obtain nor yet digest the food required to maintain its constant temperature, hence no mammals and no birds are as small as the smallest frogs or fishes. The disadvantage of small size is all the greater when loss of heat is accelerated by conduction as in the arctic, or by convection as in the sea. The far north or far south is home to large birds and large mammal; not to mice. The smallest of the dolphins live in warm climates and there are no small mammals in the sea" (Thompsen 35). This principle is commonly referred to as Bergman's Law.
These relationships help explain why elephants have thick skin, why insects could never survive without a non-porous exoskeleton, why humans sweat, and why certain shapes must be. The shapes that pertain to movement, like arms, legs or fins are secondary; important, true, but even these shapes have an underlying pattern that can be visualized upon graphing.
Let's look at fish. In contradiction to the similitudes of land based animals, Thompsen pointed out that fish have an advantage by being bulky.
"The larger a fish grows the greater is its speed. For its available energy depends on the mass of its muscles, while its motion through the water is opposed not by gravity, but by 'skin friction', which increases only as the square of its linear dimensions. The bigger the fish the faster it tends to go, but only in the ratio of the square root of the increasing length. For the velocity (V) which the fish attains depends on the work (W) it can do and the resistance (R) it must overcome. Now, we have seen that the dimension of W are L(cubed) and of R are L(squared), and by elementary mechanics therefore:(Thompsen 31)."This is Froudes law of the correspondence of speeds--a simple and most eloquent instance of 'dimensional theory' [which is Thompsen's jargon for scaling symmetry]"
Let's look at the internal components of (wo-) Man and other life forms. Thompsen enjoyed analyzing skeletal structures in the same way an engineer would study physical laws pertaining to the design of a bridge. He also concerned himself with the requirements for motion and activity. In Froude's "Law of Steamship Comparison", another vector of organic influence comes to light. In a large animal, lungs, the heart and other "boiler-type" components must increase proportionately to a creature's size.
Thompsen did not have the powerful microscopes of our day to look at these internal organs nor the computers to generate the results. Had he, he would have been astounded by the networks found on the microscopic scale which makes use of very small spaces to accomplish a great deal of work. Gleick points out that blood vessels "branch and divide and branch again until they become so narrow that blood cells are forced to slide through single file--the nature of their branching is fractal. As a matter of physiological necessity, blood vessels must perform a bit of dimensional magic. Just as a Koch curve [a specific type of fractal] squeezes a line of infinite length into a small area, the circulatory system must squeeze a huge surface area into a limited volume. In terms of the body's resources, blood is expensive and space is at a premium. The fractal structure nature has devised works so efficiently that, in most tissue, no cell is ever more than three or four cells away from a blood vessel. Yet the circulatory system takes up little space. It is, as Mandelbrot [chaos researcher at Los Alamos National Laboratory] put it, the Merchant of Venice Syndrome--not only can't you take a pound of flesh without spilling blood, you can't take a milligram" (Gleick 108). One who looks at the capillaries in a finger tip, due to the system's fractal nature, can infer the features of the veins and arteries nearer to the heart; they're just bigger in the dimensional sense.
These analogies and similitudes can be found in the other body organs, such as the intestines, lungs, testicles, brain, liver and so on. Thompsen was convinced that the external shapes of the body's organs followed patterns observed in physics. In fact, he saw no difference between the shapes which a glass blower could derive than those which evolution has formed. By placing the cross section of the slightly elliptical human stomach on an x, y graph, for example, he was able to show just what it would take to make the more bulbus shape which gravity would prefer. With only a slight variation in the equation, he could make the shape that defines a kidney, or virtually any other organ. "Material things, be they living or dead, shows us but a shadow of mathematical perfection" (Thompsen 1030). "The bean and the human kidney owe their 'reniform shape' to the existence of anode or 'hilus', about which forces of growth are radially and symmetrically arranged. We can illustrate the shapes of leaves by mea ns of radial coordinates, and even attempt to define them by polar equations" (Thompsen 1044).
If one were to physically restrict the growth of a plant, such as a tree or gourd, that plant's natural inclination toward symmetrical growth would only be restricted by the restraining force. "It's clear, therefore, that we may account for many biological processes or transformations of form [evolutionary change] by the existence of lines of constraint which limit and determine the action of the expansive forces of growth" (Thompsen 1049)
This is why a clear understanding of the fundamental nature of matter, energy and time is so essential. And it's not so much a matter of isolating these forces as conventional physicists have done but it's a matter of understanding them in their underlying and interactive sense.
Now that we've looked at shapes, let's look at process, specifically, natural selection. Thompsen grasped the general gist of the underlying mathematics of evolution when he wrote:
"The characteristics of Mendelian genetics show no fault; tall and short, rough and smooth, plain or colored are opposite tendencies or contrasting qualities. But when the morphologist compares one animal with another, point by point or character by character, these are too often the mere outcome of artificial dissection and analysis. Rather is the living body one integral and indivisible whole, in which we cannot find, when we come to look for it, any strict dividing line even between the head and the body, the muscle and the tendon, the sinew and the bone. Characters which have differentiated insist on integrating themselves again; and aspects of the organism are seen to be conjoined which only our mental analysis had put asunder. The coordinate diagram throws into relief the integral solidarity of the organism, and enables us to see how simple a certain kind of correlation is which had been apt to seem a subtle and complex thing.(Thompsen 1036, 1037)"If on the other hand, diverse and dissimilar fishes [as an example] can be referred as a whole to identical functions of very different coordinate systems, this fact will of itself constitute a proof that variation has proceeded on definite and orderly lines [as with Punctualism], that a comprehensive 'law of growth' has pervaded the whole structure in its integrity, and that some more or less simple and recognizable system of forces has been in control"
Thompsen goes on to graph the femur bone of an ox within a system of two-dimensional coordinates. By transferring the"same drawing, point for point, to a system in which for the "x" of the original diagram we substitute (x1 = 2x/3), we obtain a drawing which is a very close approximation to the cannon [femur] bone of the sheep. Similarly, the long slender cannon bone of the giraffe is referable to the same identical type, subject to a reduction of breadth, or increase of length, corresponding to
(x1 = x/3)" (Thompsen 1039) Thompsen called these types of evolutionary correlations "transformations".
What about temporal evolution? "Another curious and important transformation is that by which a system of straight lines become transformed into a conformal system of logarithmic spirals: the straight line (Y = Y - AX = C)corresponds to the logarithmic spiral (@ - A [Log r] = C).This beautiful and simple transformation lets us at once convert, for instance, the straight conical shell of the Pteropod or the Orthoceras into the logarithmic spiral of the Nautiloid" (Thompsen 1047).
Precisely these same types of coordinate transforming equations can convert any part of a chimpanzee, including its skull, to that of the corresponding part of a human. In fact, the chimp's entire body can be converted to a human with a very simple equation. During a time when the genetic proximity of the human to the chimp was not known, Thompsen, using his coordinate system indicated that man's closest biological ally was the chimpanzee.
Are these similitudes and transformations the mere progression of random activity within the DNA and the self-cycling reinforcement once the creatures manifest and subsequently interact within their respective environments? Thompsen, and many others these days, don't think so.
"These various systems of coordinates, which we have now briefly considered, are sometimes called 'isothermal coordinates', from the fact that when employed in this particular branch of physics, they perfectly represent the phenomena of the conduction of heat, the contour lines of equal lines temperatures appearing under appropriate conditions, as the orthogonal lines of the coordinate system. And it follows that the 'Law of Growth' which our biological analysis by means of orthogonal coordinate systems presupposes, or at least foreshadows, is one according to which the organism grows or develops along stream lines, which may be defined by a suitable mathematical transformation"(Thompsen 1048).
In essence, what's being said here is that biological form and its evolutionary transformations are caused by forces that become apparent after corollary graphing is done. This book was written prior to the discovery of DNA, but the edition I read was published after, so there was an editorial addendum which hinted that particularly in the DNA one finds symmetries which hint at underlying attractors. The very word "double helix" should be a dead give-away. Also, it is important to note that Thompsen did not at all refute Darwin's and Mendel's theories; he merely expressed the nonlinear belief that randomness was not the driving force behind adaptation.
Thompsen, as most nonlinear scientists agree, believed that life was always in motion and unceasingly responding to rhythms--"the deep seated rhythms of growth, which create universal form and dynamics; the interpretation, in terms of force, of the operations of energy" (Gleick 202). "Every natural phenomenon, however simple, is really composite, and every visible action and effect is a summation of countless subordinate actions" (Thompsen 1028).
Sometime in the winter of 1961, Dr. Edward Lorenz of M.I.T.'s meteorological sciences department was inputting weather variants into his self-created, room sized computer. Basically, what the crude computer did was analyze input data consisting of temperature and atmospheric pressure gradients, meager satellite reporting, wind velocities, weather station sightings of cloud conditions; cross referencing this data with past chronologies; and ultimately outputting with a line graph representation of the projected weather for a given location.
Often, Lorenz simply inputted hypothetical variants in order to quantify probability. That's what he was doing on this particular day; he was redoing an old experiment. But instead of inputting 0.506127 for a particular gradient variable, he dropped the decimal points smaller than ten-thousandths. After all, he reckoned to himself, weather data generally is only accurate to the hundredth's decimal point. Don't forget that he was entering arbitrary numbers.
He went out to get some coffee while the computer clicked and clacked for fifteen minutes, crunching the numbers. Upon retrieving the long graph which the computer spewed out, Lorenz was shocked! At first, he thought it had broken down. The graph projected a completely different weather pattern than had earlier been forecast. So he tried it again, this time entering values which included the hundred thousandth's decimal points. Now the projections looked familiar. For Lorenz, who was something of a mathematical enigma to his colleagues at M.I.T, this was the beginning of an awe-inspiring saga into the science of chaos and nonlinearities.
"The type of error in Lorenz's computer was comparable to a small puff of wind, which ought to cancel out--yet, it created a catastrophic new pattern" (Gleick 17). As I've already explained, Lorenz called this the "butterfly effect". The ability of predicting a pattern out of random probabilities--as in weather forecasting, or natural selection--becomes increasingly problematic. "Any prediction deteriorates rapidly. Errors and uncertainties multiply, cascading upward through a chain of turbulent features"(Gleick 20).
Most scientific theories are based on patterns of observations which can be repeated with a relatively great deal of accuracy. But as I've already explained, accuracy is very subjective; it is, in fact, absolutely impossible. Within the framework of natural compartments for which science has found some degree of pattern, such theories have been instrumental in the human capacity for affecting change in the environment. However, there is so much which science cannot look at in this linear fashion; and what it does look at is inaccurate. To me, that's why nonlinearity and the science of chaos seems like the next step in scientific abstraction.
Lorenz recognized this. He was the first to see it since D'Arcy Thompsen. After his initial shock wore off, Lorenz began experimenting with other systems. He built numerous models based on random, changing variables "which signaled pure disorder, since no point or pattern of points ever recurred. Yet, they also signaled a new kind of order. Few laymen realized how tightly compartmentalized the scientific community had become, a battleship with bulkheads sealed against leaks. Biologists had enough to read without keeping up with the mathematics literature. --For that matter, molecular biologists had enough to read without keeping up with population biology etc. A decade later [after 1972], biologists and other scientists began seeking something like Lorenz's systems. But Lorenz was a meteorologist, and no one thought to look for chaos on page 130 of Volume 20 of the 'Journal of the Atmospheric Sciences'" (Gleick 31). Lorenz's work, though noble, went into obscurity for nearly two decades.
There is an old paradigm in science: simple systems behave in simple ways; complex behavior implies complex causes; and different systems behave differently. With the advent of chaos as a science, "all that has changed. Scientists have created an alternative set of ideas: simple systems give rise to complex behavior; complex systems give rise to simple behavior; and most importantly, the laws of complexity hold universally, caring not at all for the details of a system's constituent atoms [not literal atoms, the author means: parts]" (Gleick 304).
One of the interesting paradoxes that one seems to learn when studying science in this way is that those compartments for which traditional science has already established patterns are in fact tiny features of a grande design. a former professor of mathematical sciences at N.Y.U., Frank Hopensteadt ten years ago entered a nonlinear equation through a computer hundreds of millions of times. "He took photographs from the computer's display screen at each thousandth's different tuning. The bifurcations appeared, then chaos--and then, within the chaos, the little spikes of order, ephemeral in their instability. Fleeting bits of periodic behavior. [The film that resulted from this photo montage was startling,] like flying through an alien landscape" (Gleick 77).
What can be gleaned from such observations is that the periodicity and order which conventional science deals with and searches for are like the fleeting bits of periodic behavior which nonlinear science uncovers when looking at a huge system.
Biological sciences, and its related sister--physical anthropology,-- is unique in that its search is upon an island of form which appears to struggle against the entropic--the chaos producing-- processes of the mechanical universe. Though this is only an illusion in time, and thus gravity, it is unique. Benoit Mandelbrot of I.B.M.'s Research and Development put it this way: "Pattern born amid formlessness; that is biology's basic beauty and its basic mystery. Life sucks order from a sea of disorder. A living organism has the astonishing gift of concentrating a 'stream of order' on itself and thus escaping the decay into atomic chaos" (Gleick 300).
Mandelbrot was educated in the tradition of plotting variation on a bell curve, with the majority trying to be in the middle. Statistics was his love. But like Lorenz, he eventually saw things differently. Rather than separating tiny changes from grand ones and plotting them, his concept bound them together. "He was looking for patterns, not at one scale or another, but across every scale. It was far from obvious how to draw the picture he had in his mind, but he knew there would have to be a kind of symmetry, not a symmetry of right or left and top or bottom--but rather a symmetry of large and small scales" (Gleick 86).
Mandelbrot unraveled the real mathematics that chaos science currently uses. Inherent in these scalings are the subtle influencing factors at each scale. In evolution, the subtleties of molecular and atomic forces are just as real as the not so subtle forces of adaptation and group theory. To a nonlinear scientist you can't separate one from the other.
This concept, which has come to be known as the Mandelbrot set, states that for evolution, "perhaps the determinants existed long ago and nature is organizing itself by means of simple physical laws, repeated with infinite patience and everywhere the same" (Gleick 240). Again we see the notion of sensitive dependence on initial conditions. Gleick states it more precisely: "Randomness is essential to obtaining images of a certain invariant measure that live upon the fractal object. But the object itself does not depend on the randomness. In natural selection, the random nature of events (even in genetics) reflects this notion. Here we have time and we're caught in its vortex" (Gleick 239).
Eventually, Mandelbrot began finding unique situations in nature that defied conventional explanation. A snowflake is a prime example because it is a complicated shape which cannot be predicted even if the physical properties regulating its formulation are understood. The snowflake begins, like rain, as a droplet of water upon an ionic particle. As it falls through a moisture laden cloud in freezing weather, molecules of water cling to the external fissures of the original particle. The random nature of where these accumulating bits of ice will adhere depends on the sensitive initial conditions.
"Snowflakes are non-equilibrium phenomena. They are the products of imbalance in the flow of energy from one piece of nature to another. The flow turns a boundary into a tip, a tip into an array of branches, the array into a complex structure never before seen. As scientists have discovered such instability obeying universal laws of chaos, they have succeeded in applying the same methods to a host of physical and chemical problems. . . algae, cell walls, organisms--budding and dividing into complexity"(Gleick 314).
Mandelbrot applied some of his logic to actually solving problems for I.B.M., particularly in the area of computer modem signal interference. He used the nonlinearity of a cantor set. What you do is start with an "interval of numbers from '0' to '1', represented by a line segment. Then you remove the middle third. That leaves two segments, from which you remove the middle third of each again (from 1/9thto 2/9th and from 7/9th to 8/9th). That leaves four segments, and you remove the middle third of each--and so on to infinity. What remains? A strange dust of points arranged in clusters, infinitely many yet infinitely sparse. Mandelbrot was thinking of transmission errors as a cantor set arranged in time" (Gleick 92). In essence, what Mandelbrot did was to create a highly abstract way of dealing with error.
"Since Euclidean measurements--length, depth and thickness--failed to capture the essence of irregular shapes, Mandelbrot turned to a different idea, the idea of dimension. [He would ask,] What's the dimension of a ball of twine? Mandelbrot says that it depends on one's point of view. From a great distance away, it's a point without a dimension. From closer still, the twine comes into view, and the object becomes one dimensional in a three dimensional matrix. In the microcosmic realm, twine turns to three dimensional columns, and these columns resolve into one dimensional fibre. The solid material dissolves into 0 dimensional points"(Gleick 96, 97).
That ball of twine is our universe.
Allied to chaos science is a field of mathematics which I've occasionally referred to: symmetry. It is also non-Euclidean, hence nonlinear, in nature but instead of looking at apparent anomalies to derive scale it looks at complex relationships. Some mathematicians consider symmetry geometry and fractal geometry as one and the same. Symmetry geometry can effectively describe anything from how black holes bend time and space to how a population of monkeys bifurcate into two species.
Dr. Joe Rosen of Tel Aviv University has catalogued several distinct types of symmetry shapes. In his view, symmetries are the result of group theory type phenomena. Group theory dynamics, without getting into some complex mathematical jargon, simply assesses a system with the presumption that there is a great deal of interactive components that are not easily recognizable. But if four basic conditions can be met, then a symmetry pattern can be ascertained. These four conditions involve the closure, associative, identity and inverse properties of mathematics.
In looking at nature he has discovered that the simplest type of symmetry is linear displacement symmetry, which "occurs in every day situations. Some examples are long traffic jams, a long train, R R tracks, telephone poles, along highway [etc]" (Rosen 30).
And then there is plane reflection, or plane inversion symmetry, which generally results in transformations such as can be evidenced by DNA replication. In such symmetries, there is always a reflection center. This type of symmetry is fundamental throughout nature and is more commonly known as bilateral symmetry. "The transformation of line reflection (or line inversion) acts on a two dimensional system by reflecting each point of the system through a line perpendicular to the plane of the system. All the geometric transformations that can be considered for 2-dimensional systems are readily applied to 3-dimensional systems. These transformations are displacement, plane and line reflection, glide and rotation. As expected, the addition of a third dimension allows applications of transformations that are inapplicable to lower dimensional systems. Of these are point reflection and the screw. We also consider the dilation transformation, applicable to systems of any number of dimensions" (Rosen 46, 47 ).
Other types of fundamental symmetries are: spherical(ball bearings etc., the center point being the inversion point), the axial (round jars, bullets, fingers) and not least, there are the approximate temporal DISPLACEMENT SYMMETRIES.
"Nature abounds with examples of approximate temporal displacement symmetry. The sleep cycles of higher animals and the opening-closing cycle of flowers are linked to the universal cycle of day-night produced by the earth's rotation about its axis. The reproductive and migratory cycles of many higher animals as well as the leaf-shedding and renewal cycles of deciduous trees are linked to the annual cycle of the revolution of the earth about the sun. The human female menstrual cycle . . . was at some stage in evolutionary history probably linked to the lunar cycle of brighter and darker nights, just as the behavior of certain sea life is at the present time. The assumption that time is essentially homogeneous is a cornerstone of Special Relativity [Einstein] and is intimated by the Law of Conservation of Energy"(Rosen 83).
In this book, Rosen goes on to describe several paradoxes in the universe. However, I've already indicated that our universe could very easily have evolved in a mechanically different way; and in fact, there is scientific speculation of the parallel existence of one or several universes that are antithetical of this one. In subatomics, Rosen points out that reflection symmetry is found in antimatter.
There are still other types of symmetries and transformations. "A transformation that completely cancels the effect of another transformation is called the inverse of the latter [anti-matter]. The inverse of the transformation of rotation by 90 degrees about a given axis, for example, is rotation by an additional 270 degrees about the same axis, producing a total rotation of 360 degrees which is no rotation at all" (Rosen 10). Often, and usually in nature, there will be many transformations interacting. "The set of all symmetry transformations of a system comprises the symmetry group of the system" (Rosen 12).
Then there is dilation symmetry, where distances, orbits or radii from a given point radiate outward in geometric multiplications. "A curious example of approximate dilation symmetry is Bodes Law of planetary distances. The distance of each planet from the sun is the square root of 3" (Rosen, J. 97) In effect, each planet outward from the sun is 1.73 (square root of "3") times more distant than it's inner neighbor. Dilation symmetry is also seen in the ratios of human appendage lengths. "Two of the fundamental symmetries of nature are dilation (R > ar) and translation (R > r + b)"(Kadanoff 6).
This is a sampling of Joe Rosen's work on symmetries. He describes others but I think I've already complicated the issue enough with these. However there is another type of symmetry which must be described and it deals with the larger scale of an interacting population and its inherent flux towards speciation. I suppose it could also deal with cells and other sorts of non-sentient populations such as viruses.
"If you were following an animal population governed by this simplest of nonlinear equations, you would think the changes from year to year were absolutely random, as though blown about by environmental noise. Yet in the middle of this complexity stable cycles return. Even though the parameter is rising, meaning that nonlinearity is driving the system harder and harder, a window will suddenly appear with a regular period: an odd period, like 3 or 7. The pattern of changing populations repeats itself on a 3 year or 7 year cycle. Then the period-doubling bifurcations begin all over at a faster rate, rapidly passing through cycles of 3, 6, 12, . . . or 7, 14, 28 . . . and then breaking off once again to renewed chaos"(Gleick 73).
Crucial to symmetry mathematics is an understanding of what happens when symmetry is lost, or as these scientist would call it, catastrophism? In fact, they have developed a set of theories to deal with it. The concepts within the theory of catastrophism deal with the extreme boundaries of analysis, as mathematical science is now understood.
Saunders explains: "A great many of the most interesting phenomena in nature involve discontinuities. These may be in time, like the breaking of a wave, the division of a cell or the collapse of a bridge, or they may be spatial, like the boundary of an object or the frontier between two kinds of tissue. Yet the vast majority of the techniques available to the applied mathematician have been designed for the quantitative study of continuous behavior. These methods, based primarily on calculus, when we turn to the biological and social sciences we generally find that we're unable to conduct the relatively complete models which would permit us to apply the same methods" (Saunders 1). Here is where there is a slight contrast between chaos theorists and symmetry theorists. "The catastrophe theory deals with singularities. When applied to scientific problems, therefore, it deals with the properties of discontinuities directly, without reference to any specific underlying mechanism" (Saunders 1). This see ms diametrically opposed to the search for an underlying pattern which chaos scientist attempt to plot. But is it?
Saunders states that the primary utility of the catastrophe mechanism is to describe how smooth potentials can abruptly give rise to discontinuities "through the disappearance of stable states" (Saunders 12). In polynomial and other differential equations, we often find curves approaching hyperbolic paraboloids or its inverted, elipticparaboloids; either opening upwards for a minimum or downwards for a maximum. "The generalization for a function of two variables is that near a non-degenerative critical point it can be closely approximated. This is the point at which most calculus texts leave the problem, on the grounds that almost all cases have been covered. However, both x, y directional plotting can lead to catastrophism" (Saunders 22); which is a collapse of the symmetry.
The seven classic types, or geometric shapes, of catastrophes are: the fold, the cusp, the swallowtail, the elliptic umbilic, the hyperbolic umbilic, the butterfly, and the parabolic umbilic. Thus, when a chaos researcher plots out a graph to analyze a system and comes up with one of the above mentioned shapes, it becomes a flag that some sort of dramatic, unpredictable behavior is about to occur.
Notwithstanding these complicated phenomena, Dr. Gruber of the Illinois Academy of Science states that "analysis has shown that the phenomena of spontaneous breakdown of symmetry is really a dynamical rearrangement of symmetry, meaning that the observable manifestations of symmetry differs from the basic symmetry" (Gruber 413). Hence, catastrophism is merely a re-scaling of an underlying pattern. Rosen concludes that "all the laws of nature responsible for bulk matter and its properties, that influence our everyday lives, are reflection symmetric. And in this sense we claim that nature possesses reflection symmetry" (Rosen, 93).
As you can see, chaos science and symmetry mathematics tends to look at the same types of systems with the same type of holistic eye.
Fractal geometry is thought to be a tool, not yet fully developed, with which to look at seemingly random or chaotic movements by analyzing them at their fractal roots and thus making deductions at the compressed level. "Fractals, figures with fractional dimensions, have become an important topic in mathematics and science in the last few years, but they have found so many applications in all branches of science that scientists are becoming accustomed to thinking in fractional dimensions" (Physics 217).
Thomsen wrote that "fractals turned out to be a mathematical curiosity that science was just waiting for. We now have fractal theories or suggested fractal explanations for all sorts of things, from the tone sequences in Gregorian chants to the power spectrum of the Sun, to the random walk of the proverbial drunk under the lamppost, to the folding of proteins" (Thomsen 184). And talking about the folding of DNA, Gleick writes that "DNA cannot specify the vast number of bronchi, bronchioles and alveoli or the particular spatial structures, but it can specify a repeating process of bifurcation and development. Such processes suit nature's purposes. Theoretical biologists have found that fractal scaling is not just common but universal in morphogenesis"(Gleick 110)
I remember taking a problem-solving class at U Mass taught by Dr. Altshuller in undergraduate school. Professor Altschuller has had the distinction of having been hired by GE, NASA, and other respectable institutions to solve engineering riddles. He helped devise a method called the "OFPISA" problem solving technique; the assumption being that the problem is so close to the linear researchers that "they can't see the trees for the forest".
OFPISA is an acronym. The first word is the exclamation "Oh what a mess?" Then you write down all the facts pertaining to the mess--not what you think is the problem. From these arbitrary facts, which often can be farfetched, you can configure the real problem. To resolve the problem, you need to get even more far-fetched. You do this by assembling as many people as possible connected to the problem and ask them to creatively generate solution Ideas to the newly uncovered problem. In an open forum, as many ideas are generated as possible, without criticism of their outlandishness. Dr. Altschuller indicated that often a solution is locked behind virtually inaccessible regions of the unconsciousness, and would remain there if the creative process were stunted by peer criticism of off-the-wall ideas.
After the creative impulses are exhausted and someone has chronicled these ideas, everyone gets into the process of weeding out the bad ideas through a selection process. This step is often the longest, but out of its debate usually comes the answer.
It is this kind of attitude which a researcher must possess to effectively deal with chaos science and symmetry mathematics in order to play a viable role in humanity's evolving role in self understanding and environmental access. Gleick states that "if the image is complicated the rules will be complicated. On the other hand, if the object has a hidden fractal order to it--and it's a central observation that much of nature does have this hidden order--then it will be possible with a few rules to decode it. Randomness has no relevance to biology. It is suspected that when the brain is understood they'll find that it's organized through a coding scheme for building the brain which is of extraordinary precision. The idea of randomness is just reflex. Chance is a tool. The results are deterministic and predictable"(Gleick 238).
Dr. Gilpin, of U.C.S.D.'s Molecular Genetics Institute, is renowned for his work on predator-prey group dynamics. His research postulates that natural selection is only one of many ingredients pushing evolution. He calls the greatest of these "other" forces group selection. "Group Selection amounts to a new paradigm of evolution that threatens the 'synthetic theory' of evolution that many accept. It adds a force that opposes, and may prevail against, the conventional forces of individual (or Mendelian) selection" (Gilpin VII).
In essence, Gilpin asserts that Group Selection regulates the intensity of resource exploitation and thereby population density. Hence, adaptations favoring an individual are selected out such that the group will better survive and pass on the genetic material.
In a field closely allied to sociobiology, Gilpin takes us into prehistory to describe a striking example of group selection. "Women of hunting and gathering groups could not have too many children too closely spaced, because they carried them around and couldn't handle more than one at a time. More often the particular population regulation act was the indirect consequence of some group-held cultural belief. Obedience to the cultural belief could not be adaptive to the individual, for by reducing the number of offspring, the individual's fitness was lowered" (Gilpin 4).Further, "the fitness of a group depends on the ability of a group's cultural traditions to regulate population growth and maintain densities that avoid overpopulation" (Gilpin 5).Joseph Campbell, the recently deceased mythologist, once said something to the effect that the reason why inner cities are in such turmoil is because of a breakdown in a viable mythology: the collective memory dealing with inter- and intra- group variance. G ilpin elaborates on this phenomenon with great precision:
"The genetic fixation of 'socially advantageous' but individually disadvantageous traits would require some form of intergroup selection. In a large population such a gene would be selected against and would go to extinction. But if this gene were present in an ensemble of small populations, the effects of genetic drift would be magnified and could possibly overcome selection for a series of generations and drive the 'socially advantageous' gene to fixation. If migration occurred between groups, and if the groups with a high frequency of the 'socially advantageous' gene were more prosperous and sent out more migrants, then it could be possible that the gene would go to fixation throughout the whole ensemble of populations"(Gilpin 6).
Gilpin goes to great length at describing specific examples of group selection. The caste system of social insects is a classic example, as is the overall reduction in canine-tooth length in the evolutionary history of most predators. Prey-effective alleles are advantageous to the individuals but disadvantageous to the predator group. Hence, group selection will be a force that will diametrically try to undermine individual selection.
The complexity of this and other interactive selection forces dictates a nonlinear approach in order to quantify the information. It is the only sort of mathematics that Dr. Gilpin and his associates at U.C.S.D. can use.
In an interview with Chris Lanken of the Los Alamos National Laboratory, he explained to me that the scaling geometry of nonlinear mathematics has helped uncover some of the forces underlying morphogenesis during different embryonic stages. But more interesting than that, he described a very simple graph which precisely quantifies the whole evolutionary process involved in speciation.
In the graph which I've drawn, individual and group selective forces maintain a circular pattern for each successive generation. But each of these cycles has a mechanical tendency to escape the orbit on an outbound vector. For the most part, these migrating vectors [like most mutations] die out; yet, a few will band together on a vector and eventually will be so far distanced from the"gravity" of the original orbit that they begin the pattern anew, as a new species. With such a symmetry diagram, a researcher can quickly see that there is an attractor at the center of each circle, holding the individuals and groups to their respective orbits of reproduction; and equally, that there are specific points outside the circle (or inertial forces within) which represent patterns that will ultimately cause bifurcation. It's a new way to describe evolution which doesn't minimize anything Darwin has done, but which provides highly effective information.
Gleick points out that "there is a study of the 3-dimensional entanglements of protein molecules; instead of static structures, biologists should understand such molecules as dynamical systems, capable of phase transition. When you reach an equilibrium in biology, you're dead"(Gleick 298).
Both the researchers at Los Alamos and Dr. Saunders of London University agree that it is the chemical and physical gradients within an embryo that produces the various processes which differentiates cell development. This was precisely Thompsen's assertion nearly one hundred years ago, when he did his primitive nonlinear calculations, and it has been the reason why I've gone to such great lengths at describing the complexities of physical phenomena in the universe.
"Let us suppose that the cells are able to measure time, in the sense that there are time-dependent changes going on within them. And also that the fate of each cell is decided by a particular substance which we may call a morphogen and whose concentration is determined by the equilibrium states of a set of ordinary differential equations. The equilibrium states will themselves depend on a number of parameters, but since there can only be four independent gradients (the three in space and one in time) there are at most four independent control variables"(Saunders 117).
Chaos science can now look at the fibrillations on the electrocardiograms of various types of heart attacks to ascertain life saving measures. "Several chaos minded cardiologists found that the frequency spectrums of heartbeat timing, like earthquakes and economic phenomena, followed fractal laws [underlying pattern caused by a sensitive dependence on initial conditions], and they argued that one key to understanding heartbeat timing was the fractal organization of this network, a labyrinth of branching pathways organized to be self-similar on smaller and smaller scales" (Gleick 109). Due to this work, special heart pacers have been developed which, through a biofeedback mechanism, measures and stabilizes arrhythmias in the heart.
Gleick describes dozens of other diseases which are capable of being viewed in a fractal way. "Systems that normally oscillate, stop oscillating, or begin to oscillate in a new and unexpected fashion, and systems that normally do not oscillate, begin oscillating. These syndromes include: breathing disorders, dynamical blood disorders such as leukemia, and even schizophrenia" (Gleick 292).
Evidence indicates that the human immune system is highly responsive to our personal outlook. Our moods and emotions are based upon the fractal scale of the brain, at least according to the evidence chaos scientists postulate. Thus, it might not be unreasonable to expect dramatic cures to result from a better understanding of the brain through fractal geometry. In fact, there is an idea I've read several researchers describe which looks at many of the so-called "new-age" remedies (e.g. homeopathy) as well as the Chinese and Japanese systems of healing through meridian points (acupuncture) to be nothing more than dealing with disorders at their fractal roots.
In the same vein, astrology--which deals with underlying patterns between things at the human scale and things at the cosmic scale--can perhaps be viewed with a new bias. In Tibet, doctors often use intricate geometric shapes, called mandalas, to realign systems which have lost their harmony with nature. Native American tribes, like the Navajo, do the same using songs and Dry Paintings. Though these various cultures use metaphors and jargon not recognized by conventional science, it appears that the premise is the same.
The paradox in using fractal geometry to describe phenomena is profound. Gleick states that the researcher has two choices in the matter. "You can wait until the biochemist [or other scientists] figure out the mechanisms of [biological] clocks and then try to derive some behavior from the known mechanisms, or you can start studying how clocks work in terms of complex systems and nonlinear and topological dynamics" (Gleick 285).
I'm describing these non-anthropological examples to point out the universal applications to fractal scaling. In getting back to evolution, I'd like to end this section by quoting Thompsen again. He said that "organic evolution has its physical analogue in the universal law that the world tends, in all its parts and particles, to pass from certain less probable to certain more probable configurations or states. This is the 2nd law of thermodynamics. It has been called 'the law of evolution of the world', and we now call it, after Clausius, The Principles of Entropy, which is a literal translation of 'evolution' into Greek" (Thompsen 11).
So, how does one really use chaos science?
In getting back to the Amazon fairy tale, if that subject who was overwhelmed by the chaos of the encompassing jungle were to have had an eye for symmetries and scale--as perhaps a child does when exploring the world--then he or she would see that the surrounding flora and fauna are but scaled expressions of the identical mechanisms which make up anything human.
If he or she were a researcher utilizing the techniques of chaos science then that person would study data with a focus on the underlying pattern. And perhaps more than any place in the world, the underlying patterns of life are expressed in a jungle. U.C.S.D.'s Dr. Sukahara would suggest that from the available data, one would ask: "are there scaling properties that illustrate patterns?" (Sukahara). Or, in other words, find out how the data properties come out at scales.
There are two fundamental concepts which chaos science shatters. Chris Lanken of Los Alamos explains that "the neat thing about the chaos model is that you can have a deterministic universe with infinite capacity for variation"(Lanken). So perhaps the next natural step in human evolution would be for the species to grasp these fundamental concepts, and thus consciously set a course for adaptation which will result in survival.
Going back, then, to that person in the jungle. It is by recognizing the beautiful patterns--both observable as well as mathematical--of the interrelated systems in the Amazon's vegetation and animals, subatomics, the cosmos, and the observer--the human factor--that perhaps that tractor will be stilled. . . and the trees left standing. . . and the atmosphere cleansed. . . and global warming halted. . .and cures for countless diseases discovered. . . .
. . . .And humanity's time-distanced and physical link with that jungle--and its link with the cosmos--are features which anthropology could perhaps better recognize with a nonlinear perspective. In studying this science's profound and provocative axioms I cannot help but wonder if there isn't some sort of simple wisdom to be found in many mythologies; fairy tales describing a universe which is both archetypally deterministic and filed with choices at the same time.
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