MacTutor
Benoit Mandelbrot, who died on October 14 aged 85, was largely responsible for developing the discipline of fractal geometry -- the study of rough or fragmented geometric shapes or processes that have similar properties at all levels of magnification or across all times.
Before Mandelbrot, mathematicians believed that most of the patterns of nature were far too complex, irregular, fragmented and amorphous to be described mathematically. Euclidian geometry was concerned with abstract perfection almost non-existent in the real world. Mandelbrot's achievement was to conceive and develop a way of describing mathematically the most amorphous natural forms -- such as the shape of clouds, mountains, coastlines or trees -- and measuring them. His work has become the foundation of Chaos theory -- the mathematics of non-linear, dynamic systems.
In the 1960s Mandelbrot, a research fellow with IBM, began a mathematical analysis of electronic "noise" which was sometimes interfering with IBM electronic transmissions, causing errors. Although the nature of these errors was not understood, IBM scientists noted that the blips occurred in clusters; a period of no errors would be followed by a period with many.
Examining these clusters, Mandelbrot noticed that they formed a pattern and that the closer they were examined, the more complex the pattern seemed to become. An hour might pass with no errors, while the next hour might pass with several errors. However, if one of the hours that contained errors was divided into 20-minute sections, there would be 20 minutes with no errors, then 20 minutes with many errors.
On any scale of magnification, Mandelbrot found, the proportion of error-free transmission to error-ridden transmission remained constant. In other words the electronic interference exhibited "self-similarity" at every scale of magnification: each small part, when magnified, reproduced exactly the larger portion.
Mandelbrot began to notice the same phenomenon of "self similarity " in other fields. For example, when he analysed statistical records of cotton prices, he noticed that, while the daily, monthly and yearly pricing of cotton was random, the curves of daily monthly and yearly price changes were identical.
Examining coastlines, he found that while the lines on maps featured bays, they did not feature the small bays that are within the bays, or the small structures within the small bays, and so on.
In a seminal essay entitled How Long Is the Coast of Britain? (1967), Mandelbrot showed that the answer to that question depends on the scale at which one measures it: the coastline grows longer as one takes into account first every bay or inlet, then every stone, then every grain of sand.
These patterns could not be explained by existing statistical methods, so Mandelbrot set about devising a system that would. Through the years that followed, he developed the concept of fractal geometry, codifying the "self-similarity" characteristics of many fractal shapes. (He coined the word "fractal" -- from the Latin verb frangere, "to break" -- in 1975). Mandelbrot's eclectic research ultimately led to a great breakthrough summarised by a simple mathematical formula: . This formula is now named after its inventor and is called the Mandelbrot set. Computer images of fractal shapes became popular on T-shirts and album covers.
First in isolated papers and lectures, then in The Fractal Geometry of Nature (1982), which has sold more copies than any other book of advanced mathematics, Mandelbrot argued that most traditional mathematical and classical geometric models were ill-suited to natural forms and processes. "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," he wrote.
Instead these phenomena and others, including variations in stock market prices, the fluctuations in turbulent fluids, geologic activity, planetary orbits, animal group behaviour, socioeconomic patterns and even music, can be modelled using fractals. The difference between the flower heads of broccoli and cauliflower, for example, can be exactly characterised in fractal theory.
The discipline of fractals came into its own in the computer age. It is now possible to create "fractal forgeries" of mountains, coastlines, trees, clouds, cell growth and other processes which bear an uncanny resemblance to the real thing. Applications range from digital compression in computers to finding the best mix of tyre ingredients, and from modelling turbulence on aircraft wing designs to texturing medical images.
Benoit B Mandelbrot (he awarded himself a middle initial, although it stood for nothing) was born on November 20 1924 in Warsaw, Poland, into a family of Lithuanian Jewish extraction. His father made his living selling clothes while his mother was a doctor, but the family had a strong academic tradition and, as a boy, Mandelbrot was introduced to mathematics by two uncles.
In 1936 Mandelbrot's family emigrated to France where one uncle, Szolem Mandelbrot, a Professor of Mathematics at the Collège de France, took responsibility for the boy's education. Mandelbrot attended the Lycée Rolin in Paris, but was not a good student; it was said that he never learned the alphabet (he could never use a telephone directory, for example), nor his multiplication tables past five.
At the outbreak of war, his family moved to Tulle, a small town to the south of Paris. There, the war, the constant threat of poverty and the need to survive kept him away from school and, in consequence, he was largely self taught.
At this time, French mathematical training and thinking was strongly analytic and abstract, dominated by an influential group of young formalist mathematicians who wrote under the pseudonym of Nicolas Bourbaki; Mandelbrot's uncle Szolem was a member of the "Bourbaki" set. In contrast to their approach, Mandelbrot visualised problems whenever possible, preferring geometry to abstract formalism.
Despite his poor performance at school, he found that he had a quite extraordinary ability to "visualise" mathematical questions and solve problems with leaps of geometric intuition rather than the "proper" established techniques of strict logical analysis. After the war he passed the entrance exams for the École Polytechnique, achieving the highest grade in Algebra by "translating the questions mentally into pictures".
After completing his studies, Mandelbrot went to America and eventually, sponsored by John von Neumann, enrolled at the Institute for Advanced Study in Princeton. He returned to France in 1955 and worked at the Centre National de la Recherche Scientifique, but, finding himself increasingly out of sympathy with the Bourbaki school, he returned to America. In 1958, he was appointed an IBM fellow at the company's laboratories at Yorktown Heights in New York State.
IBM presented Mandelbrot with an environment which allowed him to choose the directions that he wanted to take in his research. They proved to be more diverse, eclectic and far-reaching than anyone could have imagined. His interest in fractals took him into many out of the way disciplines. He became expert in certain areas of linguistics; game theory; aeronautics; engineering; economics; physiology; geography; astronomy and physics. He was also an avid student of the history of science.
As most of the rest of the academic world at this time was heading towards ever greater specialisation, Mandelbrot's concern with such a broad spectrum made him unpopular in some establishment circles, and earned him a reputation as an outsider. Sceptics pointed to his arrogant reluctance to prove his theories, and his preference for "making bold and crazy conjectures".
Nevertheless, in 2004 he predicted the looming global financial meltdown, comparing bankers to "mariners who heed no weather warnings".
After retiring from IBM, Mandelbrot became a Professor of Mathematical Sciences at Yale, and later held appointments as Professor of the Practice of Mathematics at Harvard University; Professor of Engineering at Yale; Professor of Mathematics at the École Polytechnique; Professor of Economics at Harvard, and Professor of Physiology at the Einstein College of Medicine.
He received numerous honours and prizes including the Barnard Medal for Meritorious Service to Science (1985). the Franklin Medal (1986); the Alexander von Humboldt Prize (1987); the Steinmetz Medal (1988); the Nevada Medal (1991) and the Wolf prize for physics (1993).
Benoit Mandelbrot's wife, Aliette Kagan, and their two sons survive him.
© Telegraph Group Limited 2010.
Before Mandelbrot, mathematicians believed that most of the patterns of nature were far too complex, irregular, fragmented and amorphous to be described mathematically. Euclidian geometry was concerned with abstract perfection almost non-existent in the real world. Mandelbrot's achievement was to conceive and develop a way of describing mathematically the most amorphous natural forms -- such as the shape of clouds, mountains, coastlines or trees -- and measuring them. His work has become the foundation of Chaos theory -- the mathematics of non-linear, dynamic systems.
In the 1960s Mandelbrot, a research fellow with IBM, began a mathematical analysis of electronic "noise" which was sometimes interfering with IBM electronic transmissions, causing errors. Although the nature of these errors was not understood, IBM scientists noted that the blips occurred in clusters; a period of no errors would be followed by a period with many.
Examining these clusters, Mandelbrot noticed that they formed a pattern and that the closer they were examined, the more complex the pattern seemed to become. An hour might pass with no errors, while the next hour might pass with several errors. However, if one of the hours that contained errors was divided into 20-minute sections, there would be 20 minutes with no errors, then 20 minutes with many errors.
On any scale of magnification, Mandelbrot found, the proportion of error-free transmission to error-ridden transmission remained constant. In other words the electronic interference exhibited "self-similarity" at every scale of magnification: each small part, when magnified, reproduced exactly the larger portion.
Mandelbrot began to notice the same phenomenon of "self similarity " in other fields. For example, when he analysed statistical records of cotton prices, he noticed that, while the daily, monthly and yearly pricing of cotton was random, the curves of daily monthly and yearly price changes were identical.
Examining coastlines, he found that while the lines on maps featured bays, they did not feature the small bays that are within the bays, or the small structures within the small bays, and so on.
In a seminal essay entitled How Long Is the Coast of Britain? (1967), Mandelbrot showed that the answer to that question depends on the scale at which one measures it: the coastline grows longer as one takes into account first every bay or inlet, then every stone, then every grain of sand.
These patterns could not be explained by existing statistical methods, so Mandelbrot set about devising a system that would. Through the years that followed, he developed the concept of fractal geometry, codifying the "self-similarity" characteristics of many fractal shapes. (He coined the word "fractal" -- from the Latin verb frangere, "to break" -- in 1975). Mandelbrot's eclectic research ultimately led to a great breakthrough summarised by a simple mathematical formula: . This formula is now named after its inventor and is called the Mandelbrot set. Computer images of fractal shapes became popular on T-shirts and album covers.
First in isolated papers and lectures, then in The Fractal Geometry of Nature (1982), which has sold more copies than any other book of advanced mathematics, Mandelbrot argued that most traditional mathematical and classical geometric models were ill-suited to natural forms and processes. "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," he wrote.
Instead these phenomena and others, including variations in stock market prices, the fluctuations in turbulent fluids, geologic activity, planetary orbits, animal group behaviour, socioeconomic patterns and even music, can be modelled using fractals. The difference between the flower heads of broccoli and cauliflower, for example, can be exactly characterised in fractal theory.
The discipline of fractals came into its own in the computer age. It is now possible to create "fractal forgeries" of mountains, coastlines, trees, clouds, cell growth and other processes which bear an uncanny resemblance to the real thing. Applications range from digital compression in computers to finding the best mix of tyre ingredients, and from modelling turbulence on aircraft wing designs to texturing medical images.
Benoit B Mandelbrot (he awarded himself a middle initial, although it stood for nothing) was born on November 20 1924 in Warsaw, Poland, into a family of Lithuanian Jewish extraction. His father made his living selling clothes while his mother was a doctor, but the family had a strong academic tradition and, as a boy, Mandelbrot was introduced to mathematics by two uncles.
In 1936 Mandelbrot's family emigrated to France where one uncle, Szolem Mandelbrot, a Professor of Mathematics at the Collège de France, took responsibility for the boy's education. Mandelbrot attended the Lycée Rolin in Paris, but was not a good student; it was said that he never learned the alphabet (he could never use a telephone directory, for example), nor his multiplication tables past five.
At the outbreak of war, his family moved to Tulle, a small town to the south of Paris. There, the war, the constant threat of poverty and the need to survive kept him away from school and, in consequence, he was largely self taught.
At this time, French mathematical training and thinking was strongly analytic and abstract, dominated by an influential group of young formalist mathematicians who wrote under the pseudonym of Nicolas Bourbaki; Mandelbrot's uncle Szolem was a member of the "Bourbaki" set. In contrast to their approach, Mandelbrot visualised problems whenever possible, preferring geometry to abstract formalism.
Despite his poor performance at school, he found that he had a quite extraordinary ability to "visualise" mathematical questions and solve problems with leaps of geometric intuition rather than the "proper" established techniques of strict logical analysis. After the war he passed the entrance exams for the École Polytechnique, achieving the highest grade in Algebra by "translating the questions mentally into pictures".
After completing his studies, Mandelbrot went to America and eventually, sponsored by John von Neumann, enrolled at the Institute for Advanced Study in Princeton. He returned to France in 1955 and worked at the Centre National de la Recherche Scientifique, but, finding himself increasingly out of sympathy with the Bourbaki school, he returned to America. In 1958, he was appointed an IBM fellow at the company's laboratories at Yorktown Heights in New York State.
IBM presented Mandelbrot with an environment which allowed him to choose the directions that he wanted to take in his research. They proved to be more diverse, eclectic and far-reaching than anyone could have imagined. His interest in fractals took him into many out of the way disciplines. He became expert in certain areas of linguistics; game theory; aeronautics; engineering; economics; physiology; geography; astronomy and physics. He was also an avid student of the history of science.
As most of the rest of the academic world at this time was heading towards ever greater specialisation, Mandelbrot's concern with such a broad spectrum made him unpopular in some establishment circles, and earned him a reputation as an outsider. Sceptics pointed to his arrogant reluctance to prove his theories, and his preference for "making bold and crazy conjectures".
Nevertheless, in 2004 he predicted the looming global financial meltdown, comparing bankers to "mariners who heed no weather warnings".
After retiring from IBM, Mandelbrot became a Professor of Mathematical Sciences at Yale, and later held appointments as Professor of the Practice of Mathematics at Harvard University; Professor of Engineering at Yale; Professor of Mathematics at the École Polytechnique; Professor of Economics at Harvard, and Professor of Physiology at the Einstein College of Medicine.
He received numerous honours and prizes including the Barnard Medal for Meritorious Service to Science (1985). the Franklin Medal (1986); the Alexander von Humboldt Prize (1987); the Steinmetz Medal (1988); the Nevada Medal (1991) and the Wolf prize for physics (1993).
Benoit Mandelbrot's wife, Aliette Kagan, and their two sons survive him.
© Telegraph Group Limited 2010.