Fractals, Hidden Order Beneath Chaos, and the Story of a Fugitive Who Changed the Statistics of Reality – The Marginalian

I've learned that the lines we draw that contain infinity end up giving out more than they hold.

I have learned that many things in life are better and more beautiful not linear but fractal. Especially love.

In evidence of Aldous Huxley's insight that “all great truths are obvious truths but not all obvious truths are great truths,” the polymathic mathematician. Benoit Mandelbrot (November 20, 1924–October 14, 2010) noted in his most famous and powerful sentence that “clouds are not spheres, mountains are not lumps, coasts are not circles, and barks are not smooth, and lightning does not travel in a straight line.”

An obvious truth that a child can tell you.

A great truth that would throw thousands of years of science into a proper chaos, came from a mind that demolished the mansion of mathematics with the tools of an outsider.

A self-described “nomad-by-choice” and “pioneer-by-necessity,” Mandelbrot believed that “rare nomadic scholars by choice are essential to the intellectual well-being of settled subjects.” He lived as a testament to his discovery of the patterned order underlying many of nature's irregularities – the growing uniformity of nested uniformity repeated in what might be read as chaos.

The transformative insight he came to while studying cotton prices in 1962 became a never-ending vector of revelation for life and destined for infinity, shining with equal illuminating power on everything from the geometry of broccoli flowers and tree branches to the behavior of earthquakes and economic markets.

Mandelbrot needed a name for his discovery – because of this amazing new geometry with its brilliant composition and its amazing disruption of the basic emotions of the human mind, this art of order named in the new mathematical language of chaos. One winter afternoon in his early fifties, poring over his son's Latin dictionary, he said fractus — an adjective from a verb frangere“you will break.” After surviving his early life as a Jewish refugee in Europe by combining languages ​​- his native Lithuanian, then French when his family fled to France, then English when he began his scientific career – he quickly realized the word's echoes in English. to break again fractionconcepts that resonate with the nature of his repetitive jagged geometry. In the dead language of ancient science he carved the vocabulary of a new model for making sense of the living world. Name a fractal was born – binominal and bilingual, both adjective and noun, the same in English and French – and everywhere the universe was new.

In her essay on the art and science anthology of artist Katie Holten, About Trees (public library) – trees are perhaps the most tangible and attractive manifestation of fractals in nature – poetics historian James Gleick describes Mandelbrot's titanic legacy:

Mandelbrot created nothing less than a new geometry, to stand next to that of Euclid – a geometry to show not the ideal forms of thinking but the real complexity of nature. He was an unrepentant mathematician… and he pretended that was fine with him… In various incarnations he taught physiology and economics. He was a non-recipient of the Wolf Prize in physics. Labels didn't matter. It turns out that he was one of the few scientists of the twentieth century who changed, as if with the push of a button, the way we see the world in which we live.

He is the one who made us appreciate chaos in all its glory, noisy, wayward and extraordinary, from the smallest to the largest. He gave a new field of study and coined a new term: “fractal geometry.”

It was Gleick, in his seminal 1980s book Chaos: The Making of a New Science (public library), did for the idea of ​​fractals what Rachel Carson did for the idea of ​​ecology, embedding it in the popular imagination as a scientific concept and as a machine for making sense of reality, full of metaphorical elements that now inhabit all cultures.

He writes of Mandelbrot's success:

Over and over again, the world shows a common anomaly.

[…]

In the mind's eye, a fractal is a way of seeing infinity.

Consider a triangle, each side of which is a meter long. Now imagine a specific revolution – a set of specific, well-defined, easily repeatable rules. Take the middle one-third of each side and attach a new triangle, the same shape but one-third the size. The result is a star of David. Instead of three one-foot segments, the frame of this shape is now twelve four-inch segments. Instead of three points, six.

As you tend to infinity and repeat this change over and over again, you attach smaller and smaller triangles to smaller and smaller sides, the shape becomes more detailed, it looks like a complete complex snowflake – but with amazing and amazing features: a continuous contour that never breaks as its repetitive length increases with the repetition of each point.

If the curve were a straight Euclidean line, its vector would reach the edge of the universe.

It is exciting and mind boggling to bend to this concept. Fractals baffle even mathematicians. But they describe many things and events that happen in the real world, from clouds to capital to cauliflower.

It took an unusual mind acquired by an unusual feeling – an ordinary experience wandering in unusual ways – to arrive at this extraordinary evolution. Gleick writes:

Benoit Mandelbrot is best understood as a refugee. He was born in Warsaw in 1924 to a family of Lithuanian Jews, his father was a clothes merchant, his mother a dentist. Aware of the reality of the world, the family moved to Paris in 1936, drawn in part by the presence of Mandelbrot's uncle, Szolem Mandelbrojt, a mathematician. When war came, the family once again stayed ahead of the Nazis, leaving behind everything but a few suitcases and joining the masses of refugees who blocked the roads south from Paris. At last they came to the city of Tulle.

For a while Benoit traveled as an apprentice toolmaker, dangerously visible for his height and educated background. It was a time of unforgettable things and fear, yet later he remembered a little personal hardship, instead he remembered the times when he became his friend in Tulle and elsewhere school teachers, some of whom were distinguished scholars, themselves trapped in the war. Overall, his learning was disorganized and discontinuous. He said he hadn't learned the alphabet or, more importantly, the multiplication tables in five years. Still, he was gifted.

When Paris was released, he took and passed the month-long École Normale and École Polytechnique exams, despite his lack of preparation. Among other things, the experiment had a groundless examination of the painting, and Mandelbrot found a hidden place to copy the Venus de Milo. In the mathematical parts of the test – exercises in systematic algebra and integral analysis – he was able to hide his lack of training with the help of his geometric intuition. He had realized that, when faced with an analytical problem, he could almost always think in terms of a situation in his mind. Given a shape, he could find ways to transform it, change its symmetries, make it fit. Often his conversion led directly to solving the same problem. In physics and chemistry, where he could not use geometry, he got poor marks. But in math, the questions he could never answer using proper techniques melted away when faced with his interaction with the situation.

At the heart of Mandelbrot's mathematical transformation, it's a nice play of mind, the idea of ​​being like yourself — the fractal curve looks exactly the same as you zoom all the way out and in, at all available magnification scales. Gleick describes the nested multiplicity of self-similarity as a “scale measurement,” a “pattern within a pattern.” In his absolute beauty Chaoshe goes on to explain that the Mandelbrot set, considered by many people to be the most complex thing in mathematics, became “a kind of social symbol of chaos,” confusing our basic ideas about simplicity and complexity, and carving from that delicate confusion a new model of the world.

Couple that with the story of the Hungarian youth who bent Euclid and equipped Einstein with the building blocks of relativity, then revisit Gleick over time with his beautiful reading and meditation on Elizabeth Bishop's ode to the nature of knowledge.