The maverick mathematician: Benoît Mandelbrot and the stunning beauty of the fractal universe

The maverick mathematician: Benoît Mandelbrot and the stunning beautyof the fractal universe

by N. Lesmoir-Gordon, United Kingdom

Benoît Mandelbrot was born in Warsaw on 20th November 1924. When he was 12 and with the threat of war looming the family had the foresight to quit Poland and escape to France. He completed his education at the University of Paris with a PhD on Zipf’s law, after which he left France to take up residence in the USA. With his development of fractal geometry the visionary Benoît Mandelbrot has given science a new language to describe roughness and nature. Fractal geometry is an entirely innovative way to study and describe the real world. The discipline has opened up a host of new directions in science. Fractals seem intimately connected to the concepts of beauty and elegance. Amazingly, until recently there was no word to describe the familiar shapes of nature. Now we can see that there are fractals everywhere. Now we can view the universe through fractal eyes. On 1st March 1980 at IBM’s Thomas J. Watson Research Center in upstate New York, Benoît Mandelbrot discovered the now iconic Mandelbrot set. With its thrilling visualizations and infinite nature, it brought the world of mathematics back into public consciousness. And without fractals, your smartphone would be silent…

Medicographia. 2012;34:354-364 (see French abstract on page 364)

“The Mandelbrot set is one of the most beautiful and remarkable discoveries in the entire history of mathematics.” Sir Arthur C. Clarke. Image courtesy of Nigel Lesmoir-Gordon.

On 1st March 1980 at IBM’s Thomas J. Watson Research Center in upstate New York Benoît Mandelbrot discovered the now iconic Mandelbrot set. With its thrilling visualizations and infinite nature it brought the world of mathematics back into public consciousness. British science fiction author, inventor, and futurist, Sir Arthur C. Clarke, famously known for his book and screenplay “A Space Odyssey,” and one of the “Big Three” alongside Isaac Asimov and Robert Heinlein, hailed the Mandelbrot set as “one of the most beautiful and remarkable discoveries in the entire history of mathematics, […] one of the seven wonders of the world,” an opinion emphatically echoed by physicist John Archibald Wheeler, protégé of Niels Bohr and friend of Albert Einstein, quoted as saying: “No one will be considered scientifically literate tomorrow, who is not familiar with fractals.”

In my film The Colors of Infinity, which Clarke presents, he says: “No matter how much we magnified the Mandelbrot set, a million times, a billion times—until the original set was bigger than the entire Universe—we would still see new patterns, new images emerging, because the frontier of the Mandelbrot set is infinitely complex. And when I say infinitely, I really mean that. Most people when they say infinitely, they mean, ‘oh, only, rather big.’ But, this is really infinity!”

Fractal mountainscape. Photo Nigel Lesmoir-Gordon.

Fractal winter trees. Photo Nigel Lesmoir-Gordon.

Clouds are not spheres…

The seeds of this discovery were in fact sown decades before the Mandelbrot set was first seen. In Paris, in 1917, two French mathematicians, Gaston Julia, a student of Henri Poincaré, and Pierre Fatou, published papers connected with complex numbers. The results of their endeavors eventually became known as Julia sets. Although Julia and Fatou never saw a Julia set! They could only guess at them. And it would not be until the advent of modern computers that Julia sets could be seen for the first time. It was Mandelbrot’s uncle Szolem, who initially directed him to the work of Julia and Fatou.

The world that we live in is not naturally smooth-edged. The real world has been fashioned with rough edges. It’s a wiggly world! Smooth surfaces are the exception in nature. Throughout recent human history mankind has accepted a geometry that only describes shapes rarely—if ever—found in the real world. The geometry of Euclid describes ideal shapes—the sphere, the circle, the cube, the square. Now these shapes do occur in our lives, but they are man-made and not natural.

« Clouds are not spheres Mountains are not cones Coastlines are not circles Bark is not smooth Nor does lightning travel in a straight line”
Benoît Mandelbrot

Benoît Mandelbrot was a visionary, who has given science a new language to describe roughness and the way things really look, feel, and sound. Fractal geometry is an entirely new way to study and describe the natural world. The discipline has opened up a host of new directions in science.

Fractals are aesthetically pleasing, frequently revealing stunning beauty in the most subtle ways. The Mandelbrot set defies verbal description. Fractals seem intimately connected to the concepts of beauty and elegance. Amazingly until recently, there was no word to describe the familiar shapes of nature. Now we can see that there are fractals everywhere. Now we can view the universe through fractal eyes.

Benoît Mandelbrot’s early years in France and the first inkling of the fractals

Benoît Mandelbrot was born in Warsaw on 20th November 1924. Benoît showed an early delight in geometry. He excelled at chess, though he admits that he did not think the game through logically, but in a geometric fashion. When Benoît was 12, and with the threat of war looming, the family had the foresight to quit Poland and escape to France. When he turned 13 his uncle Szolem was appointed a professor of mathematics at the Collège de France in Paris, proving that a career in this subject was a real possibility. In 1937 Benoît attended the Lycée, but when Paris fell to the Nazis in 1940 the family fled further south to Tulle. Benoît went to Lyon for a year of post–high school study. It was here that he first discovered that he had a remarkable and extraordinary visual ability. He found that he could transform a mathematical problem into shapes.

Benoît Mandelbrot as a schoolboy in Tulle in 1940. Photo courtesy of Aliette Mandelbrot.

After the liberation of France at the end of World War II Benoît returned to Paris to prepare for his examinations. Extraordinarily, he ended up passing the entrance exams without the usual two years of preparation!

Benoît started to study mathematics at the École Normale Supérieure, which trained university and high-school professors. But mathematics was dominated then by the Bourbaki group—a group of mainly French mathematicians founded in 1935—who wrote highly influential books on advanced mathematics based on set theory under the collective madeup pseudonym of Nicolas Bourbaki. (The group is still extant today, and true to its tradition, the names of currentmembers are kept secret). Ironically Benoît’s uncle Szolem was one of the founders of the movement. The group sought to keep mathematics in the realm of the abstract and separate from real life. They based their ideas on the precepts of Plato. After just two days at The École Normale Supérieure something very significant happened to Benoît. He looked around and asked, “What am I doing here?” He spent a day agonizing and then resigned. This was a decision that set in train the path his life was to follow. He had been accepted into the most exclusive school, which would certainly have given him a guaranteed future. The alternative was to go to the École Polytechnique, which had a much less cohesive plan.

When he finished at the Polytechnique he moved to Caltech where he studied aeronautics, encountering the daunting complexity of turbulence. After Caltech Benoît went back to France to study at the University of Paris. He was a disappointed man “because the miracle of finding something special to do in America had not happened.”

He was looking for a PhD subject, which was not an easy task for Benoît! When visiting his uncle Szolem Benoît he asked him for something to read on the metro. Szolem reached into his waste-paper basket and pulled out a document. Benoît recalled him saying, “Somebody sent me this paper, which is crazy, but you like crazy things. So here it is.” Benoît took the paper with him. It was on Zipf’s law, which became the subject of his thesis. Once completed, he became a lecturer at the University of Lille and then went to the Massachusetts Institute of Technology to focus on information theory. After about six months at MIT he decided to move on to Princeton’s Institute for Advanced Study where he was sponsored by John von Neumann. Von Neumann introduced Benoît to the idea of the Hausdorff-Besicovitch dimension—the revelation that there were phenomena that existed outside onedimensional space, but in somewhat less than two dimensions. Benoît adopted the Hausdorff-Besicovitch dimension on the spot. It was an almost ubiquitous tool and a special example of the wider concept of fractal dimension, which was to come.

IBM and the price of cotton

In 1958, he went to IBM as a faculty visitor for the summer. It was there that his ideas started to bear fruit and he decided to stay at IBM. The corporation gave Mandelbrot the funding, the facilities, a research team, and the mental space in which to work. The powers-that-be at IBM had vision, unlike the reactionary management of mainstream academia. It was a gamble, but it paid off. In just three years he had made his first major discovery.

Benoît was visiting Harvard to give a seminar in economics. His host had a drawing on the blackboard and Benoît asked, “How come you have the same drawing as I’ll be using in my lecture?” His host replied, “This drawing is on the behavior of cotton prices.” Benoît decided on the spot that cotton prices deserved extremely careful attention.When he got back to IBM he made a number of tests and those tests were successful. In a very short time he had a model for the variations of speculative prices. It was innovative and daring.

Economics, like most of the younger sciences, was attempting to imitate physics. In physics, the motion of a body must be continuous. It can’t just jump from one place to another, but economics can be very jagged and irregular and not continuous. In economics, Benoît argued, there’s no reason why prices should be continuous. If a piece of news arrives, a share price can go from, say, £100 to £10 or to £3. Benoît accepted the idea of discontinuity in his model of prices and strangely that idea had not occurred to anyone else.

Building up the Sierpinski gasket. Image courtesy of Nigel Lesmoir-Gordon.

2-Dimensional Cantor set (5 iterations). Image courtesy of Nigel Lesmoir-Gordon.

Menger sponge (3-D extension of the Cantor set and Sierpinski carpet). ©

Carpets, gaskets, and sponges: a bizarre world of arcane mathematical objects

Right at the beginning of Benoît’s career at IBM he tackled a practical issue, which directly involved and concerned his employers. Inside the company, data were being lost or corrupted when passing between computers by random noise bursts, which they could not get rid of or predict. He used the same tools he had learned as a student and which had been introduced into mathematics a hundred years before as being so-called “pathologies.” Around 1900 objects like the Cantor set, the Sierpinski triangle, the Koch curve and the Menger sponge were known by a few, but they were mostly pure mathematicians, who were convinced that those objects or shapes proved that pure mathematical thought was quite separate from reality because they believed that those ideas had no practical implementation in nature. Benoît found that to be quite the contrary. Benoît grasped that the noise in the IBM system was deeply embedded in nature and impossible to drive out. He instantly doomed any attempts to predict, suppress, or eliminate it. This noise issue was an early and very clear example of the strange logic of fractals—the unruly collection of irregular geometric phenomena that only Benoît comprehended.

The Polish mathematician Vaclav Sierpinski introduced his fractal in 1916. The Sierpinski triangle or gasket is obtained by starting with a filled equilateral triangle, which is then divided into four smaller equilateral triangles, of which the middle one is removed, leaving a triangular hole. The three remaining filled equilateral triangles are then divided in exactly the same fashion, so that three smaller triangular holes appear. Conceptually, we can repeat this process indefinitely, at smaller and smaller scales, reaching, in the limit, Sierpinski’s gasket. The 3-dimensional version (illustrated here) constitutes a “Sierpinski pyramid.”

Georg Cantor’s quest for the meaning of continuity led him in 1883 to the set that is now named after him. It was one of the first fractals to be studied mathematically. The Cantor set: take a line and remove the middle third leaving two equal lines. Likewise remove the middle thirds from each of these two lines. Repeat this process an infinite number of times, and you are left with the Cantor set.

The snowflake-like Koch curve.
Image courtesy of Nigel Lesmoir-Gordon.

Austrian mathematician Karl Menger first described what became known as the Menger sponge in 1926. Menger was working in topology and was trying to develop a definition of dimension. It turns out that the sponge, which he created, is in fact a 3-dimensional fractal. The sponge has an infinite surface, but contains zero volume. All these shapes are incredibly complicated to describe in Euclidean terms, yet share an affinity with many shapes of modern mathematics, displaying an endless series of motifs within motifs repeated at all scales.

One such shape is the snowflake curve, devised by Helge von Koch in 1904. He defined the curve as the limit of an infinite sequence of increasingly wrinkly curves. The finished curve is infinitely long, despite being contained in a finite area. What von Koch did not realize however was that such curves with infinite length would make ideal models for the shapes of the real world, like coastlines and arteries.

How long is the coast of Britain?

Acting like an 18th-century naturalist, Benoît scoured through forgotten and obscure journals in his quest for insight. Fortunately he uncovered the work of an eccentric and unremembered mathematician called Lewis Fry Richardson. Benoît had struck a rich seam and he knew it. The library at IBM was discarding books that nobody ever looked at. Fortunately Benoît was friendly with the librarian and she regularly told him when a truck was going to come and pick up discarded books. Benoît would look through them. Most of them were the proceedings of meetings that nobody was at all interested in and were only suitable for pulping. There was one periodical, however, which he opened more or less at random where he saw the name Lewis Fry Richardson. This name was already known to him through Richardson’s work on turbulence in the 1920s. Richardson was a great hero of Benoît. He was a very strange character. He was a very great man in many ways and was a true English eccentric.

Richardson’s coastline paradox.

The length of the coast of Britain depends on the scale of measurement. In this scale model of Britain, ruler A measures 200 km; ruler B is 100 km, and
ruler C is 50 km; at each measurement, the two ends of the ruler must touch the coast. The measurements found here are 2400, 2800, and 3450 km: empirical evidence shows that instead of eaching a finite number representing the true length of the coastline of Britain, the length of the coastline continues to increase and tends toward infinity as the length of the ruler gets smaller and tends toward 0. © Acadac/Creative Commons.

What really struck Benoît were Richardson’s ideas on the lengths of coastlines. What appeared to be a simple question of geography exposed some of the essential features of fractal geometry. The closer you come, the longer the coastline becomes. And that’s where the idea of fractional dimension really came alive. Benoît found that his theory worked beautifully with coastlines.

He produced a paper called “How Long is the Coast of Britain?—Statistical Self-Similarity and Fractional Dimension” (Science. 1967;156:636-638). In 1973 he started to develop an algorithm for creating real-looking coastlines. He was preparing to publish his first book in France in 1975 and when the book was close to completion he knew he needed a name for his creation. One day he was thumbing through his son’s Latin dictionary, idly looking for a word he might adopt. He came across the word “fractus” which meant broken-up, fractional, irregular. He coined the word fractal. Though the idea of fractals in the form of iteration and self-similarity is an ancient one, it took this wanderer-by-choice to give the idea a name. Once named, the field took off.

The Mandelbrot set

On March 1st 1980 at IBM Benoît discovered what is now called the Mandelbrot set. The irony is that Mandelbrot’s uncle Szolem had strongly suggested that Benoît look at the papers which Julia and Fatou had published in 1917, with a view to making them the subject of his PhD. Reexamining the maths behind the Julia sets again in 1980 using a computer led Mandelbrot directly to the discovery of the Mandelbrot set.

During the First World War, Julia and Fatou had studied the rational mappings of the complex plane. They had also looked at the process of iteration. Although their work remained largely unknown to most mathematicians, we now know that this was because without modern computer graphics it was almost impossible to communicate their subtle ideas. Self-similarity was also well-known to Julia and Fatou. It would not be until the advent of modern computers though that Julia sets could be seen in their full glory.

Mathematician Benoît Mandelbrot in the Computer Art Gallery at the SIGGRAPH Conference in San Francisco, July 24, 1985.

© Roger Ressmeyer/Corbis.

Benoît: “For me, the first step with any difficult mathematical problem was to program it and see what it looked like. We started programming Julia sets of all kinds. It was extraordinary great fun! At one point we became particularly interested in the Julia set of the simplest possible transformation: z goes to z squared plus c, .”

The Julia sets for this mapping depend only on the value of the parameter c.When c is small, they are simple loops, like wrinkled circles. For large values of c, the fractal consists of innumerably many discrete points, spread out and dust-like. When the first picture of the set rolled off the printer, Benoît and his colleagues’ first reaction was that there must be some mistake, a fault in the program perhaps. The picture was extremely strange and unexpected. Over a period of weeks, working late into the night in the basement of a laboratory in Harvard University, Benoît and his assistant explored the astonishing new world they had discovered. By feeding new coordinates into their program, they made successively deeper zooms into the boundary of the set. One of the most striking discoveries was that buried deep within the seething froth of the object’s boundary, were tiny replicas, almost identical to the original set. Zooming further into these baby Mandelbrot sets they encountered further variations of these very same patterns with added frills and embellishments.

The initial, ghost-like image of the Mandelbrot set (left) and the ultimate refined version (right). Image courtesy of Nigel Lesmoir-Gordon.

Benoît recalled: “We made many pictures of it. The first one was very rough. But the very rough pictures were not the answer. Each rough picture asked a question. So we made another picture, another picture. And after a few weeks we had this very strong, overwhelming impression that this was a kind of big bear we had encountered!” Benoît and his team pushed the computers of the day to their extreme limits to refine their images. The beauty of the set, which revealed itself, was all the more extraordinary and exciting because it was totally unexpected.

There is an interesting parallel here with the equation that almost everybody is familiar with: E = mc². Albert Einstein’s equation, which states that matter and energy are equivalent to each other. That was a very simple equation, but with very far-reaching consequences. And the equation for the Mandelbrot set is equally simple z ↔ z² + ci. The letters in the Mandelbrot equation, though, stand for numbers, unlike those in Einstein’s equation where they stand for the physical quantities: mass, velocity, and energy. The Mandelbrot set equation numbers are coordinates, positions on the plane, defining the location of a spot. Another difference from Einstein’s equation—and a very important one—is the two-way arrow in the middle.

z ↔ z² + ci

It is a kind of two-way traffic sign. The numbers flow in both directions, constantly feeding back on themselves. This process of going round and round a loop is called iteration. It is rather like a dog chasing its own tail: the output of one operation becomes the input of the next and so on and so on. When the Mandelbrot equation is given a number representing a point and that number is iterated through the equation one of two things happen. Either the number gets bigger and bigger and runs out to infinity or it shrinks down to zero. Depending on which happens, the computer then knows where to draw a boundary line. So, what is obtained from this basic iteration is a map, dividing this world into two distinct territories. Outside it are all the numbers that have the freedom of infinity. Inside it, numbers that are prisoners, “trapped and doomed to ultimate extinction.”

No matter how much we magnify the set, a million times, a billion times until the original set is bigger than the entire Universe— we would still see new patterns, new images emerging because the frontier of the Mandelbrot set is infinitely complex. This set is the most famous fractal of all. It’s not easy to describe it visually. It looks like a man, a cat, a cactus, or a cockroach. It has little bits and pieces that remind us of almost anything we can see out in the real world, particularly living things. There is indeed an infinite variety in the Mandelbrot set just as there is in the world of nature. We see shapes that remind us of elephant trunks, tentacles of octopi, sea horses, and compound insect eyes. There is certainly some connection between the Mandelbrot set and the way nature operates. The Mandelbrot set is one of the few discoveries of modern mathematics to be assimilated by society as a whole. It has appeared on mugs, T-shirts, record sleeves, and in pop videos, even in cinema and television commercials.

Mandelbrot set detail, showing spike and repetition of successively smaller versions of the original version, stretching away into infinity. Image courtesy of Nigel Lesmoir-Gordon.

Clouds, Hokusai, and smartphone antennas

When Benoît was exploring this set he never felt that he had invented it. He never felt that his imagination was rich enough to invent all those extraordinary things. He knew he was making a discovery. “They were simply there, even though nobody had ever seen them before. It’s marvelous,” he said, “that such a very simple formula explains all these very complicated things! So the goal of science is starting with mess to explain by simple formulas. It’s the kind of dream of science. And in this case the dream was implemented in a fantastic fashion.” Though the stunning beauty of the images the Mandelbrot set generates appeal to us on many levels, the psychological reasons for this appeal are still a mystery. Perhaps there is some structure deep in the human mind that resonates to the patterns in the set. Like the Mandelbrot set, life is richest on the boundaries between land and sea. Between earth and sky. Consciousness and life itself exist at the edge of chaos. Nature often finds the same solution to many different problems. Like how to drain water from the land into the oceans, and how to get blood from our hearts to our fingertips and back again. And the templates that nature uses are fractals. Clouds look the same at all scales. It is impossible to determine the size of a cloud from a photograph of it. Clouds have now been shown to have the same dimension over 10 orders of magnitude, making them the most uniform fractal objects on the planet.

“Clouds are not spheres….” No matter how magnified, the contour of clouds remains
similar, and the scale of the photo cannot be determined.

“Clouds over Paris.” Photo courtesy of Marc Bertrand, July 2012.

We may expect fractals to be found more often in nature, and Euclidean shapes more often in manufacture, but fractal designs have always appeared in architecture, whether for motives of utility, aesthetics, or the desire to mimic nature. They can be found in Islamic, Roman, Egyptian, Celtic, and Japanese paintings, in carpets and sculptures and in the work of Escher, Latham, and Hokusai. In the 1830s and 40s the Japanese artist Katsushika Hokusai produced marvelous woodblock prints, which clearly captured fractal aspects of nature. This woodblock “In the hollow of a wave off the coast of Kanagawa” from his Thirty-Six Views of Mount Fuji, is familiar to many.

Over his long life Hokusai observed that nature shows patterns that were then nameless, but which we now call self-similar, and these are frequently reflected in his work. Hokusai saw these patterns in nature and to us they certainly look like fractals related to the Mandelbrot set. Perhaps the resemblance to nature accounts for some of our pleasure in seeing mathematical fractals.

Benoît found his most enthusiastic acceptance first among applied scientists working with oil, rock, or metals. Fractals have become an organizing principle among physicist in the study of polymers and in the design of smartphone antennas. Earlier mobile phones had conspicuous antennas that had to be pulled out: now they are invisible, tightly packed inside your smartphone, folded according to a symmetrical and self-repetitive pattern at all scales, allowing their size to be shrunk up to 4 times and multiband and broadband functioning, without the need for multiple antennas. In addition, there is no need for additional components, as the fractal design creates a virtual combination of capacitors and inductors.

Mount Fuji seen in the hollow of a wave off the coast of Kanagawa, by Hokusai.

From: Thirty-Six Views of Mount Fuji. 1826/1833.© Corbis.

Fractal design. © Bill Ross/Corbis.

The way that a virus binds to a living molecule relies on the fractal structuring of both components. Fractal geometry is helping in the detection of cancerous cells in vitro and has provided the answer to the unusually long incubation period of the AIDS virus. Fractal growth methods are now used to model marine organisms such as sponges and coral. Scaling is also of interest in the study of vegetative ecosystems, earthquake data, and in the behavior of density-dependent populations.

Benoît was always willing to do things that others looked down on and dismissed. He had faith in his own insights and for many years it was not shared. But that did not daunt him. It was not easy to stick to it and to persist, but he always held fast to his vision and that vision has subsequently been wholly justified. Many people might have taken a glancing look at these things and when they did not yield or when others said it was nonsensical they might have gone on into other things. But not Benoît. He had vision and persistence and it is those qualities that made all the difference.

Fractal antenna. US Patent and Trademark Office. Patent 7088965.

Fractal geometry is the language of nature, of the familiar and apparently random forms like trees, coastlines, rivers, lightning, the human body, a winding coastline, the branching structure of a fern, the spacing of stars in the night sky. With fractal geometry, the maverick Benoît Mandelbrot has given us a new language, which is applicable throughout all the sciences, and is one which has a mind-opening effect on most people who come across it. This new language is changing our lives and the world of scientific endeavor. _