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Mandelbrot’s door to infinity

Back in the 1960s, IBM – like the majority of technology businesses – was faced with one issue above any other. How could it prevent distortion in the analogue transmission of data? Little did the ‘Big Blue’ suspect that the answer would not only transform technology, but provide the essential key to describing (and ultimately predicting) a huge spectrum of apparently random occurrences.

To find that answer, IBM approached the French-American mathematician, Benoit Mandelbrot. In his early career he had already published a dazzling series of academic papers that used complex mathematical and geometrical equations to detect repetition patterns in apparently random activity. For example, using these techniques, he could effectively analyse and predict events in the formation of coastlines and river basins (both of which typify what Mandelbrot called ‘roughness’), as well as in stock market trading and fluctuation.

Now at IBM, he was quick to spot similar patterns in the distortion waves created by analogue data – where the waveform of the distortion signals apparent in just a second of distortion were remarkably similar to those lasting 20 minutes or more. Moreover, in the same way as coastlines and market fluctuation, there were nano-patterns that, while invisible to the naked eye, recurred time and time again, forming the structure of the entire activity.

Mandelbrot could detect these by using a branch of mathematics known as fractals. He didn’t invent fractal mathematics – that was created by Gaston Julia in the 1920s – but was the first person to use computer technology to look at how fractals work over not just millions, but trillions of repetitions, and apply these principals to the analysis of apparently mundane, adhoc phenomena. Why use fractals? Because Gaston Julia had shown that despite the great complexity of the numbers involved, at certain critical moments (eg, with every several million calculations), there was an exact repetition – in other words, he had created a branch of mathematics which while apparently random, was itself based on repetition sequences and the ability to see order in chaos.

A doorway to infinity: the Mandelbrot Set

Quite early on in his work with IBM, Mandelbrot was able to create the fractal formulas for wave-generation that would effectively block excessive analogue distortion. He then went much further. Taking Gaston Julia’s work, he decided to use computer graphics to create and display a complete universe of fractal geometric images, showing the endless sequence of ‘returns’ that even the most open-ended complex formula included. This graphical representation was possible because fractals were based on ‘complex’ equations that were generally represented in two dimensional drawings. In these drawings, the final sum of the equation was measured against an axis and plotted as a real or minus sum. So, Mandelbrot took each fractal equation and plotted it with computer graphics, eventually generating an overall shape that repeats itself, potentially forever. This defining shape – encapsulating all possible variants – is called the Mandelbrot Set.

But there’s more. Since that shape is constructed with trillions of equations, you can enter it at any point and see a never-ending universe of fractal geometric images. A great example of this is shown here –

https://www.youtube.com/watch?v=gEw8xpb1aRA

Mandelbrot himself claimed that if you started off with the overall shape being only one centimeter across, even if you expanded it to be the size of the entire universe, it would still be opening out with infinite complexity. Quite literally, it is a man-made doorway to infinity.

The Mandelbrot Set is also the most powerful tool we have for identifying the actual behaviour of seemingly random events – and thereby for minimizing their Disruptive power. It can also provide a predictive analysis that empowers us to create structures that co-exist with, and surpass, the most extreme and critical change.

This article was attributed and provided by PG International

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