Not in chaos, at the λ=0.273 vicinity of it

Photo – “O caos é uma ordem por decifrar” (Portuguese), that is… “Chaos is an order yet to be deciphered“, a quote from the Nobel Prize in Literature (1998) José Saramago [Lisbon, V. Ramos, 2013].

In 1990 (*), on one of his now famous works, Christopher Langton (link) decided to ask an important question. In order for computation to emerge spontaneously and become an important factor in the dynamics of a system, the material substrate must support the primitive functions required for computation: the transmission, storage, and modification of information. He then asked: Under what conditions might we expect physical systems to support such computational primitives?

Naturally, the question is difficult to address directly. Instead, he decided to reformulate the question in the context of a class of formal abstractions of physical systems: cellular automata (CAs). First, he introduce cellular automata and a simple scheme for parametrising (lambda parameter, λ) the space of all possible CA rules. Then he applied this parametrisation scheme to the space of possible one-dimensional CAs in a qualitative survey of the different dynamical regimes existing in CA rule space and their relationship to one another.

By presenting a quantitative picture of these structural relationships, using data from an extensive survey of two-dimensional CAs, he finally review the observed relationships among dynamical regimes, discussing their implications for the more general question raised above.  Langton found out that for a 2-state, 1-r neighbourhood, 1D cellular automata the optimal λ value is close to 0.5. For a 2-state, Moore neighbourhood, 2D cellular automata, like Conway’s Life, the λ value is then 0.273.

We then find that by selecting an appropriate parametrisation of the space of CAs, one observes a phase transition between highly ordered and highly disordered dynamics, analogous to the phase transition between the solid and fluid states of matter. Furthermore, Langton observed that CAs exhibiting the most complex behaviour – both qualitatively and quantitatively- are found generically in the vicinity of this phase transition. Most importantly, he observed that CAs in the transition region have the greatest potential for the support of information storage, transmission, and modification, and therefore for the emergence of computation. He concludes:

(…) These observations suggest that  there is  a fundamental connection between phase transitions and computation, leading to the following hypothesis concerning the emergence of computation in  physical systems: Computation may emerge spontaneously and come to dominate the dynamics of physical systems when those systems are at or near a transition between their solid and fluid phases, especially in the vicinity of a second-order or “critical” transition. (…)

Moreover, we observe surprising similarities between the behaviours of computations and systems near phase transitions, finding analogs of computational complexity classes and the halting problem (Turing) within the phenomenology of phase transitions. Langton, concludes that there is a fundamental connection between computation and phase transitions, especially second-order or “critical” transitions, discussing some of the implications for our understanding of nature if such a connection is borne out.

The full paper (*), Christopher G. Langton. “Computation at the edge of chaos”. Physica D, 42, 1990, is available online, here [PDF].

P(n+1)=r.P(n).[1-P(n)]

Figure (clik to enlarge) – Applying P(0)=0.6; r=4; N=100000; for(n=0;n<=N;n++) { P(n+1)=r*P(n)*(1-P(n)); } Robert May Population Dynamics equation [1974-76] (do check on Logistic maps) for several iterations (generations). After 780 iterations, P is attracted to 1 (max. population), and then suddenly, for the next generations the very same population is almost extinguish.

“Not only in research, but also in the everyday world of politics and economics, we would all be better off if more people realised that simple non-linear systems do not necessarily possess simple dynamical properties.” ~ Robert M. May, “Simple Mathematical models with very complicated Dynamics”, Nature, Vol. 261, p.459, June 10, 1976.

(…) The fact that the simple and deterministic equation (1) can possess dynamical trajectories which look like some sort of random noise has disturbing practical implications. It means, for example, that apparently erratic fluctuations in the census data for an animal population need not necessarily betoken either the vagaries of an unpredictable environment or sampling errors: they may simply derive from a rigidly deterministic population growth relationship such as equation (1). This point is discussed more fully and carefully elsewhere [1]. Alternatively, it may be observed that in the chaotic regime arbitrarily close initial conditions can lead to trajectories which, after a sufficiently long time, diverge widely. This means that, even if we have a simple model in which all the parameters are determined exactly, long term prediction is nevertheless impossible. In a meteorological context, Lorenz [15] has called this general phenomenon the “butterfly effect“: even if the atmosphere could be described by a deterministic model in which all parameters were known, the fluttering of a butterfly’s wings could alter the initial conditions, and thus (in the chaotic regime) alter the long term prediction. Fluid turbulence provides a classic example where, as a parameter (the Reynolds number) is tuned in a set of deterministic equations (the Navier-Stokes equations), the motion can undergo an abrupt transition from some stable configuration (for example, laminar flow) into an apparently stochastic, chaotic regime. Various models, based on the Navier-Stokes differential equations, have been proposed as mathematical metaphors for this process [15,40,41] . In a recent review of the theory of turbulence, Martin [42] has observed that the one-dimensional difference equation (1) may be useful in this context. Compared with the earlier models [15,40,41] it has the disadvantage of being even more abstractly metaphorical, and the advantage of having a spectrum of dynamical behaviour which is more richly complicated yet more amenable to analytical investigation. A more down-to-earth application is possible in the use of equation (1) to fit data [1,2,3,38,39,43] on biological populations with discrete, non-overlapping generations, as is the case for many temperate zone arthropods. (…) in pp. 13-14, Robert M. May, “Simple Mathematical models with very complicated Dynamics“, Nature, Vol. 261, p.459, June 10, 1976 [PDF link].

10 PRINT CHR$(205.5+RND(1)); : GOTO 10

 

“Coders are now habitat providers for the rest of the world.” ~ Vitorino Ramos, via Twitter, July, 17, 2012 (link).

Video lecture – Casey Reas (reas.com) at Eyeo2012 (uploaded 2 days ago on Vimeo): From a visual and conceptual point of view, the tension between order and chaos is a fertile space to explore. For over one hundred years, visual artists have focused on both in isolation and in tandem. As artists started to use software in the 1960s, the nature of this exploration expanded. This presentation features a series of revealing examples, historical research into the topic as developed for Reas‘ upcoming co-authored book “10 PRINT CHR$(205.5+RND(1)); : GOTO 10″ (MIT Press, 2012, book link; cover above), and a selection of Casey‘s artwork that relies on the relationship between chance operations and strict rules.

The North, the East, the West, and the South

(…) The Four Mists of Chaos, the North, the East, the West, and the South, went to visit Chaos himself. He treated them all very kindly and when they were thinking of leaving, they consulted among themselves how they might repay his hospitality. Since they had noticed that he had no holes in his body, as they each had (eyes, nose, mouth, ears, etc.), they decided each day to provide him with an opening. At the end of seven days, Kwang-tse tells us, Chaos died. (…)  in Indeterminacy – Ninety Stories by John Cage (Transcript of story number 27), With Music, ca. 26’00” to 27’00”, From John Cage’s [1958] Lecture ‘Indeterminacy’, 26’00” to 27’00”, in Die Reihe No. 5, English edition on p.120.

Sand piles

Video -“Sand, sand, pile of sand” by Robert Proch (2009) – Impression without an idea, about trying to catch one (music by Pinkfreud).

Complex Adaptive Systems (CAS) are dynamic developing systems which arrange themselves according to external influences and to their own inner current state. If you are familiar with Conway’s “Game of Life“, you have an example of such a system. Complex adaptive systems arrange themselves around one or more critical factors (like in a sandpile, for example, where the pile will rearrange itself when you drop additional sand grains onto it). The theory behind these systems is related to chaos theory, but the systems are said to be “on the edge of chaos“, because they have the ability to adjust themselves (around the critical factors), unlike truly chaotic systems. One interesting thing with CA-systems is that they can suddenly rearrange themselves rather violently or criticality. Like in a sandpile which has a grain added to it, which topples onto other grains, which in turn topple onto other grains, etc. In physics, the Bak-Tang-Wiesenfeld sandpile model is the first discovered example of a dynamical system displaying self-organized criticality and is named after Per Bak [1] [2] [3], Chao Tang [1] [2] and Kurt Wiesenfeld [1] [2]. While running the model, you soon then have an avalanche effect and signatures produced equivalent to those found in nature. In fact, many phenomena in daily life are complex adaptive systems, like weather, traffic, earthquakes, eco-systems, or the stock market, and many of them share this precise 1/f noise pattern. As our brains.

[1] Per Bak, Chao Tang and Kurt Wiesenfeld (1987). “Self-organized criticality: an explanation of 1/f noise“. Physical Review Letters 59: pp. 381-384.
[2] Per Bak, Chao Tang and Kurt Wiesenfeld (1988). “Self-organized criticality“. Physical Review A 38: pp. 364-374.
[3] Per Bak (1996). How Nature Works: The Science of Self-Organized Criticality. New York: Copernicus. ISBN 0-387-94791-4.

Technique is just a means of arriving at a statement

Video – Jackson Pollock (USA, 1912-1956) painting outside his house in 1950. “Technique is just a means of arriving at a statement” [from YouTube]. Pollock was probably the first and still unique man on planet earth able to continuously  increase his fractal dimension signature along his life work. He call it “Action Painting“. In a way, canvas was an habitat for him.

[…] The simulated ecology of different stimuli response threshold organisms, triggered by the seeds of these stigmergic processes, whether in the form of 3D local configurations, or by the qualitative values of any conceptual data items, must not be overestimated. Above all, the behaviour that emerges from all these spatial-temporal relationships conduct us into the realm of what nature is about: dynamical patterns of complexity. Not chaotic or purely rendered at random, but at the edge of chaos (Langton), where creative and autonomous aLife survives. As reported by Nature magazine (Sept., 13, 2000), research suggests that the abstract works of artists such as Jackson Pollock are esthetically pleasing because they obey fractal rules similar to those found on the natural world. Pollock was known to have swung his paint back and forth like a pendulum, using a can on the end of a string with a hole punched in it. Researchers (Jensen) have found that if a swinging pendulum is hit with a hammer at just the right frequency (slightly less than the natural rhythm of the pendulum), its motion becomes chaotic and the paint traces out very convincing “fake Pollocks”. However, the artist had no idea of fractals or chaotic motion. This seems to be in line with the actual synthetically computational art, where there is a need to reference some kind of external artifact or mechanism, but nevertheless and as it appears, not those of the self whether they are conscious, unconscious, intuitive or not. Synthetically generative art, and above all, artificial systems of morphogenesis of any kind, should be much more about what scientists call “complexity”, and rely on nature as a physical generative force of ontological significance. Moving on to the implicit, rather on the specific. […]

in Vitorino Ramos, On the Implicit and on the Artificial – Morphogenesis and Emergent Aesthetics in Autonomous Collective Systems, ARCHITOPIA Book, Art, Architecture and Science, INSTITUT D’ART CONTEMPORAIN, J.L. Maubant et al. (Eds.), pp. 25-57, Chapter 2, ISBN 2905985631 – EAN 9782905985637, France, Feb. 2002.

Pollock’s most famous paintings were made during the “drip period” between 1947 and 1950. He rocketed to popular status following an August 8, 1949 four-page spread in Life Magazine that asked, “Is he the greatest living painter in the United States?” At the peak of his fame, Pollock abruptly abandoned the drip style. ^{(image above)}

Pollock’s work after 1951 was darker in color, including a collection painted in black on unprimed canvases. This was followed by a return to color, and he reintroduced figurative elements. During this period Pollock had moved to a more commercial gallery and there was great demand from collectors for new paintings. In response to this pressure, along with personal frustration, his alcoholism deepened. (controversial Wikipedia entry)

Order out of Disorder

“Chaos theory has a bad name, conjuring up images of unpredictable weather, economic crashes and science gone wrong. But there is a fascinating and hidden side to Chaos, one that scientists are only now beginning to understand. It turns out that chaos theory answers a question that mankind has asked for millennia – how did we get here?

Over this 2010 BBC 4 documentary “The Secret Life of Chaos“, Professor Jim Al-Khalili sets out to uncover one of the great mysteries of science – how does a universe that starts off as dust end up with intelligent life? How does order emerge from disorder? It’s a mind bending, counter-intuitive and for many people a deeply troubling idea. But Professor Al-Khalili reveals the science behind much of beauty and structure in the natural world and discovers that far from it being magic or an act of God, it is in fact an intrinsic part of the laws of physics. Amazingly, it turns out that the mathematics of chaos can explain how and why the universe creates exquisite order and pattern. The natural world is full of awe-inspiring examples of the way nature transforms simplicity into complexity. From trees to clouds to humans – after watching this film you’ll never be able to look at the world in the same way again.” (description at YouTube).

[1 hour documentary in 6 parts: Part I (above), Part II, Part III, Part IV, Part V and Part VI. Even if you have no time, do not miss part 6. I mean, do really not miss it. Enjoy!]