Benoit Mandelbrot finally converges on infinity

Benoit Mandelbrot finally converges on infinity

Summary: You don't get many new shapes in a lifetime. Many new combinations, true, but break the visual world down and it's built out of bits of regular polygons, curves and a smattering of randomness - as it has forever, and as we've learned to look at it since the ancient Greek philosophers.

TOPICS: Emerging Tech

You don't get many new shapes in a lifetime. Many new combinations, true, but break the visual world down and it's built out of bits of regular polygons, curves and a smattering of randomness - as it has forever, and as we've learned to look at it since the ancient Greek philosophers.

Benoit Mandelbrot, who died on Thursday at an integral approximation of 85, gave us new shapes. A Jewish genius who fled the Nazis —€” and how many times will the technological history of our age include that phrase? — he most famously named and created the field of fractal mathematics. He didn't discover the basic maths of fractals, but he took an obscure, unfashionable concept, that you can explore the space between two dimensions and three dimensions, and showed its fundamental role in the fabric of the world.

You can read many fine explanations of what fractals and Mandelbrot's other work have done for mathematics. You can also read how he did it, eschewing the rigorous path of carefully nuanced proof in favour of the intuitive and practical.

This, however, is a more personal note.

It was the latter end of the 80s. I was working for a start-up, writing OS-level 386 assembler during the day, writing about computers in the evening, and intensely interested in the nature of thought and the intersection between numbers, reality and the self. All those good things that rightly concern young geeks who find themselves in the middle of a revolution and possessed of a copy of Godel, Escher, Bach.

If you like rock and roll, you'll have heard how it got into the psyche of the post-war nation. John Peel got infected by listening to American armed forces radio late at night; others with just one sting, the opening chord to Hard Day's Night. So often, it was an epiphany —€” a chance encounter on the wireless with one piece of music that was simultaneously alien, exciting and bursting with the energy of new ideas.

For me, that moment was running Fractint on a 386 for the first time. It wasn't just that the chip had the performance to encourage real-time exploration of the Mandelbrot set's baroque coastline: the PC also had VGA graphics that could animate thousands of colours. Zoom in on a sumptuous, suggestive, endlessly morphing landscape and hit palette rotate. Bang! You're falling into a psychedelic rabbit hole —€” just one of an infinite number, stretching in all directions.

Which would be gorgeous enough, except I knew how it worked. The maths behind fractals is simple, provided you're not frit by complex numbers: zn+1 = zn^2 + c. An early 386 VGA computer running DOS —€” that was probably the last time I was deluded enough to think I could take any aspect of the hardware and software and understand it down to gate and opcode level. But that a simple system like that could not only create profoundly complex, unknown yet intuitively real patterns, but make them so accessible... what on earth was going on? My mind was comprehensively blown, man.

So, while many of my peers were weekending it on Es and repetitive beats, I gave the warehouses full of light and DayGlo a miss (well, mostly) and spent my time in 640x480xinfinity pixels. I read voraciously, discovering the connections between Mandelbrot's maths, emergent complexity, theories of mind, Turing, and the increasingly persuasive idea that when the early personal computer pioneers called the microprocessor a thought amplifier they were spot on.

Until Fractint, I had always thought of computers in the abstract as having the potential for being a combination of universal microscope and telescope: where you reached a human limit of handling data and needed more detail, more speed, more capacity, you could build a tool out of computing that let you go further. It was a nice, exciting thought, but not very involving — after all, who could afford a supercomputer any more than an amateur astronomer could afford Palomar?

Fractint was the Hubble for such ideas. It doesn't matter how well or badly you understand the science behind it; if you have the basic human endowments of curiosity and beauty, you will be changed when you look at Hubble's pictures of cosmic creation and destruction. And, if you let yourself, you will be led by the nose into the world of "If that's so, then what if...", and may that never, ever stop. This was a Hubble in my front room. (It was only later that I realised that Fractint itself was an early example of the idea of collaborative, open, Net-mediated software development that has itself changed the world in subtle and powerful ways. What a time to be alive.)

The concepts that Fractint demonstrated, thanks to the work of Mandelbrot, remain with me still and keep me hungry for more. It can be hard to stay mindful of what we're doing in technology, swamped as we are in the noxious fumes thrown up by competing commercial and social interests. But that computers can guide us through new vistas of creativity, new ways of extending our web of ideas and new understandings of fundamental reality — these are new shapes indeed, and they're still growing.

Thank you, Professor Mandelbrot.

Topic: Emerging Tech

Rupert Goodwins

About Rupert Goodwins

Rupert started off as a nerdy lad expecting to be an electronics engineer, but having tried it for a while discovered that journalism was more fun. He ended up on PC Magazine in the early '90s, before that evolved into ZDNet UK - and Rupert evolved with them into an online journalist.

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  • One of the reasons I'm rather excited about GPU computing at the moment is the mix it offers of horsepower to churn through enormous data sets to deliver some fundamental developments in computing science (the computer vision problems I struggled with in my degree are snap-of-the-finger commonplace building blocks for the hard vision problems and the same seems true in so many areas) and then to visualise results. I'm truly grateful to everyone enabling the sense of wonder with all this.
    Simon Bisson and Mary Branscombe
  • @Rupert: your comment about hit home: I remember running (I don't know what program it was - I probably typed it in from a copy of Acorn User) on a BBC Micro back in the early-mid 1980s and being blown away by the concept and beauty of infinite fractals. Incredible.
    Manek Dubash
  • I first came across the Mandelbrot set as one of the pieces of computationally intensive code that was easy to parallelise and small enough to fit into the on-chip memory of the transputer. We had some devices that had been miswired i.e. the wrong i/o pins, but we were able to put together boards that had nothing but CPUs connected by serial links. In one of those wonderful coincidences, the maximum number we could get on a board was 6x7 = 42 - answer to life, the universe etc (see Hitchikers Guide to the Galaxy). Running nearly 400 processors (9 boards) in a 6U rack gave an almost real time calculation, which was staggering back in 1986.
    It was magic to interactively "explore" mathematics; people were fascinated by the fact that no matter how far you zoomed in, you never reached the end - it sort of made the concept of infinity easier to understand (except to my father, who could never understand infinity, and who I spent hours trying to explain infinity when he was doing an Open University maths course)
  • Well done Rupert, this was a brilliant piece of writing.
  • Many thanks for an excellent article that brought back some very nice memories. In the late 1980's I was using, learning, working with and teaching about the X Window System, and sending tapes of the source code to anyone in Europe who wanted it. Sometime around V10R3 or V10R4, I came across a Mandelbrot program, and I typed it into a Vaxstation 2000, alas with only a monochrome display... but it fascinated me, endlessly.

    Thanks for the memories...

  • It was the beauty of the Mandelbrot set that made me start with computer programming.
    You can explore the Mandelbrot set online here: