Benoît Mandelbrot’s Asymmetry
Goethe arrives at the botanical garden in Palermo and reaches the conclusion that the answers are to be found in the world of plants. It is here that he conceives the idea of redefining the concept of metamorphosis:
“It was revealed to me that in that organ of the plant which we ordinarily call the leaf lies the true Proteus, which can conceal itself and reveal itself in every possible form. Forwards and backwards, the plant is always nothing but leaf, so inseparably bound up with the future germ that we are not even permitted to think of them separately.”
And after writing Versuch, die Metamorphosen der Pflanzen zu erklären (The Metamorphosis of Plants) in 1790, Goethe takes a position contrary to Isaac Newton’s conception in Zur Farbenlehre (Theory of Colours) in 1810, attempting to demonstrate the unity of spirit and matter in natural phenomena.
Newton, by contrast, approaches the natural world through a mechanistic framework centred on the law of gravity. Newton’s influence extended far beyond physics. Roger Babson, the founder of the Gravity Research Foundation, in turn incorporates Newton’s third law into economics and creates the Babsonchart of Economic Indicators, which helps him predict the stock market crash of 1929.
And yet it is another mathematician who succeeds in explaining what Goethe attempts to grasp qualitatively and Newton quantitatively: Benoît Mandelbrot.
Trees offer perhaps the most intuitive example of fractal structures: asymmetrical patterns that repeat across scales.
A fractal is, by definition, “a fragmented or broken geometric figure that can be divided into parts, each of which is (at least) a miniature copy of the whole”. The term, introduced by Benoît Mandelbrot in 1975, derives from the Latin fractus, meaning “fractured” or “broken”.
But if we have spoken of trees to illustrate what fractals are, we need not confine ourselves to this example. Most forms of life in the plant kingdom exhibit fractal structures. And these are far more robust than any structure created by human beings. The plant kingdom may appear fragile – its leaves fall, only to reappear in spring. Yet this is precisely what demonstrates its resilience and continuity.
Given this robustness, it is reasonable to ask whether stability in more complex organisms likewise emerges from the principles governing their structure and function.
In this way, stability emerges from the interaction of different components, allowing a degree of harmony and coherence to develop. If, however, these components are based on different – or even erroneous – systems and principles, it will be impossible for the society in question to display the same robustness as the plant kingdom.
To summarise, human beings have climbed to the top of the food chain and now look down upon the other species. Yet a historical analysis reveals that there is no form of social organisation that has endured as long as plant species have. Their long-term resilience appears closely linked to the fractal patterns that characterise their growth.
Yet alongside the formulation and observance of principles, such as those set out in the Code of Hammurabi, there is something else that is equally important to survival: adaptability.
For when we find ourselves within an artificially created sequence, the principles must be modified rapidly in order to preserve robustness. In other words, principles lead to equilibrium – but, importantly, an equilibrium that is constantly in motion. That is why we have also discussed adaptability.
Nassim Nicholas Taleb dedicates his most important book, The Black Swan, to Mandelbrot.
Randomness is neither a theory nor a concept, but rather the world’s capacity to generate unpredictable events and situations, Taleb suggests in his first book, Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets. He explains why it is called Mandelbrotian, or fractal, randomness. The subject had previously been studied by Pareto, Yule and Zipf, but Mandelbrot was the one who connected the dots, establishing the link between randomness and geometry and drawing the natural conclusion.
It is worth adding, however, that Charles Darwin – the man credited with the idea that it is the fittest who survive – writes in the introduction to On the Origin of Species that the facts he presents are not necessarily original, but that their implications seemed to him “interesting”.
Taleb points out that triangles, squares and circles seem to exist more readily in the minds of architects, builders and teachers than they do in nature. Mountains are not triangles or pyramids; trees are not circles. Mother Nature never attended geometry lessons. The geometry of nature is not Euclidean geometry.
This blindness is akin to the misconception that randomness is synonymous with casinos.
Photo source: pxhere.com.
