The Santa Fe Institute was founded in 1984 to study complex adaptive systems – a large part of the view under the Fractal Lens. Two of the founders were Nobel Prize winners Kenneth Arrow (economics) and Philip Anderson (physics), who started the modern revolution against strict reductionism as we discussed in The Emergence of Emergence and the Limitations of Reductionism: An Exploration of Hierarchies.
Geoffrey West served as the president of the Santa Fe Institute from 2005 to 2009. We might anoint him as a Junior Lens Crafter under the line of Master Crafter Benoit Mandelbrot. In 2017 West published a wonderful book summarizing some of the findings and applications of fractals and complexity theory entitled Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life in Organisms, Cities, Economies, and Companies. Scale reveals how fundamental the Fractal Lens is to understanding how our world is organized and operates. It also sharpens the focus. West first defines what we are talking about: "A typical complex system is composed of myriad individual constituents or agents that once aggregated take on collective characteristics that are usually not manifested in, nor could easily be predicted from, the properties of the individual components themselves. For example, you are much more than the totality of your cells and, similarly, your cells are much more than the totality of all of the molecules from which they are composed . . . In a similar fashion, a city is much more than the sum of its buildings, roads, and people, a company much more than the sum of its employees and products, and an ecosystem much more than the plants and animals that inhabit it. The economic output, the buzz, the creativity and culture of a city or a company all result from the nonlinear nature of the multiple feedback mechanisms embodied in the interactions between its inhabitants, their infrastructure, and the environment. . . . In general, then, a universal characteristic of a complex system is that the whole is greater than, and often significantly different from, the simple linear sum of its parts. In many instances, the whole seems to take on a life of its own, almost dissociated from the specific characteristics of its individual building blocks. . . .This collective outcome, in which a system manifests significantly different characteristics from those resulting from simply adding up all of the contributions of its individual constituent parts, is called an emergent behavior. It is a readily recognizable characteristic of economies, financial markets, urban communities, companies, and organisms. The important lesson that we learn from these investigations is that in many such systems there is no central control. So, for example, in building an ant colony, no individual ant has any sense of the grand enterprise to which he is contributing. Some ant species even go so far as to use their own bodies as building blocks to construct sophisticated structures: army ants and fire ants assemble themselves into bridges and rafts for use in crossing waterways and overcoming impediments during foraging expeditions. These are examples of what is called selforganization. It is an emergent behavior in which the constituents themselves agglomerate to form the emergent whole, as in the formation of human social groups, such as book clubs or political rallies, or your organs, which can be viewed as the selforganization of their constituent cells, or a city as a manifestation of the selforganization of its inhabitants. . . . It is only over the last thirty years or so that scientists have started to seriously investigate the challenges of understanding complex adaptive systems in their own right and seeking novel ways of addressing them. A natural outcome has been the emergence of an integrated systemic transdisciplinary approach involving a broad spectrum of techniques and concepts derived from diverse areas of science ranging from biology, economics, and physics to computer science, engineering, and the socioeconomic sciences. An important lesson from these investigations is that, while it is not generally possible to make detailed predictions about such systems, it is sometimes possible to derive a coarsegrained quantitative description for the average salient features of the system. For example, although we will never be able to predict precisely when a particular person will die, we ought to be able to predict why the life span of human beings is on the order of one hundred years. Now let’s look at a few of the findings." We then learn: Fractals are Ubiquitous in Nature and Optimize Energy by Effectively Creating Additional Dimensions in Space [A]ll of the laws of science must be expressible as relationships between scaleinvariant dimensionless quantities, even though conventionally they are not typically written that way. This principle reveals that there are universal constants, the most famous of which is pi. Similitude is characteristic of many scaling arguments: general results can be derived, but details of their mechanistic origins sometimes remain hidden. From this we have the Fractal Lens concept of emergence. Systematic repetitive behavior is called scale invariance or selfsimilarity and is a property inherent to power laws. It is closely related to the concept of a fractal. To varying degrees, fractality, scale invariance, and selfsimilarity are ubiquitous across nature from galaxies and clouds to your cells, your brain, the Internet, companies, and cities. In other words, the view through the Fractal Lens is everywhere and is a fundamental characteristic of most systems that exist in the real world. Now do we know this? It all comes back to Benoit Mandelbrot and how he brilliantly exposed and destroyed the 2000 yearold erroneous assumptions of Euclid, Aristotle and Plato about the nature of reality in one fell swoop. But how did that happen? Scale tells us: "Mathematicians had recognized for a long time that there were geometries that lay outside of the canonical boundaries of the classical Euclidean geometry that has formed the basis for mathematics and physics since ancient times. The traditional framework that many of us have been painfully and joyfully exposed to implicitly assumes that all lines and surfaces are smooth and continuous. Novel ideas that evoked concepts of discontinuities and crinkliness, which are implicit in the modern concept of fractals, were viewed as fascinating formal extensions of academic mathematics but were not generally perceived as playing any significant role in the real world. It fell to the French mathematician Benoit Mandelbrot to make the crucial insight that, quite to the contrary, crinkliness, discontinuity, roughness, and selfsimilarity— in a word, fractality— are, in fact, ubiquitous features of the complex world we live in. In retrospect it is quite astonishing that this insight had eluded the greatest mathematicians, physicists, and philosophers for more than two thousand years. Like many great leaps forward, Mandelbrot’s insight now seems almost “obvious,” and it beggars belief that his observation hadn’t been made hundreds of years earlier. After all, “natural philosophy” has been one of the major categories of human intellectual endeavor for a very long time, and almost everyone is familiar with cauliflowers, vascular networks, streams, rivers, and mountain ranges, all of which are now perceived as being fractal. However, almost no one had conceived of their structural and organizational regularities in general terms, nor the mathematical language used to describe them. Perhaps, like the erroneous Aristotelian assumption that heavier things “obviously” fall faster, the Platonic ideal of smoothness embodied in classical Euclidean geometry was so firmly ingrained in our psyches that it had to wait a very long time for someone to actually check that it was valid with reallife examples. . . . Mandelbrot’s insights imply that when viewed through a coarsegrained lens of varying resolution, a hidden simplicity and regularity is revealed underlying the extraordinary complexity and diversity in much of the world around us. Furthermore, the mathematics that describes selfsimilarity and its implicit recursive rescaling is identical to the power law scaling discussed in previous chapters. In other words, power law scaling is the mathematical expression of selfsimilarity and fractality. Consequently, because animals obey power law scaling both within individuals, in terms of the geometry and dynamics of their internal network structures, as well as across species, they, and therefore all of us, are living manifestations of selfsimilar fractals. . . . In the natural world almost nothing is smooth— most things are crinkly, irregular, and crenulated, very often in a selfsimilar way. Just think of forests, mountain ranges, vegetables, clouds, and the surfaces of oceans. Consequently, most physical objects have no absolute objective length, and it is crucial to quote the resolution when stating the measurement. The Euclidean world of straight lines and smooth shapes is actually not the natural world, but a world of human invention. As Mandelbrot succinctly put it: “Smooth shapes are very rare in the wild but extremely important in the ivory tower and the factory.” . . . Perhaps of greater importance is that he realized that these ideas are generalizable far beyond considerations of borders and coastlines to almost anything that can be measured, even including times and frequencies. Examples include our brains, balls of crumpled paper, lightning, river networks, and time series like electrocardiograms (EKGs) and the stock market. For instance, it turns out that the pattern of fluctuations in financial markets during an hour of trading is, on average, the same as that for a day, a month, a year, or a decade. They are simply nonlinearly scaled versions of one another. Thus if you are shown a typical plot of the Dow Jones average over some period of time, you can’t tell if it’s for the last hour or for the last five years— the distributions of dips, bumps, and spikes is pretty much the same, regardless of the time frame. In other words, the behavior of the stock market is a selfsimilar fractal pattern that repeats itself across all timescales following a power law that can be quantified by its exponent or, equivalently, its fractal dimension." Fractals thus increase the number of Euclidean dimensions by one. This is an extraordinary finding that is hard to wrap one's head around. "[A] crinkly enough line that is space filling can scale as if it’s an area. Its fractality effectively endows it with an additional dimension. Its conventional Euclidean dimension, discussed in chapter 2, still has the value 1, indicating that it’s a line, but its fractal dimension is 2, indicating that it’s maximally fractal and scaling as if it were an area. In a similar fashion an area, if crinkly enough, can behave as if it’s a volume, thereby gaining an effective extra dimension: its Euclidean dimension is 2, indicating that it’s an area, but its fractal dimension is 3." That's enough to contemplate for now. We will see how this all applies to biological systems and companies in Part II . . .
0 Comments
Leave a Reply. 
I have always been curious about the way the world works and the most elegant ideas for describing and explaining it. I think I have found three of them. I was very fond of James Burke's Connections series that explored interesting intersections between ideas, and hope to create some of that magic here. Archives
February 2019
Categories
All
