======================================================================== Cybernetics in the 3rd Millennium (C3M) --- Volume 5 Number 2, Mar. 2006 Alan B. Scrivener --- www.well.com/~abs --- mailto:abs@well.com ========================================================================

Even Better Than the Real Thing

Lock your wigs, inflate your shoes, and prepare yourself for a period of simulated exhilaration. -- Firesign Theatre, 1971 "I Think We're All Bozos On This Bus" ( www.amazon.com/exec/obidos/ASIN/B00005T7IT/hip-20 ) Last time's 'zine was quite long -- 16,275 words (a record) for the last installment and 31,620 for the whole trilogy -- and somewhat tangential to classical cybernetics (though it did deal with important issues like the digital revolution, disruptive technology, media ownership and the democratization of moviemaking). But this time I want to be more short and to the point, and also to return to fundamentals, and discuss something at the core of the methodologies of cybernetics and systems theory: simulation. A few years ago I got an email that asked this question: Are there any books (or web pages, etc.) in reference to applied cybernetics? That is, quantified cybernetic relationships in a real world application used to solve a problem or accomplish a goal. Examples outside of the science/engineering world, and cybernetics applied to business would be of particular interest. to which I replied: These days what used to be called cybernetics is usually just folded into "applied math." In the business world, they used to call it "operations research" and now they call it "management science." Here is the strategy I recommend: use visually programmed simulation packages. Go to google.com and search for: easy5 vissim ITI-SIM AMESim and you'll get links like: www.idsia.ch/~andrea/simtools.html which point to vendor web sites. Pick a package you can afford and use it to build models and play with them. Try to model as many different types of systems as you can. Then get ahold of real data, say from an e-commerce sales database, and see if you can build models that produce similar data. ... Happy studies! Since sending this email, I have come to believe it may be the best advice I have ever given to any of my readers. (I wonder if he followed it?) So I have decided to expand upon the idea for this issue.


Yes, but as a noted scientist it's a bit surprising that the girl blinded ME with science... -- Thomas Dolby, 1982 "She Blinded Me With Science" on "The Golden Age of Wireless" (music CD) ( www.amazon.com/exec/obidos/ASIN/B000007O19/hip-20 ) First, to put things in context, we need a quick review of the history of the Scientific Method:
  • The original methodology is usually attributed to Galileo, who established this pattern: create a mathematical model of a natural system (his used simple algebra), make quantitative measurements, and compare them with the theory. If the experiments repeatably give different answers than the theory, modify the theory and repeat. In this way Galileo was able to establish that a sufficiently dense body (so that air friction can be ignored), dropped from slightly above the Earth's surface, will have traveled a distance of 16*t^2 feet after t seconds have passed. (Here I write t^2 to indicate t squared, i.e., t*t or t times t, following the syntax of many computer languages, from BASIC to Java.) For example, after one second the distance traveled will be 16*1*1 = 16 feet; after two seconds the distance traveled will be 16*2*2 = 64 feet; after three seconds it will be 16*3*3 = 144 feet, and so on. (Note that you don't need a computer or a 3D graphics system to get these answers.) Graph this data and you start to get a parabola, or as Thomas Pynchon called it, "Gravity's Rainbow."
  • Isaac Newton made a huge contribution when came up with the main set of mathematical tools used to model deterministic systems with a number of interconnected, continuous (i.e., "smooth when plotted") variables. Newton's toolbox is built on algebra and geometry, and includes calculus, so-called "linear algebra" and systems of Ordinary Differential Equations (ODEs).
  • Using his tools Newton was able to prove that -- if you make some simple assumptions about forces and masses and gravity -- a single planet orbiting the sun must follow the shape of an ellipse, a conclusion that matched the detailed observations of Tycho Brahe, which Kepler had analyzed and formulated into Kepler's Laws. (Note that you don't need a computer or a 3D graphics system to get these answers either.)
This much of the story is pretty widely known, representing huge victories for quantitative science. But the next developments have been less trumpeted, for they represent defeats.
  • It was quickly discovered that Newton's equations did not yield ready answers when THREE bodies interacting gravitationally are analyzed. This story is told in the wonderful paper "A Mathematics for Physiology" by Alan Garfinkel (1983) which appeared in "The American Journal for Physiology." Dr. Garfinkel explains: The motion of two mass points is then described by a curve ... the solution curve to this differential equation for a given set of initial conditions. Newton's achievement was to show that that this model yielded Kepler's three laws of planetary motion: that the planets move in ellipses [&etc]. Before this derivation, Kepler's Laws had been entirely empirical, so the explanation that Newton contributed was very profound. It became the basic paradigm for a rigorous physical theory: a reduction to a differential equation, with the hypothesized forces appearing as the right-hand side of the equation, and the resulting motion of the system given by the integral curves. But the beauty of Newton's solution to the two-body problem did not seem to be extendable. In the typical cases, even in systems slightly more complex than the two-body problem, one could write equations based on first principles, but then it was completely impossible to say what motions would ensue, because the equations could not be solved. A classic case was the three-body problem. This is a more realistic model of the solar system, because it can take into account the non-negligible gravitational effects of Jupiter. A great deal of attention was focused on this problem, because it expressed the stability of the solar system, a question that had profound metaphysical, even religious, consequences. Mathematicians attempted to pose and answer this, some spurred on by a prize offered by King Oscar of Sweden. [link added -- ABS] ( www.sciencenews.org/pages/sn_arc99/11_13_99/mathland.htm ) Several false proofs were given (and exposed), but no real progress was made for 150 years. The situation took a revolutionary turn with the work of Poincare and Bruns circa 1890, which showed that the equations of the three-body problem have no analytic solution. ... They showed that the usual methods of solving differential equations could not solve this problem. In addition, because the write-a-differential-equation-and-solve-it method had become the normative ideal of an explanation, the proof that no such solution existed (not just that people had not found none) had revolutionary consequences: it represented the defeat of Newton's program. The revolution was resolved by Poincare. It was his genius to reevaluate the question and ask what we really wanted from a mathematical model of nature. Consider the problem of the stability of the solar system: what are we really asking for when we are asking if it is stable? To Poincare, it meant asking whether the orbit of the earth, for example, was closed, spiraled into the sun, or escaped into space. He was that the fundamental difference between the closed orbit and the other two was qualitative: the closed orbit is essentially a circle, and the other two are essentially lines. To make this distinction precise required the invention of a new subject, topology. In the topological view, only breaks and discontinuities are meaningful; two figures, such as the ellipse and the circle, which can be deformed into each other without discontinuities, are equivalent. However, the circle and the line are not equivalent, because we must the break the circle somewhere to map it smoothly and one-to-one onto the line. Poincare then reposed the fundamental question of dynamics. No longer was it a request for analytic solutions. Now it was asking for the qualitative FORMS of motion that might be expected from a given kind of system. This was the idea that was to revolutionize dynamics, an idea that requires a radically different view of dynamics, in which we imagine it pictorially instead of symbolically.
This definitely made ME go "wow" when I read it. I'd known since boyhood about the field of mathematics known as topology; I'd read about it in "The Time-Life Book of Mathematics" (1963) by the editors of Time-Life -- ( go to ebay.com and search for "Time Life Mathematics" ) and looked at the terrific pictures -- ( www.well.com/~abs/Cyb/4.669211660910299067185320382047/Sim/time-life_topology.jpg ) but I didn't know it was INVENTED to solve problems in systems theory. Elsewhere in the book was a turgid discussion of the unsolved "three body problem," which I had no clue was connected to topology. ( www.well.com/~abs/Cyb/4.669211660910299067185320382047/Sim/time-life_3body.jpg ) After winning the King Oscar prize Poincare ruminated over what he had learned by all this, and eventually published his conclusions in "Science and Method" (1914). ( www.amazon.com/exec/obidos/ASIN/0486432696/hip-20 ) A more thorough academic study of this event and its consequences appeared just a decade ago, in "Poincare and the Three-Body Problem (History of Mathematics, V. 11)" ( book, 1996 ) by June Barrow-Green. ( www.amazon.com/exec/obidos/ASIN/0821803670/hip-20 )


A differential equation gives the rule by which the state of the system determines the changes of state of the system, which then determine its future evolution. -- Alan Garfinkel, 1983 "A Mathematics for Physiology" in "American Journal of Physiology" Pardon me as I take a short respite from the main exposition to encourage you to read Dr. Garfinkel's paper. After years of wishing it was available on the internet, I found it is now on-line at the web site of the original journal it appeared in ( www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=6624944&dopt=Abstract ) as a PDF document for an $8.00 charge (one day access), and also temporarily available for free (I'm only telling you what Google told me) in a separately scanned PDF version ( www.as.wm.edu/Faculty/DelNegro/cbm/GarfinkelAJP1983.pdf ) at the web site for a class called "Cellular Biophysics and Modeling" ( www.as.wm.edu/Faculty/DelNegro/cbm/cbm.html ) taught by Christopher Del Negro ( www.as.wm.edu/Faculty/DelNegro.html ) at the College of William and Mary Dept. of Applied Sciences. ( www.as.wm.edu ) A check of Dr. Del Negro's CV shows that he obtained his PhD in Physiological Sciences in 1998 from UCLA, where Dr. Garfinkel teaches, ( www.physci.ucla.edu/physcifacultyindiv.php?FacultyKey=965 ) ( www.cardiology.med.ucla.edu/faculty/garfinkel.htm ) (it's a sure bet Del Negro was Garfinkel's student there) and is now building his expertise in the area of chaos and respiration, leading him to study sleep apnea -- which is coincidentally the field I have worked in for the last year, at ResMed Corp. ( resmed.com ) I first ran across Dr. Garfinkel's paper in the late 1980s, when I worked for Stardent Computer and he was a customer, using his new supercomputer to analyze cardiac chaos. I ended up handing out photocopies of "A Mathematics for Physiology" to other Stardent customers, acting as a "pollinating bee" for his ideas (or a "supernode" in network theory). Overjoyed as I am that it is now available electronically, I urge you now to take a respite from this 'zine and read the paper -- at least the first five pages. I'll wait.


Is dis a system? -- "Mr. Natural" ( www.toonopedia.com/natural.htm ) a character created by underground cartoonist R. Crumb ( www.toonopedia.com/crumb.htm ) Welcome back. Stymied by the three-body problem and other complications, systems theory stalled for a while. Poincare's topology solved some simple problems that were intractable under analysis, but did not offer a general program for proceeding with many other real world problems. In assembling his writings on a generalized theory systems, Ludwig Von Bertalanffy wrote in "General System Theory" (book, 1968) ( www.amazon.com/exec/obidos/ASIN/0807604534/hip-20 ) about the problems of using Newton's paradigm: Sets of simultaneous differential equations as a way to "model" or define a system are, if linear, tiresome to solve in the case of a few variables; if non-linear, they are unsolvable except in special cases. and then provides a rather alarming table of how things really are. ( www.well.com/~abs/equations.html ) In the same year, ecological cyberneticist Ramon Margalef wrote in "Perspectives in Ecological Theory" ( www.amazon.com/exec/obidos/ASIN/0226505065/hip-20 ) Almost inadvertently we have been shifting from the consideration of elementary relations in the ecosystem, like the response of a population to an environmental change or the interaction between a predator and its prey, to elementary cybernetic feedback loops and to the multiplicity and organization of a great number of such feedback loops. Almost everyone would agree it would be difficult, but theoretically feasible, to write down the interactions between two species, or possibly three, according to the equations suggested by Volterra and Lotka. This can be done in ordinary differential form as expressions, or in the more fashionable cybernetic form. But it seems a hopeless task to deal with the actual systems; first because they are so much too complex, and second because we need to know many parameters which are unknown. When major portions of this book were reprinted in the Summer 1975 issue of the "CoEvolution Quarterly," a diagram was included that sheds light on cybernetic relationships between predator and prey levels. ( www.well.com/~abs/Cyb/4.669211660910299067185320382047/Sim/margalef.jpg ) Both authors, Von Bertalanffy and Margalef, end up concluding that systems must be studied, even if the tools are inadequate, and both go on to advocate that HEURISTICS be used as stop-gap measures, such as simplifications, analogies, intuition, etc., nibbling on the corners of problems that can't be solved outright.


When I came to M.I.T. around 1920, the general mode of putting the questions concerning non-linear apparatus was to look for a direct extension of the notion of impedance which would cover linear as well as non-linear systems. The result was that the study of non-linear electrical engineering was getting into a state comparable with that of the last stages of the Ptolemaic system of astronomy, in which epicycle was piled on epicycle, correction upon correction, until a vast patchwork structure ultimately broke down under its own weight. -- Norbert Wiener, 1961 "Cybernetics, Second Edition" ( www.amazon.com/exec/obidos/ASIN/026273009X/hip-20 ) Perhaps the most heroic efforts against the unsolvability of general systems theory was the prodigy genius Norbert Wiener. Recently an informative biography of him was published, "Dark Hero Of The Information Age: In Search of Norbert Wiener The Father of Cybernetics" (book, 2004) by Flo Conway and Jim Siegelman. ( www.amazon.com/exec/obidos/ASIN/0738203688/hip-20 ) From it I learned that, sitting at MIT and traveling to collaborate with a long, impressive list, Wiener had his hand in the rebirth of Fourier and Laplace Transform techniques, the framing of the Uncertainty Principle in quantum physics, the selection of base 2 for use in information measure (definition of a bit) and processing (binary computers), and some other cool stuff I don't remember right now, but he spent quite a bit of his time trying to beat nonlinearity, hurling himself at the door over and over again trying to get it to open. (He noticed that random functions survived some nonlinear transforms the way sine and cosine survived adding 360 degrees to angles in linear equations, and thought he was on to something. As it turned out, not so much.) At the time it seemed Wiener's most promising approach was the so-called frequency methods, which used Fourier's Integral and related tools to spread data out on a "Procrustean Bed" of frequency analysis (sort of like today's Graphic Equalizers and Spectrum Analyzers in audio systems) ( www.audiofilesland.com/company/axis-software-company/axis-spectrum-analyzer.html ) and characterize it by frequency distributions and how they evolve over time. (Sort of like studying motors by listening to the sounds they make.) One of his more fruitful approaches was to use analog computers, and when they proved unbuildable for some nonlinear problems he proposed that they needed to start building DIGITAL computers and using them to SIMULATE nonlinear systems. What he wanted to do was to use the oldest method for calculating values from differential equations, Euler's Method, also known as Euler integration, which just starts with the initial conditions and approximates the evolution of the system in small steps, adding tiny amounts to each state variable based on the change equations. ( en.wikipedia.org/wiki/Euler%27s_method ) It is pretty effective, but takes a huge amount of computation. And more recent refinements that estimate and correct for error due to the "chunky" time steps, such as the Runga-Cutta method, are even more accurate and require even more computation.


"I been in the trade forever. Way back. Before the war, before there was any matrix, or anyway before people knew there was one." He was looking at Bobby now. "I got a pair of shoes older than you are, so what the #### should I expect you to know? There were cowboys [i.e., system crackers] ever since there were computers. They built the first computers to crack German ice [ICE = Intrusion Countermeasure Electronics]. Right? Codebreakers. So there was ice before computers, you wanna look at it that way." He lit his fifteenth cigarette of the evening, and smoke began to fill the white room. -- William Gibson, 1987 "Count Zero" (sci-fi novel) ( www.amazon.com/exec/obidos/ASIN/0441117732/hip-20 ) ( project.cyberpunk.ru/lib/count_zero/ ) There are three main "creation myths" associated with computers. Retired admirals still like to tell each other that these oversized adding machines were funded to crank out ballistics tables faster, so the sailors could fire their 16" guns in any wind conditions. There may still be some veterans of the US and British codebreaking efforts in World War Two, who chuckle over how everyone else thought the "Turing Machine" was a mathematical fiction, but Alan Turing built one at Bletchley Park to crack the German "Enigma" code. The third group was the practitioners of systems theory -- and applied physics -- who wanted to use the new high-speed computers to simulate systems of ordinary differential equations. I've described in a previous 'zine the "Fermi-Pasta-Ulam problem" which was an early simulation of a nonlinear system done at Los Alamos, in C3M Volume 2 Number 1, Jan. 2003, "Why I Think Wolfram Is Right". ( www.well.com/~abs/Cyb/4.669211660910299067185320382047/c3m_0201.txt ) They were the pioneers who helped establish a new "simulation paradigm" after World War II. By the time I was in the supercomputer biz in 1988 I found people using simulations to study diffusion of ground water (for Yucca Mountain studies), sloshing of fluids (in missile fuel tanks), cracking of metal at high heat and pressure (in rocket nozzles), and diffusion of heat (in a reactor). May years ago at some aerospace company I saw a picture that summarized the paradigm. I was able to get a photocopy, and I've scanned it for you: ( www.well.com/~abs/Cyb/4.669211660910299067185320382047/Sim/scientific.jpg ) And they would have probably gone on indefinitely building better bombs this way except that some funky things kept happening, like turbulence.


The simple linear feedbacks, the study of which was so important in awakening scientists to the role of cybernetic study, now are seen to be far less simple and far less linear than they appeared at first view. -- Norbert Wiener, 1961 "Cybernetics, Second Edition" ( www.amazon.com/exec/obidos/ASIN/026273009X/hip-20 ) One of things that seemed to amaze everybody when chaos showed up in the 1980s is why they never noticed it before. In each field the evidence had been piling up and was being ignored. Weather patterns, dripping faucets, epileptic seizures, lemming populations -- all had chaotic modes. When people tell me they want to learn about chaos I sometimes suggest they start at a casino. Rolling dice, shuffling cards, churning Keno balls in a fishbowl -- all are examples of chaos being used to generate apparent randomness. Of course chaos would never have been discovered without computers. We were perfectly willing to blame erratic events on "noise" and such until computers showed us that deterministic, non-periodic systems could exist in our mathematical models. The tale is still told best in "Chaos -- The Making of New Science" (book, 1988) by James Gleick, ( www.amazon.com/exec/obidos/ASIN/0140092501/hip-20 ) which showed up at about the same time as my entry into the supercomputer world. The university research centers, aerospace companies and national labs I had as customers were all abuzz about this stuff. Everyone knew the oft-told tale of Lorenz's discovery of the attractor that bears his name, and the so-called "Butterfly Effect" it illustrated. ( en.wikipedia.org/wiki/Lorenz_attractor ) One well-researched web site, "Hypertextbook" gives a detailed account. ( hypertextbook.com/chaos/21.shtml ) The article describes the process Lorenz went through trying to simplify a set of equations that showed instabilities in atmospheric convection. He finally reduced to only three differential equations. Although greatly simplified, we have here a model that is still impossible to solve analytically and tedious to solve numerically. One that would require an army of graduate students scribbling on hundreds of pages of paper working around the clock. It probably wouldn't have been solved in 1960 if it weren't for the fact that Lorenz had something better than an army of human computers -- the improbably named Royal McBee -- an early electronic computer whose vacuum tubes could perform sixty multiplications a second, round the clock, without taking a break or asking for time off. The Royal McBee made it possible to do numerical calculations that would have been cruel and unusual punishment to the human calculators. One could configure it in the morning and let it run for hours or days, printing out solutions for later analysis. This is how Lorenz discovered chaos. In the course of doing this I wanted to examine some of the solutions in more detail. I had a small computer in my office then so I typed in some of the intermediate conditions which the computer had printed out as new initial conditions to start another computation and then went out for awhile. When I came back I found that the solution was not the same as the one I had before. The computer was behaving differently. I suspected computer trouble at first. But I soon found that the reason was that the numbers I had typed in were not the same as the original ones. These were rounded off numbers. And the small difference between something retained to six decimal places and rounded off to three had amplified in the course of two months of simulated weather until the difference was as big as the signal itself. And to me this implied that if the real atmosphere behaved in this method then we simply couldn't make forecasts two months ahead. The small errors in observation would amplify until they became large. In order to conserve paper, the computer was instructed to round the solutions before printing them. Thus, a solution like 0.506127 was printed as 0.506. Even in 1960 computers gave answers with more significant digits than were required for most problems. An error of one part in four thousand should hardly have been significant. Tolerances aren't anywhere near this tight in construction or manufacturing or life in general. If you built your home using a meter stick that was 999.7 millimeters long would your house collapse? Would it be askew? Would you ever notice anything was wrong with it? When it comes to the weather, the answer to that last question was "yes." After enough time had elapsed, the tiny error introduced by dropping the digits after the thousandths place became an error as large as the range of possible solutions to the system. Lorenz called this the Butterfly Effect. This watershed event showed most of the enduring features of chaos: deterministic behavior without periodic behavior, sensitive dependence on initial conditions, and hidden beauty with fractal properties. The best way I have found to appreciate the Lorenz Attractor is to play with a simulation. Several are on-line. ( www.geom.uiuc.edu/java/Lorenz ) ( www.falstad.com/mathphysics.html ) The best way to learn about chaos in detail, in my opinion is to study the so-called "chaos comics" by Abraham & Shaw, "Dynamics: The Geometry of Behavior" (multi-volume book, 1982). ( www.amazon.com/exec/obidos/ASIN/0201567172/hip-20 ) Its hand-drawn color diagrams of a whole rogues gallery of strange attractors help develop an intuition for just how mind-bogglingly weird the pure mathematics of systems theory can be.


It's so simple, so very simple, that only a child can do it. -- Tom Lehrer, 1964 "New Math" (novelty song) on "That Was the Year That Was" ( www.amazon.com/exec/obidos/ASIN/B000002KO7/hip-20 ) ( www.sing365.com/music/lyric.nsf/SongUnid/EE27EF26A4F581BE48256A7D002575E1 ) Tell me honestly, how many of you stumbled over the word "pedagogic" above? Let's see a show of hands. You can look it up in Wikipedia: ( en.wikipedia.org/wiki/Pedagogic ) One reason kids are such great learners is they don't mind feeling stupid so much; they just go right on learning anyway. I firmly believe you could teach kids systems theory all the way up to ODEs without much arithmetic or algebra, and with no proofs at all, but WITH interactive simulation software that works from block diagrams and produces graphs and animations of the system behaviors. But meanwhile, I also have a very simple example that yields surprisingly subtle insights if you think about it enough. So reach into yourself and invite your inner child to ponder this: A Very Simple Game Have you ever seen the prank where you hand someone a card that says "how to keep an idiot occupied for hours (see over)" printed on both sides? Well this idiotic game is like that, only it's played on a checkerboard. Each square is numbered 1 to 64. On each square is a small card thats says something like "go to square 18" or some other number from 1 to 64. In each round of the game, there is a different set of cards on the checkerboard. You play by placing your marker (perhaps a miniature Empire State Building) on one of the squares (called the 'current state of the system'), and then following the instructions on the cards one after another. Imagine in one round every card says "go to square 1" and so clearly you have one square that you always end up on, and then you stay there. In systems theory is this is called an "attractor." Imagine if in another round the left half of the board pointed to square 1, and the right half pointed to the opposite corner, square 64. Now the state space is divided into two "basins" each with its own attractor. Or imagine if each square pointed to the one above or to the right or both, until all jumps ended up on the top or right side. The square in the lower left corner with so many jumps leading away from its its vicinity is called a "repellor." Or imagine that all squares in the interior point to an edge square, and all the edge squares are joined in a chain that goes around the perimeter clockwise (i.e., on the bottom row each square points to the one to the left, meanwhile on the left edge each square points to the one above it, and so on). Now we have an "orbit" which in this case is also an attractor. It is amazing the number of distinctions that can be drawn by studying this idiotic little game. This approach is largely the one in Ross Ashby's classic "An Introduction to Cybernetics" (book 1956) ( www.amazon.com/exec/obidos/ASIN/0416683002/hip-20 ) which is back in print and also free on-line. ( pcp.lanl.gov/ASHBBOOK.html ) And unlike me, he continues the analogy while generalizing to the continuum (infinitely many states) thereby deriving the whole of cybernetics.


Gentlemen! You can't fight in here. This is the War Room! -- Stanley Kubrick, Terry Southern and Peter George, 1964 screenplay for "Dr. Strangelove" (movie) ( www.amazon.com/exec/obidos/ASIN/B0002XNSY0/hip-20 ) The deep and thought-provoking museum exhibit and book of the same name, "A Computer Perspective" (1983) by the incomparable Charles and Ray Eames, ( www.amazon.com/exec/obidos/ASIN/0674156269/hip-20 ) tells the story of a pioneer of applied systems theory, Lewis Fry Richardson. The founder of scientific weather prediction, Richardson's equations for the behavior of air under all conditions of pressure, flow, moisture, etc., are still used today. His 1922 text "Weather Prediction by Numerical Process" ( www.amazon.com/exec/obidos/ASIN/0521680441/hip-20 ) is still a classic. (This picture taken from that book resembles my checkerboard analogy.) ( www.well.com/~abs/Cyb/4.669211660910299067185320382047/Sim/richardson.jpg ) But as the Eames explained: Richardson was a Quaker and conscientious objector. His wife recalled, "There came a time of heartbreak when those most interested in his 'upper air' researches proved to be the 'poison gas' experts. Lewis stopped his meteorology researches, destroying such as had not been published. What this cost him none will ever know!" He devoted the rest of his life to the mathematical study of the causes of war, publishing several books on the subject and single- handedly founding quantitative sociology. His "Arms and Insecurity" (written in 1953 -- the year of his death -- and published in 1960) ( www.amazon.com/exec/obidos/ASIN/0835703789/hip-20 ) presents a model of an arms race between two nations. Let's map it on to the checkerboard game. Say that the lower left corner (square 1) represents each nation spending nothing on arms. Then motion to the right represents nation A spending more on arms, in increments of 2% of its Gross National Product (GNP), while moving up represents nation B spending more on arms. Let's assume the nations behave in a perfectly symmetrical manner. Each time step will be a year, a typical government funding cycle. Now you have to ask yourself, if you were Minister of Finance for nation A, and you were spending 6% of your GNP on arms while your hostile neighbor nation B spent 10% of their GNP, what would do next year? Then you have to do it again for every combination of A and B's spending on the board. It quickly become obvious that for "reasonable" human responses to each scenario you end up with a system in which the state moves up and to the right, in an unending arms race (at least until you run out of checkerboard. I think the U.S. and its allies won the Cold War and beat the U.S.S.R. because they ran out of checkerboard first.) But up until that time things were pretty dicey. Richardson quoted Sir Edward Grey, British Foreign Secretary a the start of World War I: The increase of armaments that is intended in each nation to produce consciousness of strength, and a sense of security, does not produce these effects. Our little checkerboard model helps us understand why.


Alice was beginning to get very tired of sitting by her sister on the bank, and of having nothing to do: once or twice she had peeped into the book her sister was reading, but it had no pictures or conversations in it, "and what is the use of a book," thought Alice, "without pictures or conversations?" -- Lewis Carroll, 1865 "Alice's Adventure in Wonderland" ( www.amazon.com/exec/obidos/ASIN/0393048470/hip-20 ) Drawing pictures of the world-lines of systems in a state space didn't used to be so popular, you know. Many of the great mathematicians of the last 300 years have taken a dim view of "visualizing" analytic results, especially among the French masters, thinking of geometry as a crutch for the weak. (Ironically, this is even true of Poincare's topology -- "real" mathematicians work it all out with symbols alone!) As R. Buckminster Fuller explained in the essay "Prevailing Conditions in the Arts" in his book "Utopia or Oblivion" (1970), ( www.amazon.com/exec/obidos/ASIN/0553028839/hip-20 ) scientists stopped using pictures so much around the time they began exploring electromagnetic forces, and ultimately the rather erudite equations of James Clerk Maxwell. ( en.wikipedia.org/wiki/Maxwell's_equations ) As Bucky describes it: They found themselves getting on very well without seeing what was going on. It was during some of these early experiments on energy behaviors that a fourth-power relationship was manifest. The equations contained a fourth power x^4. You can make a model of x^3, e.g. a cube, and you can make a model of x^2 (X to the second power) and call it a square, and a model of x^1 and call it a line; but you could not make a geometric model of x to the fourth power. The consequences of this unmodellable fourth-power . . . are tied up with other events. At the same historical period literary men were trying to explain the new invisible electrical energy, which could do yesterday's tasks with new and miraculous ease, to their scholarly readers and the public in general. They began to use visually familiar analogies to explain the invisible behaviors. The concept of a current of water running through a pipe as analogous to electricity running through wires was employed by the nonscientific writers. The scientists didn't like that at all, because electricity really doesn't behave as water. Electricity 'ran' uphill just as easily as 'downhill' (but it did not really 'run'). The scientists felt that analogies were misleading and they disliked them. When the experiments that showed a fourth power relationship occurred, the scientists said 'Well, up to this time we have felt that visual models were legitimate (though not always easy to formulate), but now, inasmuch as we can't make a fourth-power energy-relationship model, the validity of the heretofore accepted generalized law of models is broken. From now on physically conceptual models are all suspect. We're going to work now entirely in terms of abstract, "empty set" mathematical expressions.' Their invisible procedures thenceforth to this day [~1970] have a counterpart in modern air transport and night fighter flying -- which is conductible and is usually conducted 'on instruments.' When you qualify for instrument flight you are to fly in fog and night without seeing any terrain. You get on very well, and arrive where you want on instruments. Scientists went 'on instruments' about 1875 -- almost entirely. By the time I went to school at the turn of the century we were taught about instruments and equations and how to conduct experiments. We were taught that the forth dimension was just 'ha ha' -- you could never do anything about it. I remember reading a story, which I can't seem to find corroboration for on the web, about John von Neumann defending his design for a digital computer (at the Institute for Advanced Studies in Princeton) against critics. They said if you had a high-speed computer it would do calculations so fast that you couldn't print them out at that speed, and even if you could nobody could read them that fast, so what was the use? Von Neumann replied that he could hook the computer up to an oscilloscope and WATCH the computations proceed. (And he did.) But of course that oscilloscope is mainly used to plot one dimensional data changing over time. (I think sometimes that after Galileo revolutionized astronomy in 1610 using a telescope to discover the moons of Jupiter, then Pasteur revolutionized biology 1862 using a microscope to discover disease germs, and then Janssen revolutionized chemistry in 1868 using a spectroscope to discover Helium, the cybernetics group were trying valiantly to discover something revolutionary with the oscilloscope.) We are fortunate that the Lorenz Attractor is only three dimensional, and so people can look at it. If it had been 4D there might not have been a Chaos Revolution in the 1980s -- it might have been too hard to "get." But the Lorenz Attractor just a pedagogic example, simplified so that we CAN see it. The real systems we have to deal with are very, very high dimension. Newton's sun-and-planet system that produces the elliptical orbit (and all of the conic sections) has eight equations and eight variables -- that's the simple, solvable textbook example. This lead us to something I call the "Cyberspace Fallacy." We've had ideas developed in science fiction about direct-brain computer interfaces into some kind of Computer Graphic representation of the thing-that-the-Internet-evolves-into. In "True Names" (sci-fi short story, 1980) by Vernor Vinge, reprinted in "True Names and the Opening of the Cyberspace Frontier" ( www.amazon.com/exec/obidos/ASIN/0312862075/hip-20 ) ( home.comcast.net/~kngjon/truename/truename.html ) The imagery of this interface is deliberately vague, suggesting some kind of high-dimensional direct-knowledge, or "grokking" of the data. They drifted out of the arpa "vault" into the larger data spaces that were the Department of Justice files. He could see that there was nothing hidden from them; random archive retrievals were all being honored and with a speed that would have made deception impossible. They had subpoena power and clearances and more. * * * * * * "Look around you. If we were warlocks before, we are gods now. Look!" Without letting the center of their attention wander, the two followed his gaze. As before, the myriad aspects of the lives of billions spread out before them. But now, many things were changed. In their struggle, the three had usurped virtually all of the connected processing power of the human race. Video and phone communications were frozen. The public data bases had lasted long enough to notice that something had gone terribly, terribly wrong. Their last headlines, generated a second before the climax of the battle, were huge banners announcing GREATEST DATA OUTAGE OF ALL TIME. Later, William Gibson sharpened the idea, and named it "cyberspace," in "Neuromancer" (sci-fi novel, 1986). ( www.amazon.com/exec/obidos/ASIN/0441569595/hip-20 ) A cowboy [cracker] is using a direct-brain-interface to defeat a company's ICE in this vivid scene: Case's virus had bored a window through the library's command ice. He punched himself through and found an infinite blue space ranged with color-coded spheres strung on a tight grid of pale blue neon. In the non space of the matrix, the interior of a given data construct possessed unlimited subjective dimension; a child's toy calculator, accessed through Case's Sendai, would have presented limitless gulfs of nothingness hung with a few basic commands. Case began to key the sequence the Finn had purchased from a mid-echelon sarariman with severe drug problems. He began to glide through the spheres as if he were on invisible tracks. Here. This one. Punching his way into the sphere, chill blue neon vault above him starless and smooth as frosted glass, he triggered a sub- program that effected certain alterations in the core custodial commands. Out now. Reversing smoothly, the virus reknitting the fabric of the window. Done. Cool as jazz, for sure, but here's the rub. The problem isn't in visualizing low-dimensional data like router topologies (how the servers are wired together) but high-dimensional data like traffic load over time at every node, broken down by data type (voice, video, machine instructions, text, etc.). The problem is two-fold: how to REPRESENT the high-dimensional data, which is pretty darned tricky, and then how to TRAIN the users to "read" the data, which they surely will not naturally understand at first viewing.


Electronic man has no physical body. -- Marshall McLuhan I was fortunate in Junior High School to have the same great math teacher in 7th and 8th grade: Aubrey Dunne, who is still my friend 40 years later. He recognized my budding curiosity about math, and recommended I read the book "One Two Three... Infinity: Facts and Speculations of Science" (1947) by George Gamow. ( www.amazon.com/exec/obidos/ASIN/0486256642/hip-20 ) One of the things this book tried to explain was the 4th dimension, using various tricks such as the "tesseract" figure and a "2-worms- in-the-apple" analogy. I worked on the problem myself, as a kid and later as an adult. One of my explorations is explored in the paper "Visualizing 4D Hypercube Data By Mapping Onto a 3D Tesseract" (1996) by Alan B. Scrivener ( www.well.com/~abs/SIGGRAPH96/4Dtess.html ) submitted for presentation as a Research Paper at the ACM/SIGGRAPH '96 Conference held August 4-9, 1996 in New Orleans, Louisiana , but not accepted. (They pointed out I was only exploring the eight cubes that form the 4-D "faces" of the hypercube. But I could solve that by slicing over time!) I later discovered the work of Alfred Inselberg, as I reported in C3M June 2003, "Steers, Beers and the Nth Dimension." ( www.well.com/~abs/Cyb/4.669211660910299067185320382047/c3m_0206.txt ) Inselberg finally has a web site with some of his research posted. ( www.math.tau.ac.il/~aiisreal ) Click on "Images" for the good stuff. I have previously compared his "parallel coordinates" to the "sick bay" readout on TV's original "Star Trek" series. ( www.amazon.com/exec/obidos/ASIN/B0002JJBZY/hip-20 ) ( www.well.com/~abs/Cyb/4.669211660910299067185320382047/SG2002/sickscan.jpg ) But the two best methods I have happened upon for visualizing higher dimensions I call the "N-Dimensional Tic Tac Toe" and "N-Dimensional Elevator" metaphors. In "N-Dimensional Tic Tac Toe" I like to play 4-in-a-row because otherwise the center square is a super-duper square, and the first player has a huge advantage. You know how to draw a 2D game. To make a 3D game just draw 4 2D games in a row, separated by a little space. Those are the 4 slices in the third dimension, like the floor plan to a low-rise building. Then duplicate this 4-in-a-row pattern of 2D game fields three more times below, making a 2D array of 2D arrays. There's your 4D game. This will easily fit on an 8.5x11 inch sheet of paper, and you can draw it and photocopy it to make playing fields for real 4D tick-tack-toe games. Great for older kids on car trips. A win is (1,1,1,1),(1,1,1,2),(1,1,1,3), (1,1,1,4) for example, or (1,4,4,1),(2,3,3,1),(3,2,2,1),(4,1,1,1). Each independent coordinate has to stay the same or rise/fall by a unit with each X or O position to form 4 in a 4D row. Now, you can make a row of 4 of these pages, and get a 5D game, or make a 4x4 array of them on the floor, and get a 6D game, and so on. You may need some sheets of poster board for the next steps. The other metaphor is the "N-Dimensional Elevator" which is much simpler. Let's say you're in a 13-story 7-dimensional building full of large, windowless apartments. Each apartment is only 3D, so you'll feel at home, and the elevator is 3D so humans can ride in it, but you get in and there are 7 vertical columns of buttons, one for each dimension, and each column has 13 buttons for the 13 "floors" in that dimension. Like a Chinese menu, you select one from each column, and then the elevator takes you there. Anything you leave in an apartment you will find there later by pushing the same buttons. But if you leave a barking dog the neighbors in 14 directions (instead of 6 directions in a 3D building) might complain. After much thought I have concluded that if a Turing Machine can simulate a system of any dimensionality (and it can, and so can any computer) then dimensions don't matter so much in finite systems. The checkerboard model I described previously sometimes acted 2D and sometimes 1D, but because in the most general case any state could lead to any other, you COULD argue that it's an INFINITE-dimensional system. Only by REMOVING transitions can you constrain it to act 2D, as in the model of the Richardson arms race. Something else I've been pondering is this: if we found chaos by going from 2D to 3D, what is waiting for us to discover in going from 101 dimensions to 102? (Not to mention all the steps along the way.) I fear that these are the sorts of questions that only Wolfram's method can answer: simulate all possibilities and look at the results.


SCENE: Jobs and Woz walking up stairs carrying a box. WHOOPIE: "Believing in things and making them happen has always been the California way. Soon a new gold rush was taking place in Northern California." JOBS (as Woz trips): "Watch it Woz... SCENE: Room of people crowded at the table looking at the computer. JOBS (arms open): "Well, here it is. The first personal computer." CROWD: Wow / Alright / Clapping... WOZ: "We could sell like a dozen of these!" JOBS: "Whadaya talkin' about? We're gonna sell one to everyone on the planet." -- "Golden Dreams" (theme park show, 2001) at Disney's California Adventure, Anaheim, California ( www.disneyfans.com/articles/2001/03-01-01_California_Apple.htm ) ( en.wikipedia.org/wiki/Golden_Dreams ) Of course, Steve Jobs is right; he IS going to sell one to everyone on the planet. If you count imbedded systems computers now outnumber people in the U.S.A. The supply of cheap computers in the 1980s made possible the switch from frequency methods to state space methods, and the resulting renaissance of applied systems simulation. The story is told in "Control Systems Design" (1986) by Bernard Friedland [revised as "Advanced Control Systems Design" (1995)]. ( www.amazon.com/exec/obidos/ASIN/0130140104/hip-20 ) This is the text I used when I studied control theory at UCLA: The history of control theory can be conveniently divided into three periods. The first, starting in prehistory and ending in the early 1940s, may be termed the PRIMITIVE period. This was followed by a CLASSICAL period, lasting scarcely 20 years, and finally came the MODERN period which includes the content of this book. The term PRIMITIVE is used here not in a pejorative sense, but rather in the sense that the theory consisted of a collection of analyses of specific processes by mathematical methods appropriate to, and often invented to deal with, the specific process, rather than an organized body of knowledge that characterizes the classical and the modern period. Although feedback principles can be recognized in the technology of the Middle Ages and earlier, the intentional use of feedback to improve the performance of dynamic systems was started at around the beginning of the industrial revolution in the late 18th and early 19th centuries. The benchmark development was the ball-governor invented by James Watt to control the speed of his steam engine. Throughout the first half of the 19th century, engineers and "mechanics" were inventing improved governors. The theoretical principles that describe their operation were studied by such luminaries of 18th and 19th century mathematical physics as Huygens, Hooke, Airy, and Maxwell. By the mid 19th century it was understood that the stability of a dynamic system was determined by the location of the roots of the algebraic equation. Rough in his Adams Prize Essay of 1877 invented the stability algorithm that bears his name. Mathematical problems that had arisen in the stability of feedback control systems (as well as in other dynamic systems including celestial mechanics) occupied the attention of early 20th century mathematicians Poincare and Liapunov, both of whom made important contributions that have yet to be superseded. Development of the gyroscope as a practical navigation instrument during the first quarter of the 20th century led to the development of a variety of autopilots for aircraft (and also for ships). Theoretical problems of stabilizing these systems and improving their performance engaged various mathematicians of the period. Notable among them was N. Minorsky whose mimeographed notes on nonlinear systems were virtually the only text on the subject before 1950. The CLASSICAL period of control theory begins during World War II in the Radiation Laboratory of the Massachusetts Institute of Technology. The personnel of the Radiation Laboratory included a number of engineers, physicists, and mathematicians concerned with solving engineering problems that arose in the war effort, including radar and advanced fire control systems. The laboratory that was assigned problems in control systems included individuals knowledgeable in the frequency response methods, developed by people such as Nyquist and Bode for communication systems, as well as by engineers familiar with other techniques. Working together, they evolved a systematic control theory which is not tied to any particular application. Use of frequency-domain (Laplace transform) methods made possible the representation of a process by its transfer function and thus permitted a visualization of the interaction of the various subsystems in a complex system by the interconnection of the transfer functions in the block diagram. The block diagram contributed perhaps as much as any other factor to the development of control theory as a distinct discipline. Now it was possible to study the dynamic behavior of a hypothetical system by manipulating and combining the black boxes in the block diagram without having to know what goes on inside the boxes. The classical period of control theory, characterized by frequency-domain analysis, is still going strong, and is now in a "neoclassical" phase -- with the development of various sophisticated techniques for multivariable systems. But concurrent with it is the MODERN period, which began in the late 1950s and early 1960s. STATE-SPACE METHODS are the cornerstone of MODERN CONTROL THEORY. The essential feature of state-space methods is the characterization of the processes of interest by differential equations instead of transfer functions. This may seem like a throwback to the earlier, primitive, period where differential equations also constituted the means of representing the behavior of dynamic processes. But in the earlier period the processes were simple enough to be characterized by a SINGLE differential equation of fairly low order. In the modern approach the processes are characterized by systems of coupled, first-order differential equations. In principle there is no limit to the order (i.e., the number of independent first-order differential equations) and in practice the only limit to the order is the availability of computer software capable of performing the required calculations reliably. Although the roots of modern control theory have their origins in the early 20th century, in actuality they are intertwined with the concurrent development of computers. A digital computer is all but essential for performing the calculations that must be done in a typical application. There is more detail on this transition in "A Brief History of Feedback Control" from "Optimal Control and Estimation" (1992) by F. L. Lewis. ( www.theorem.net/theorem/lewis1.html ) Of course the cheap computers in the 1980s also made possible the breakthroughs of Stephen Wolfram, and the methodology he used of doing extremely long-duration computer-simulated mathematical experiments.


Q: How many surrealists does it take to change a light bulb? A: A fish. -- joke But let us always remember, a model is just a model; the map may not match the territory. This is easy to pledge, but when we get into the thick of things, it's harder to do. I remember Bateson used to talk about 'diachronic' systems which repeat themselves, and 'synchronic' systems which don't. (I think the terms arose with anthropologists analyzing myths, contrasting the non-repeating cultures such as Messianic Judaism and Oceana's Cargo Cults, vs. the repeating cycle cultures like the Mayans with their calendar and the Tibetans with their prayer wheels.) In the middle of debating whether a certain system "is" diachronic, Bateson would stop to remind us that these terms apply only to MODELS OF REALITY. A certain MODEL is one or the other, but not real reality. This concept of "real reality" is a tough one. It's easy to argue that it's just another model. One begins to appreciate why Lao Tzu wrote: The way that can be spoken is not the eternal way. in the "Tao Te Ching" (4th Century BC). ( www.amazon.com/exec/obidos/ASIN/014044131X/hip-20 ) When doing science, the thing to remember is the bug may not be in the simulation or the data, it may be in the theory. Remember what Isaac Asimov said: The most exciting phrase to hear in science, the one that heralds new discoveries, is not Eureka! (I found it!) but rather, "hmm.... that's funny...."


People only remember: * 10% of what they read; * 20% of what they hear; * 30% of what they see; * 50% of what they see and hear together; * And 80% of what they see, hear, and do. -- attributed to Dr. Mehrabians ( www.getworldpassport.com/UK/marketplace.aspx?ID=MW ) There's no doubt. You have to do it yourself. Find a way to play with simulations. Some of Richardson's critics (and more recent critics of computer modeling) said that you only get out of a computer a confirmation of the assumptions that you put in. Oh that it were so! I have found that to be a rare occurrence. More frequently I find that my assumptions DO NOT lead to my conclusions, and I have to re-examine them. Modeling is humbling, and it reminds me there are many mysteries in those uncharted waters we call COMPLEXITY. Here are some resources to help you find some sim software that suits you. Summary pages: Individual packages (my favorites are in the section following this one):


A program is like a nose; Sometimes it runs, sometimes it blows. -- Howard Rose (quoted in "Computer Lib/Dream Machines" (book, 1974) by Ted Nelson) ( www.amazon.com/exec/obidos/ASIN/0893470023/hip-20 ) ( www.digibarn.com/collections/books/computer-lib ) There are a few programs I've played with myself, that I can highly recommend:
  • Stella ( www.iseesystems.com/softwares/Education/StellaSoftware.aspx ) very good for learning general systems theory included in the book "Modeling Dynamic Systems: Lessons for a First Course" by Diana Fisher ( www.iseesystems.com/store/modelingbook/default.aspx ) and in "Dynamic Modeling (Modeling Dynamic Systems)" by D. H. Meadows ( www.amazon.com/exec/obidos/ASIN/0387988688/hip-20 )
  • VisSim - Modeling and Simulation of Complex Dynamic Systems ( www.vissim.com ) ( https://journal.fluid.power.net/issue2/software.html ) written by a fiend of mine, Peter Darnell, who left Stellar to "do something with visual programming" and ended up as CEO of Visual Solutions - read his amazing account of controlling a prototype Antiskid Braking System (ABS) with a laptop running VisSim ( www.adeptscience.co.uk/products/mathsim/vissim/apps/gm.html )
  • Mathematica ( amath.colorado.edu/computing/Mathematica/basics/odes ) when I was first learning how to solve linear ODEs at UCLA course taught by a satellite dynamics engineer, and working at Stellar with my own graphics supercomputer running mathematica, Craig Upson suggested I use the graphics in Mathematica to visualize the ODEs behaviors -- it was great advice
  • AVS5 ( www.avs.com ) I worked for this company for four years and used its products for three years before that and for ten years since; what can I say, I love this software -- it's visually programmed visualization software, and you can add module in C I did some visualization of 2D and 3D ODEs with it, and posted some of it on-line (also did some videos which I ought to digitize) ( www.well.com/~abs/math_rec.html )
  • NeatTools ( www.pulsar.org/neattools ) ( www.pulsar.org/images/neatimages/index.html ) this free tool created by my friend Dave Warner's company Mindtel is very handy for hooking up real time data for low-cost Digital Signal Processing (DSP) prototyping -- you can add modules in C++
  • write your own code nothing beats REALLY doing it yourself here's some (buggy) C code: /* simulate linear n-dimensional system with Euler's method */ main() { int t, v; float state_array[NUMBER_OF_VARIABLES] float transition_matrix[NUMBER_OF_VARIABLES][NUMBER_OF_VARIABLES]; void initialize_change_rules(transition_matrix); void initialize_state(state_array); void display(state_array); float apply_change_rules(int v, state_array, transition_matrix); for (t = 0; t < MAX_TIME STEPS; t++) { for (v = 0; v < NUMBER_OF_VARIABLES; v++) { state_array[v] = state_array[v] + apply_change_rules(v, state_array, transition_matrix); display(state_array); } } } float apply_change_rules(int v, state_array, transition_matrix) { int n; float sum = 0; for (n = 0; n < NUMBER_OF_VARIABLES; n++) { sum = sum + transition_matrix[v][n]*state_array[n]; } I leave it to the reader to write the routines: initialize_change_rules(transition_matrix) initialize_state(state_array) display(state_array)


The thing that got me started on the science that I've been building now for about 20 years or so was the question of okay, if mathematical equations can't make progress in understanding complex phenomena in the natural world, how might we make progress? -- Stephen Wolfram ( www.brainyquote.com/quotes/authors/s/stephen_wolfram.html ) While working on this article I got an email from Wolfram's crew inviting me to the New Kind of Science (NKS) NKS summer school. ( www.wolframscience.com/summerschool ) I was interested, but the course is three weeks long and I don't have that much time off accrued. But it was fun to imagine anyway. Skimming the email I noticed this passage: The core of the Summer School is an individual project done by each student. The project is chosen on the basis of each student's interests, in discussion with Stephen Wolfram and the Summer School instructors. I thought to myself, "What would my project be?" And the answer I came up with was "Simulate all possible systems." So I did. First I worked out some simple cases by hand. ( www.well.com/~abs/Cyb/4.669211660910299067185320382047/Sim/system_sketch1.jpg ) Then I wrote a program in the C language (simple and portable) that I called "all_systems" which I have posted in source code form on my web site. ( www.well.com/~abs/swdev/C/all_systems ) You might think that this is like trying to get a million monkeys with typewriters to write Shakespeare (though I think James Joyce might be easier) but in fact I learned quite a bit from the exercise. First I looked at all possible deterministic finite systems -- so called "finite state machines" -- of a given size, having each start in state 1. (By symmetry arguments I believed I didn't have to check any of the other starting states to see all modes of behavior.) Going in increasing order of size, here is what I found. Systems having one state have only one behavior: the system stays in state 1 with each time step. (I believe I could have solved this analytically.) I call this an orbit with period 1. Systems having two states can stay in one state (orbit of period 1), or oscillate between two states (orbit of period 2), or start in one state, move to the other and stay there (I call this a head of length 1 followed by an orbit of period 1. Control engineers call it a transient followed by a stable mode.) At this point we seen most of the behavioral modes. There is a head of length L followed by an orbit of period P, and L + P <= N, the number of states. Here is how I would plot this, for example in a 3-state system: +---+---+---+ | 2 | 3 | 1 | +---+---+---+ | 1 | | | | | 1 | | | | | 1 | | 1 | | | +---+---+---+ Along the top row is the transition table (states are number left to right): from 1 to 2, from 2 to 3 and from 3 to 1. Below are the steps, starting in state 1, the system goes into state 2, then state 3, then back to state 1. The head is length zero and the orbit is period three. Looking at systems of three, four and five states I began adding some automatic analysis to my program, that would count the times I had heads an orbits of various lengths. I wasn't able to get much bigger, because the number of different systems with N states (before attempting to eliminate symmetries) is N^N -- a function I mentioned in passing in C3M Volume 2 Number 9, Sep. 2003, "Do Nothing, Oscillate, or Blow Up: An Exploration of the Laplace Transform" ( www.well.com/~abs/Cyb/4.669211660910299067185320382047/c3m_0209.txt ) although I had no idea at the time that it appears in systems theory. Here's a table: N N^N - ------- 1 1 2 4 3 27 4 256 5 3,125 6 46,656 7 823,543 Obviously it grows faster than e^N, so it is super-exponential. I knew I couldn't go much higher, so I looked for something else to do. I took the results of my analysis of head and orbit lengths: head: 0 orbit: 0 count: 0 percent: 0.0 head: 1 orbit: 0 count: 0 percent: 0.0 head: 2 orbit: 0 count: 0 percent: 0.0 head: 3 orbit: 0 count: 0 percent: 0.0 head: 0 orbit: 1 count: 27 percent: 33.333333 head: 1 orbit: 1 count: 18 percent: 22.222222 head: 2 orbit: 1 count: 6 percent: 7.407407 head: 3 orbit: 1 count: 0 percent: 0.0 head: 0 orbit: 2 count: 18 percent: 22.222222 head: 1 orbit: 2 count: 6 percent: 7.407407 head: 2 orbit: 2 count: 0 percent: 0.0 head: 3 orbit: 2 count: 0 percent: 0.0 head: 0 orbit: 3 count: 6 percent: 7.407407 head: 1 orbit: 3 count: 0 percent: 0.0 head: 2 orbit: 3 count: 0 percent: 0.0 head: 3 orbit: 3 count: 0 percent: 0.0 loaded it into AVS and visualized a 2D array of data as bar charts. ( www.well.com/~abs/Cyb/4.669211660910299067185320382047/Sim/percents.jpg ) I showed it to some people. Nobody cared. Then I got to thinking. This is such a simple model, maybe initial condition is a more important part of it. Within the N^N different systems at each size I decided to look at how the choice of initial condition influenced system behavior. Now I was looking for families of trajectories, N of them, making the total "runs" I had to compute grow to N*N^N, or N^(N+1). But maybe eliminating these combinations before with "symmetry" arguments had been a bad move. I had to find a way to plot the families of of trajectories. This is what I came up with (shown for a six state system): +--------+--------+--------+--------+--------+--------+ | 2 | 2 | 1 | 1 | 1 | 1 | +--------+--------+--------+--------+--------+--------+ | 1 | 2 | 3 | 4 | 5 | 6 | | 3456 | 12 | | | | | | | 123456 | | | | | | | 123456 | | | | | | | 123456 | | | | | | | 123456 | | | | | +--------+--------+--------+--------+--------+--------+ Actually, I drew this representation by hand for the N=3 case ( www.well.com/~abs/Cyb/4.669211660910299067185320382047/Sim/system_sketch2.jpg ) before I wrote the code. (The calculations on the side are computing the entropy loss -- i.e. information gained -- if you learn the state at each step, assuming equal distribution of start probablities.) From these three experiences of doing it by hand, writing a program to do it, and then looking at the output, I learned more: one of these systems is always doing one of two things: either destroying information, or preserving it. This corresponds to entropy increasing or staying the same. In the above 6-dim example, from step 1 to 2 an initial condition of 1 or 2 both jump to state 2, destroying the information of which state you came from. In the next step trajectories from 3 through 6 join in, destroying more information. After that the system preserves the remaining information that it is in state 2. I also realized that being in orbit of period greater than one means a system also has "phase information." Consider this 6-state system, looking only at the initial states of 1 & 6: +--------+--------+--------+--------+--------+--------+ | 6 | 6 | 3 | 1 | 1 | 1 | +--------+--------+--------+--------+--------+--------+ | 1 | 2 | 3 | 4 | 5 | 6 | | 456 | | 3 | | | 12 | | 12 | | 3 | | | 456 | | 456 | | 3 | | | 12 | | 12 | | 3 | | | 456 | | 456 | | 3 | | | 12 | +--------+--------+--------+--------+--------+--------+ One goes to six and six goes to one. But depending on where you start there are two orbits that crisscross like shoe laces. This system can store a "bit" of information in just where it is at a given time in this orbit. (The longer the orbit period the more information can be stored.) This is what I've discovered in a few days' time. (Okay, I'll grant it must be a rediscovery, but I've VERY THOROUGHLY learned this by doing it myself.) I wonder YOU can discover?


Though you all would want to know about "Sonoma 2006: The 50th Annual Meeting of the International Society for the Systems Sciences" ( www.isss.org/conferences/sonoma2006 ) being held at Sonoma State University, Rohnert Park, California, USA July 9th - 14th 2006. From the web site: The 50th anniversary conference of the International Society for the Systems Sciences offers an opportunity to celebrate a half-century of theory and practice in the broadly defined field of systems, honoring the vision of the founders (Ludwig von Bertalanffy, Kenneth Boulding, Ralph Gerard, James Grier Miller, and Anatol Rapoport) and recognizing the contributions of leading systems thinkers. It is also a time to reflect upon what we have learned, and to collaboratively envision future directions. * * * * * Pre-Conference Workshop: Mind in Nature: Gregory Bateson and the Ecology of Experience, July 7 - 9


Last time's issue was emailed out with a header that said "Volume 4 Number 8, Nov. 2005" by mistake instead of the correct "Volume 5 Number 1, Jan. 2006." Also, I misspelled the names Lasseter and Catmull (as Lassiter and Catmul, my spell-checker didn't catch them); both these errors have been corrected in the archives. ======================================================================== newsletter archives: www.well.com/~abs/Cyb/4.669211660910299067185320382047 ======================================================================== Privacy Promise: Your email address will never be sold or given to others. You will receive only the e-Zine C3M from me, Alan Scrivener, at most once per month. It may contain commercial offers from me. To cancel the e-Zine send the subject line "unsubscribe" to me. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I receive a commission on everything you purchase from Amazon.com after following one of my links, which helps to support my research. ======================================================================== Copyright 2006 by Alan B. Scrivener