Blending Math With Beauty

How a quest to combine aesthetics with mathematics produced one of math’s most famous, and gorgeous, images: the Mandelbrot set.
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TRANSCRIPT

Question: What was the discovery process behind the Mandelbrot set?

Benoit Mandelbrot: The Mandelbrot set in a certain sense is a **** of a dream I had and an uncle of mine had since I was about 20.  I was a student of mathematics, but not happy with mathematics that I was taught in France.  Therefore, looking for other topics, an uncle of mine, who was a very well-known pure mathematician, wanted me to study a certain theory which was then many years old, 30 years old or something, but had in a way stopped developing.  When he was young he had tried to get this theory out of a rut and he didn’t succeed, nobody succeeded.  So, there was a case of two men, Julia, a teacher of mine, and Fatu, who had died, had a very good year in 1910 and then nothing was happening.  My uncle was telling me, if you look at that, if you find something new, it would be a wonderful thing because I couldn’t – nobody could. 

I looked at it and found it too difficult.  I just could see nothing I could do.  Then over the years, I put that a bit in the back of my mind until one day I read an obituary.  It is an interesting story that I was motivated by an obituary, an obituary of a great man named Poincaré, and in that obituary this question was raised again.  At that time, I had a computer and I had become quite an expert in the use of the computer for mathematics, for physics, and for many sciences.  So, I decided, perhaps the time has come to please my uncle; 35 years later, or something.  To please my uncle and do what my uncle had been pushing me to do so strongly.  

But I approached this topic in a very different fashion than my uncle.  My uncle was trying to think of something, a new idea, a new problem, a new way of developing the theory of Fatu and Julia.  I did something else.  I went to the computer and tried to experiment.  I introduced a very high level of experiment in very pure mathematics.  I was at IBM, I had the run of computers which then were called very big and powerful, but in fact were less powerful than a handheld machine today.  But I had them and I could make the experiments.  The conditions were very, very difficult, but I knew how to look at pictures.  In fact, the reason I did not go into pure mathematics earlier was that I was dominated by visual.  I tried to combine the visual beauty and the mathematics. 

So, I looked at the picture for a long time in a very unsystematic fashion just to become acquainted in a kind of physical fashion with those extraordinary difficult and complicated shapes.  Two were extraordinarily difficult.  Computer graphics did not exist back then, but to have a machine which was – made it seem doable.  And I started finding extraordinary complications, extraordinary structure, extraordinary beauty of both a theoretical kind, mathematical, and a visual kind.  And collected observations of my trip in this new territory.  When I presented that work to my colleagues, it was an explosion of interest.  Everybody in mathematics had given up for 100 years or 200 years the idea that you could from pictures, from looking at pictures, find new ideas.  That was the case long ago in the Middle Ages, in the Renaissance, in later periods, but then mathematicians had become very abstract.  Pictures were completely eliminated from mathematics; in particular when I was young this happened in a very strong fashion. 

Some mathematicians didn’t even perceive of the possibility of a picture being helpful.  To the contrary, I went into an orgy of looking at pictures by the hundreds; the machines became a little bit better.  We had friends who improved them, who wrote better software to help me, which was wonderful.  That was the good thing about being at IBM.  And I had this collection of observations, which I gave to my friends in mathematics for their pleasure and for simulation.  The extraordinary fact is that the first idea I had which motivated me, that worked, is conjecture, a mathematical idea which may or may not be true.  And that idea is still unproven.  It is the foundation, what started me and what everybody failed to **** prove has so far defeated the greatest efforts by experts to be proven.  In a certain sense it’s a very, very strange because the object itself is understandable, even for a child.  If the object can be drawn by a child with new computers, with new graphic devices, and still the basic idea has not been proven. 

But the development of it has been extraordinary, then it was slowed down a bit, and now again it is going up.  New people are coming in and they prove extraordinary results which nobody was hoping to prove, and I am astonished and of course, very pleased by this development.

Question: What is the unproven conjecture that drove you?

Benoit Mandelbrot: The conjecture itself consists in two different issues in Mandelbrot set – two alternative definitions which are too technical to describe without a blackboard, but which are both very simple and which I assumed naively to be equivalent.  Why did I assume so?  Because on the pictures I could not see any difference.  Obtaining pictures in one way or another way, I couldn’t tell them apart.  Therefore, I assumed they were identical and I went on studying this piece.  I found that, again, many interesting observations of which most were very preferred by many other very, very skilled mathematicians.  But the idea that these two conditions, definitions, are identical is still open.  So there are two definitions in Mandelbrot set, the usual one and another one, and they may theoretically be different.  People are getting closer, but have not proven it completely.

Recorded on February 17, 2010
Interviewed by Austin Allen