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Author Interview: Michael F. Barnsley

Michael F. BarnsleyDr. Michael Barnsley is a British mathematician and entrepreneur whose main research has been on fractal compression. Dr. Barnsley holds several patents in the technology. Dr. Barnsley received his BA in Mathematics at Oxford University and his PhD in Theoretical Chemistry from the University of Wisconsin.

After many years on the faculty at Georgia Tech, followed by positions at the University of New South Wales and the University of Melbourne, he has been based at the Mathematical Sciences Institute of the Australian National University since 2004. As an outgrowth of his work on fractals, Dr. Barnsley founded Iterated Systems Incorporated in 1987 (sold to Interwoven in 2003) and has published two well-known books on fractals: Fractals Everywhere (1988), which has just been reprinted with updates by Dover Publications (2012), and SuperFractals (2006). He is credited for discovering the Collage Theorem.

Recently, Dover Math and Science Editor Shelley Kronzek sat down with Dr. Barnsley to discuss his work and the book.

Shelley Kronzek: Good Morning, Michael.
Michael Barnsley: Hello, Rochelle. It's a cool and frosty morning here.
Shelley Kronzek: A bit of the opposite here for much of the United States right now. While you are experiencing winter weather, we've got a heat wave here.
Michael Barnsley: I'm very pleased with how the Dover edition of my book Fractals Everywhere has turned out.
Shelley Kronzek: Thanks for agreeing for Dover to publish it and for visiting with me.
Michael Barnsley: It's a pleasure.
Shelley Kronzek: Please tell me a bit about your beginnings.
Michael Barnsley: I was born and raised in Maidstone, Kent, England, to a very artistic and decidedly non-mathematical household.
Shelley Kronzek: Is Maidstone a small town surrounded by countryside?
Michael Barnsley: Yes. The River Medway runs through the middle of the town. Maidstone is in the agricultural area of Kent known as the "Garden of England." After WWII, when I was a boy, the countryside was lush with plants, fields, orchards, copses, hedgerows, and endless butterflies, birds, and flowers. A lovely, peaceful, safe place right after the war, as children we could wander and bicycle everywhere freely, climb trees, and swim in lakes and rivers, usually without anyone telling us we should not.
Shelley Kronzek: How about your parents? What were they like?
Michael Barnsley: My father was Gabriel Fielding (the pen name of Alan Gabriel Barnsley), who was a well-known novelist. He wrote eight novels, three books of poetry, and lots of short stories. My paternal grandmother was a descendant of Henry Fielding (author of Tom Jones) so dad took his pen name from our famous ancestor and we have literature in our veins! My father was a physician with a private practice and he provided medical services at the local prison. He was in the Royal Air Force Medical Corps during WWII. He did his writing in the afternoon, after visiting patients and running a surgery. He was quite successful and gave up medicine to write full time. He was appointed Artist in Residence and then Professor of Literature at Washington State University, in Pullman, Washington — totally different from Kent. He also became an artist and had a great love of graphic art. In 1963 dad was awarded the W. H. Smith Literary Award for The Birthday King. In 1967 the degree of Doctor of Literature was conferred on him by Gonzaga University, Spokane, Washington. At that time various novelists became artists in residence, for example Ernest Hemingway was on the faculty at a nearby university.
Shelley Kronzek: How about your mum?
Michael Barnsley: Mum served in the British Women's Air Force during WWII. She had five children, loved poetry, looked after the household, and played hostess to the literary elite that came to visit us.
Shelley Kronzek: Who were your heroes as a boy?
Michael Barnsley: I'd imagine that my first hero was my dad. But I very much admired Sir Lawrence Bragg after I heard him lecture at the Royal Institute in 1961. But my real heroes were ideas. Honestly, I think that I was more interested in books than people. I read a lot of books. We all read, all the time, everything.
Shelley Kronzek: What books do you remember from your youth?
Michael Barnsley: I absolutely loved Ralph Buchsbaum's Animals without Backbones, a celebrated textbook that fascinated me. We received a weekly comic at home, Eagle Comics, and my favorite was Dan Dare, who was a science fiction hero. My dad bought me the London Observer's books of nature. I also read Strand magazines, Punch, and we were encouraged to refer to encyclopedias. I once received a gift of Scientific American for a year, where I read Martin Gardner's puzzle corner. I loved the books of Martin Gardner and Dan Pedoe and loved science fiction and, especially, nature books.
Shelley Kronzek: How did you first become interested in mathematics within this "non-mathematical" household?
Michael Barnsley: I was interested in everything — physics, biology, and math — but could only study one thing at the university so I chose math, which is important to all scientific disciplines as well as to nature and art. Mathematics got me to the bottom of things.
Shelley Kronzek: So your father moved to the States and became faculty at Washington State. Did you go with your parents?
Michael Barnsley: No, I was older than my younger brothers and sisters who did go with them, and I was already at Oxford University.
Shelley Kronzek: Tell us a little about your studies and teachers or researchers who influenced you during your higher education at Oxford.
Michael Barnsley: My tutor was John Lewis, at Brasenose College. He was a wonderful, clear-thinking mathematician with deep interest in quantum mechanics and statistical physics. He was a strong influence, leading me to do a doctorate at the University of Wisconsin in quantum mechanical perturbation theory.
Shelley Kronzek: How did you first become interested with fractals? Which scientists were influential in moving you in that direction?
Michael Barnsley: After completing my Ph.D. I went back to the UK and did various postdoctoral things, including working at Saclay (C.N.R.S) with Daniel Bessis, Pierre Moussa, Henri Cornille, and Giorgio Turchetti. Then, having accepted a position at Georgia Tech, in 1979 I went to a conference in Corsica organized by Jacques-Louis Lions and Daniel Bessis, where I met with Mitchell Feigenbaum. Mitchell was bubbling over with his theories about iterated unimodal maps, especially quadratic maps, and chaos theory. I was very excited and thought that a good way to understand what was going on was to think of things from the point of view of the complex plane. Then the cascades of bifurcations could be seen as part of bigger, more organized pictures. I called these pictures, hopefully B-sets, and sent a paper on the topic to David Ruelle, who bounced it back pretty quickly, saying that these objects were Julia sets and referring me to Benoit Mandelbrot. That was how I first met Benoit and his wife, Aliette, in Scarsdale, New York. I went to visit them one winter's day; and while the light gleamed off the snow outside into his tight office packed with Xerox copies of articles, I was introduced officially to "fractals," but, in fact, I had been working in the area for years, with Bessis and Moussa.
Benoit Mandelbrot is definitely one of my heroes. He was a loner and an individual. He wasn't afraid to think and work outside the mainstream. He moved between departments, explaining ideas from pure mathematics, which, until he brought them out into the open and illustrated them with pictures, had remained hidden. He put these ideas to work in geology, economics, biology, and art, in a free-thinking and individualistic manner, often confounding speculation with theory, but always pushing forward his vision of this new area, fractal geometry.
Shelley Kronzek: You were at Georgia Tech for quite some time. What are some highlights of your time there?
Michael Barnsley: I had a wonderful time, research-wise, at Georgia Tech. There were a number of young, enthusiastic, unselfconscious mathematicians, all willing to work together in a cooperative manner. I mention Marc Berger, John Elton, Jeff Geronimo, Steve Demko, Tom Morely. We were sort of led by Les Karlovitz, who was very supportive of my ideas about the importance of fractal geometry.
Shelley Kronzek: You have a fern on the cover of your new Dover book. Why is the fern such an important image with fractals?
Michael Barnsley: Because it shows that fractal geometry could be used deliberately as a modeling tool, rather than accidentally. It shows that the inverse problem of finding "formulas" for natural shapes and forms can be solved in the same sort of way that calculus is used to model smooth things.
Shelley Kronzek: Why and how are chaos theory and fractals linked?
Michael Barnsley: There are two bits to the answer, probably lots more actually, but two bits that come to my mind. First, chaos theory is related to modeling limiting behavior of dynamical systems, yielding patterns associated for example with turbulence (clouds, swirling patterns in a mixtures of paints) and, in particular, objects called "strange attractors and repellers": Such objects are often fractals. Second, the easiest algorithm to implement, that I know of, to calculate a picture of a fractal is to use a random iteration algorithm, that I dubbed "the chaos game." Now, thirty years later, I still use it to guide my intuition about what is going on in a particular system.
Shelley Kronzek: Many of our readers will not have much knowledge about fractals. I'd like to throw a couple of concepts out at you and ask you to explain as simply as possible what the terms mean. What are iterations?
Michael Barnsley: Doing the same thing over and over again. Precisely, repeating the same process applied to something, over and over. For example: Iron a shirt in a certain way, then pick up the same ironed shirt and iron it again in exactly the same way, and do this over and over again. For example: Make a copy of this article on a photocopy machine, then make a copy of the copy, then make a copy of that copy. For example: Replace a number by its square and subtract one, yielding a new number, then replace that number by its square and subtract one, and keep on doing this . . . millions of times. The sequence of successive ironed shirts, copies of this article, and the numbers are the iterations.
Shelley Kronzek: Self-similarity?
Michael Barnsley: The meaning depends on the context. Mathematically, working in fractal geometry, an object is said to be self-similar if it is the union of transformed copies of itself, where the transformations are compositions of elementary rescalings and shifts.
Shelley Kronzek: Fractal geometry?
Michael Barnsley: Fractal geometry is the mathematical study of the properties of objects that are, in one way or another, self-similar; it generalizes classical projective geometry.
Shelley Kronzek: What are fractal transformations?
Michael Barnsley: The right way to think of a transformation is as a geometrical object in its own right. Students of calculus, given a function, are often asked to "graph the function." When they do so they create a picture of a geometrical object. In calculus this object is always very smooth. If you magnify it up, it looks like a straight line, the tangent line approximation. A fractal transformation is a function whose graph is a self-similar object!
Shelley Kronzek: What is fractal continuation?
Michael Barnsley: It is a generalization of analytic continuation that applies to fractal transformations. It is a new and very exciting area of research.
Shelley Kronzek: Iterated Function Systems or IFS?
Michael Barnsley: A collection of continuous functions acting on a topological space. Instead of, as in dynamical systems theory, considering the iterative action of all of the functions, one studies the collective behavior of the system. An IFS may have one, or many, attractors: they are the fractals, the self-similar objects, generated by the IFS. The fractal fern is an example of an attractor of an IFS.
Shelley Kronzek: What is chaotic generation and what does that have to do with fractals?
Michael Barnsley: I think I sort of explained this above. By the way, all of this is not tight, careful mathematical explanation; it is aimed at giving you the feel for the meanings. Read Fractals Everywhere for more precision!
Shelley Kronzek: Enough of terms, what is new? What are you working on now?
Michael Barnsley: My mathematical work, with Andrew Vince, is on fractal continuation. This is a really exciting area. If you know a tiny bit of a fractal, can you predict what the rest of it looks like? Think of having a tiny bit of the graph of an analytic function: You can use that little bit, by means of repeated Taylor series expansions, to build up the function, creating a wonderful object called the Riemann surface. It turns out you can do the same with the attractor of an analytic iterated function system.
Louisa, my wife, Brendan Harding, and Neville Smythe at the Mathematical Sciences Institute of the Australian National University have been building a fractal camera. There are some pictures taken with the camera in the Dover Edition of Fractals Everywhere. The camera is called FrangoCamera and is available for the iPad from the App Store. It costs $3.00!
Shelley Kronzek: Now that Dover has published your Fractals Everywhere, what would you like to say about the book and the free generator that you've included?
Michael Barnsley: I think all students who have done calculus, and who are involved in any way with mathematical modeling, should read it and be familiar with the Collage Theorem and the Chaos Game Algorithm. These are great techniques to know, and they lead into many interesting applications and play into more advanced ideas. Fractal geometry is an area of mathematics under construction. These are the early days and there are lots of discoveries nearby. The new edition includes a lovely piece of software written by my student Brendan Harding. It runs very, very fast on most up-to-date Windows laptops: you have graphical control (and numerical control) of an IFS and interact with a fractal attractor, generated using the chaos game. You can also work with associated invariant fractal measures. It is very simple to use and holds in your hand: try it!
Shelley Kronzek: So what brought you to Australia, and what is life now for you and your family there?
Michael Barnsley: That sounds like an invitation to write a novel. We came to Australia primarily because of the large unspoilt patches of nature, and to work with John Hutchinson and others at the Australian National University. We live on the edge of Mt. Majura but can bicycle into the university. Our daughter, Rose, is in the last year of high school: she is keen on literature, writing, and media. We have two small nature reserves, one in Victoria and one in Georgia USA. We are hopefully saving something for the future.
Shelley Kronzek: Thank you for sharing with our readers and me. What is next on your agenda?
Michael Barnsley: There are two upcoming articles in the Notices of the American Mathematical Society that I edited with Michael Frame. They concern the work and life of Benoit Mandelbrot, and are memorial articles in his honour. I am working intensely on fractal continuation and will be presenting results at several meetings on fractals, including an important one in Hong Kong in December. I have various collaborators coming to work with me: Lesniak from Poland, Massopust from Germany, and Vince from Florida State University.
Shelley Kronzek: Cheers!

By Michael Barnsley: Fractals Everywhere: New Edition
Fractals Everywhere: New Edition
This new edition of a highly successful text constitutes one of the most influential books on fractal geometry. An exploration of the tools, methods, and theory of deterministic geometry, the treatment focuses on how fractal geometry can be used to model real objects in the physical world. Two sixteen-page full-color inserts contain fractal images, and a bonus CD of an IFS Generator provides an excellent software tool for designing iterated function systems codes and fractal images. Suitable for undergraduates and graduate students of many backgrounds, the treatment starts with an introduction to basic topological ideas. Subsequent chapters examine transformations on metric spaces, dynamics on fractals, fractal dimension and interpolation, Julia sets, and parameter spaces. A final chapter introduces measures on fractals and measures in general. Problems and tools emphasize fractal applications, and an answers section contains solutions and hints.
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