Tag Archives: math

Visualization

Remembering Benoît Mandelbrot

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Benoît Mandelbrot the discover/inventor of fractals has died at 85. His work has been hugely influential in areas as diverse as computer graphics, finance and ecology.

In computer graphics fractals have been used to produce more realistic landscapes and vegetation, in finance his work as inspired people such as Nassim Taleb and others to think about the distribution of events, and in ecology fractals have been extensively used to understand the scaling of landscapes.

Mandelbort described his own career as a fractal:

“If you take the beginning and the end, I have had a conventional career,” he said, referring to his prestigious appointments in Paris and at Yale. “But it was not a straight line between the beginning and the end. It was a very crooked line.”

There have been obituaries in the New York Times Benoît Mandelbrot, Novel Mathematician, Dies at 85, The Telegraph (UK), the Guardian, NPR, and The Atlantic.

Below are some links to his work:

Some Mandelbrot obituaries and appreciations have been published in The Telegraph (UK), The New York Times and The Atlantic.

Mandlebrot is probably most famous for the Mandelbrot set seen above and many version of which are seen below. R code to generate a mandelbrot set is here.

Regime Shifts

Systems theorist Vladimir Arnold has died

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Vladimir Arnold from WikipediaVladimir I. Arnold one of the major creators of dynamical systems theory used to represent ecological regime shifts died June 3rd this year.

He was one of the creators of the mathematics behind what is known as catastrophe theory and singularity theory which are used to represent regime shifts.  The New York Times writes:

Singularity theory predicts that under certain circumstances slow, smooth changes in a system can lead to an abrupt major change, in the way that the slipping of a few small rocks can set off an avalanche. The theory has applications in physics, chemistry and biology.

“He was a genius and one of the greatest and most influential mathematicians of our time,” said Boris A. Khesin, a former student of Dr. Arnold’s and now a professor of mathematics at the University of Toronto.

One of Dr. Arnold’s biggest contributions was applying the methods of geometry and symmetry to the motion of particles. Dr. Arnold work on how fluids flow was applied to the dynamics of weather, providing a mathematical explanation for why it is not possible to make forecasts months in advance. Infinitesimal gaps or errors in information cause forecasts to diverge completely from reality.

A similar approach can also be applied to the motion of planets. If Earth were the only planet to circle the Sun, its orbit would follow a precise elliptical path, but the gravity of the other planets disturbs the motion. Scientists found that it impossible to calculate the precise motion of the planets over very long periods of time or even prove that Earth will not one day be flung out of the solar system.

Understanding the subtle and difficult-to-predict boundary between stability and instability is important not only in the study of planetary dynamics but also in other endeavors, like designing a nuclear fusion reactor.

In 1954, the Russian mathematician Andrey Kolmogorov figured out a key insight to calculating whether such systems are stable. Dr. Arnold provided a rigorous proof in 1963 for one set of circumstances. Another mathematician, Jürgen Moser, provided the proof for another. The work is now collectively know at the KAM theory.