Over twenty years ago, I read James Gleick’s Chaos: Making a New Science and was instantly gripped by the exotic world of strange attractors, Cantor sets, Julia sets – and, of course, the celebrated Mandelbrot set. This was simple mathematics revealing deep and unexpected wonders – exotic, yes, but immediately resonant of the world around us. You could see nature’s mathematics just by looking out the window. And, of course, a significant component of nature’s design style is fractal, and it was Benoit Mandelbrot who revealed this to us. Mandelbrot died on the 14th October and there has been, appropriately, an outpouring of praise for this truly extraordinary man and his brilliant work. I won’t attempt to add to this, but I will tell a story: the story of when I met Benoit Mandelbrot.
Twenty years ago, I was far from alone in my enthusiasm for fractals – the concept is powerful and the mathematics shows up in so many, often unexpected, places. Like many others, I became somewhat carried away with my enthusiasm, seeking fractal geometries in every data set. I departed from what seemed to me a rather boring topic that I had been invited to address at a Financial Times Conference in London: having briefly prognosed the outlook for oil and gas exploration in southeast Asia, I launched into a (reasonably well documented) lesson in how the diviners of oil prices should seek help from fractal analysis in their hopeless task. This was not unreasonable, since Mandelbrot had demonstrated how other commodity prices demonstrate fractal characteristics, but a standing ovation was conspicuous by its absence and I was never invited back.
Fractals, however, show up in many aspects of geology – bed thicknesses in sedimentary piles, grain sizes, the spaces between sand grains (vital for the porosity and permeability of a sandstone reservoir), and characteristics of faults and fractures, to name but a few. At that time, I was working in international exploration for “big oil,” the US company ARCO, to be specific, subsumed not long after into – oh dear – BP (but I had left by then. One of the great pleasures of that job was that the company had wisely – or, more probably, through an accident of real estate management – located my group in the same building as the research labs. The stuff that went on in those labs was fascinating, and I spent a lot of time keeping track of what they were up to and staying in touch with some of the really smart people working there. It was one of my friends in the labs who called me up one day, and told me that Mandelbrot was coming in – would I like to join the discussion? The question was one of those where the Pope and bears figure in the answer.
There was, sensibly, little in the way of a formal agenda for the meeting, the intention being to show Mandelbrot some of the problems vexing the research group and open up the space to see how this remarkable mind viewed the problem – the discussion was wide-ranging and certainly not focussed on fractals. I will have to admit that I found the prospect somewhat intimidating – after all, this was the Benoit Mandelbrot. But the reality was a delight, a gentle, thoughtful, man who put effort into listening before he would begin to ask questions. And, of course, it was in his questions that the value lay, the possibility of insight; the science we were talking about was often not familiar to him, and so his questions were intriguing – and challenging. The conversation moved on to oil and gas explorationists think about the earth’s subsurface, and the term “play” came up. “What do you mean by that?” asked Mandelbrot, and all the heads in the room turned to me. So, there I was, explaining something to Benoit Mandelbrot....
Now, a “play” is a concept that helps the evaluation of how likely it is that, below the earth’s surface, an area, or a given group of rocks will have the components of the geological conspiracy that is necessary to create oil and gas accumulations. There are a number of key ingredients, perhaps the most important being a source rock – most commonly a shale, a rock that contains sufficient organic material to brew hydrocarbons; no source rock, no play. And then there’s the maturity of those source rocks, how well they have been cooked in the earth’s kitchen; immature source rocks, no play. Add in all the other components – a plumbing system for oil and gas to flow through, then something that stops that flow – a trap – and something sufficiently porous to contain the trapped oil and gas – a reservoir – and you have a means of describing a play. Each of the components can be assigned a risk of whether or not it is present, and the chances of it working, and so the further work needed to define these components can be identified and different plays compared. All this was not easy to explain, but Mandelbrot’s questions along the way were not only illuminating in making me think about what I was really trying to describe, but they were thought-provoking in general. I don’t, today, remember many of the details, but I certainly remember the experience, the rare opportunity to be challenged by the thinking of a brilliant individual.
It’s interesting how, in the meantime, the power of fractal geometry has been increasingly recognised, and how more and more examples have been identified. I was just reading a recent New Scientist article on “The chaos theory of evolution” that commented that “the history of life is fractal. Take away the labelling from any portion of the tree of life and we cannot tell at which scale we are looking. This self-similarity also indicates that evolutionary change is a process of continual splitting of the branches of the tree.” But we still haven’t figured out exactly what to do with fractals. They have revolutionised computer graphics, but their deeper meaning continues to elude us. However, the idea of scale invariance and self-similarity in nature’s designs continues to provoke and stimulate. Take, for example, this work by the artist Alison Cornyn; titled Sand and Moon: “The top image is a portrait of two grains of Coney Island sand. Below it is a NASA image of Phobos, one of the moons of Mars.” This is part of Cornyn’s larger project, Sand Portraits, of which probably more later. But for now I just wanted to mark my gratitude to Benoit Mandelbrot for an unforgettable and intellectually enriching experience.
[There’s a great video from February this year of Mandelbrot talking at a TED conference on “Fractals and the art of roughness” here (the site includes a number of other videos). Amongst many obituaries, the one from The Economist is a good starting point. For a description of her work by Alison Cornyn, and full credits for the images, see http://www.mcsweeneys.net/books/everythingthatrises.contest35.html. The photograph of desert boulders comes from a 1933 newspaper report of one of Ralph Bagnold’s expeditions.]
Memorabilia
Ah, did you once see Shelley plain,
And did he stop and speak to you,
And did you speak to him again?
How strange it seems and new!
But you were living before that,
And also you are living after;
And the memory I started at—
My starting moves your laughter!
I crossed a moor with a name of its own
And a certain use in the world, no doubt,
Yet a hand's-breadth of it shines alone
'Mid the blank miles round about.
For there I picked upon the heather
And there I put inside my breast
A moulted feather, an eagle-feather!
Well, I forget the rest.
Robert Browning
Many thanks for the story. Your citation mentions self-symmetry of scale: famously, this does not work in the three-dimensional world (as Galileo noted), or elephants could pronk. Yet much of the world is scale-symmetrical; fractals validate such uniformitarianism. The only remaining difficulty, for non-mathematicians, is to visualize a world between two and three dimensions. We need a new Mr. Tompkins, or Flatland.
Posted by: Richard Bready | April 29, 2011 at 10:29 AM