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Benoît Mandelbrot: ad infinitum

Benoît Mandelbrot: ad infinitum

Why is geometry often described as “”cold” and “”dry?” One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.

Benoît Mandelbrot, a name that has become synonymous with the beauty of complexity, was a mathematical maverick whose work has left an indelible imprint on the tapestry of scientific thought. Born into a Lithuanian-Jewish family in Warsaw, his early love for geometry and his uncanny ability at chess laid the groundwork for his future pursuits. Mandelbrot’s life was shaped by the winds of history, with his family moving to France in 1936 to escape the rise of Nazism in Germany. This move marked the beginning of Mandelbrot’s exceptional academic journey, which would see him traverse across continents and disciplines, leaving a trail of innovative ideas in his wake.

Mandelbrot’s intellectual lineage can be traced back to his two uncles, one of whom, Szolem Mandelbrojt, was a Professor of Mathematics at the Collège de France. Szolem Mandelbrojt was an admirer of Hardy and his philosophy of mathematics. This admiration, however, did not resonate with the young Mandelbrot, who developed a visceral reaction against pure mathematics. The influence of this early rejection would prove to be significant in shaping Mandelbrot’s distinct approach towards mathematics, one that was rooted more in the real world than in abstract platonic ideals.

His mathematical philosophy was greatly influenced by his atypical education and the world-shaping events of his formative years. The war and his personal experiences led him to appreciate the inherent complexity and chaos in the natural world. This appreciation, in turn, informed his view of mathematics as a tool for understanding, rather than abstracting, the complexity of nature.

Mandelbrot’s academic journey is as unconventional as his ideas. He attended the Lycée Rolin in Paris until the start of World War II. After the war, Mandelbrot attended the École Normale in Paris, but left after just one day. His brilliant performance in the entrance examinations of the École Polytechnique led him to study there under the direction of Paul Lévy. From there, he went to the California Institute of Technology, and later to the Institute for Advanced Study in Princeton, sponsored by none other than John von Neumann. It was during these years that Mandelbrot’s unique insights into mathematical problems began to take shape.

In 1955, Mandelbrot returned to France and worked at the Centre National de la Recherche Scientifique. However, he found the dominance of the French foundational school of Bourbaki not to his taste, and in 1958, he moved permanently to the United States. He began a fruitful collaboration with IBM, working as an IBM Fellow at their world-renowned laboratories in Yorktown Heights, New York. This move to IBM was a turning point in Mandelbrot’s career, providing him with an environment that encouraged him to explore a wide variety of different ideas.

Mandelbrot’s philosophical underpinnings were rooted in a deep-seated aversion to the abstract and the absolute. He was not content with mathematics that dwelt in the realm of the ideal; he was drawn instead to the irregular, the complex, and the chaotic. It was this philosophical standpoint that led him to chart his own course in mathematics, one that was starkly different from the path laid out by his uncle and the prevailing French school of thought.

Mandelbrot’s philosophy found expression in his unique approach to mathematics, which was highly geometrical. Unlike many mathematicians of his time, he did not see geometry as a sterile field of cold and dry shapes, but as a vibrant language capable of describing the complex forms of the natural world. His work was grounded in a belief that the abstract, idealized world of Euclidean geometry was insufficient to describe the rough-edged, wrinkled, crinkled, and irregular shapes found in nature.

One of the key features of Mandelbrot’s philosophical approach was his embrace of the power of visualisation. He recognised early on that the visualisation of mathematical concepts could provide unique insights that were often obscured by formal mathematical notation. This belief led him to become one of the pioneers in using computer graphics to visualise complex mathematical ideas, a practice that has now become standard in the field.

Mandelbrot’s philosophy also placed a high value on the interconnection between different fields of science. He was not content with remaining within the confines of pure mathematics but sought to make meaningful contributions to a wide range of scientific fields. This interdisciplinary approach was not just a reflection of his intellectual curiosity, but also a manifestation of his belief in the universal applicability of mathematical ideas.

Mandelbrot set © Ted Burke

Mandelbrot’s most significant contribution to the field of geometry, and indeed to the world of science, was the creation of fractal geometry. This groundbreaking work began with his study of Julia sets, named after the French mathematician Gaston Julia. Julia’s work, which dealt with complex numbers and iterative functions, would form the basis of Mandelbrot’s revolutionary ideas about fractal geometry.

In the 1970s, with the aid of computer graphics, he was able to visualise Julia sets and show how they were a source of some of the most beautiful fractals known today. This work led him to develop not only new mathematical ideas, but also some of the first computer programs to print graphics. It was this combination of mathematics and computer science that enabled Mandelbrot to bring the intricate beauty of fractals to the world.

Mandelbrot’s work on fractals was first elaborated in his book “Les objets fractals, form, hasard et dimension” (1975), and more fully in “The fractal geometry of nature” in 1982. In these works, Mandelbrot introduced the world to the concept of fractals, geometric shapes that are self-similar at different scales. His revolutionary idea was that these complex shapes could be generated by simple mathematical formulas, offering a powerful tool for understanding the complex forms found in nature.

One of Mandelbrot’s most famous contributions is the Mandelbrot set, a set of complex numbers defined by a simple iterative algorithm. The boundary of the Mandelbrot set, when visualised, reveals an incredibly intricate fractal structure. This set, which is now named after him, is one of the most famous images in the world of mathematics, and serves as a testament to Mandelbrot’s belief in the beauty of complexity.

Mandelbrot’s work extended far beyond the confines of pure mathematics. He applied his ideas about fractals to a wide range of fields, from physics to economics to physiology. His work has had far-reaching implications, opening up new avenues of research and providing new tools for understanding the complex patterns found in these diverse fields.

For a complex natural shape, dimension is relative. It varies with the observer. The same object can have more than one dimension, depending on how you measure it and what you want to do with it. And dimension need not be a whole number; it can be fractional. Now an ancient concept, dimension, becomes thoroughly modern.

― Benoît B. Mandelbrot, The (Mis)Behavior of Markets

By introducing the concept of fractals, Mandelbrot provided a new way of conceptualizing and visualizing complex geometric structures, thereby expanding our cognitive toolkit for understanding the geometric world around us.

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Fractal geometry, with its inherent self-similarity, presents a challenge to traditional cognitive models that rely on Euclidean geometry. This has sparked new research into how our brains process fractal patterns. The concept of fractals, with their infinite complexity and repetition at all scales, offers a powerful metaphor for understanding the recursive nature of cognitive processes.

Mandelbrot’s work has also contributed to the field of cognitive geometry, which investigates how geometric structures can be used to model cognitive processes. His pioneering work on fractals has provided a rich source of geometric structures that can be used to model complex cognitive phenomena. For example, fractal patterns have been used to model the temporal dynamics of cognitive activities such as memory recall and decision-making, his work finding also applications in the study of neural networks, both biological and artificial.

The complex branching structures of neurons bear a striking resemblance to fractal patterns, from which a few notable examples are:

  1. Understanding neural connectivity: the fractal weaving of the natural branches is balancing the ability of neurons to connect with their neighbors to form natural electric circuits while balancing the construction and operating costs of the circuits.
  2. The fractal-hyperbolic geometry of networks:  The fractal geometry of networks enables casting the self-similar symmetries underlying the organization of complex systems under the three pillars of scaling, universality, and renormalization.
  3. Fractal patterns preferred by children under 3 years: The discovery was made among children raised in an environment of Euclidean geometry, such as buildings with rooms built with straight lines in a basic non-repeating way, according to the study’s primary author Kelly E. Robles.
  4. Understanding the fractal brain: monitoring signals from a single neuron may be as effective as gathering information from numerous neurons at once using large, expensive arrays of electrodes.

With all this and much more, Mandelbrot’s work has had a broader impact on our cognitive understanding of the world around us. The complex, irregular shapes that fractal geometry describes are ubiquitous in nature, from the branching of trees to the shape of coastlines, reminding us that the world is not composed of simple, regular shapes, but of complex, irregular forms that can nevertheless be understood through the lens of mathematics.

This is a natural conclusion of his approach to mathematics, a radical departure from the Platonic idealism that had long dominated the field. Instead of seeking truth in the realm of perfect forms and abstract concepts, Mandelbrot looked for it in the rough, irregular shapes of the real world. His mathematics was not a pursuit of absolute certainty, but an exploration of uncertainty and complexity. This approach aligns closely with the philosophy of science known as scientific realism, which posits that our scientific theories do not capture the absolute truth about the world, but are simply models that approximate reality.

Also, in line with the philosophical currents of postmodernism, Mandelbrot’s work embraces the complexity, uncertainty, and diversity of the world. His fractals challenge the binary logic of Euclidean geometry, introducing a new kind of geometric logic that can accommodate ambiguity and paradox. This reflects a broader philosophical shift towards a more nuanced understanding of truth, one that acknowledges the multiplicity of perspectives and the indeterminacy of knowledge.

Mandelbrot’s genius lies not only in his mathematical innovation but also in his philosophical insight. He recognized that the world is not composed of simple, regular shapes, but of complex, irregular forms that nonetheless follow certain patterns. His fractal geometry is not just a mathematical tool, but a philosophical framework for understanding the complex patterns of the world around us. His work reminds us that the world is not a machine governed by deterministic laws, but a complex system of interconnected parts, constantly evolving and adapting in response to changing conditions. In doing so, Mandelbrot has not only transformed our understanding of geometry, but also our understanding of the world. His legacy, like the fractals he studied, is infinitely complex and endlessly fascinating.

I found myself in the position of that child in a story who noticed a bit of string and – out of curiosity – pulled on it to discover that it was just the tip of a very long and increasingly thick string… and kept bringing out wonders beyond reckoning.

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