Up here in my garret the walls are covered in a variety of pictures. Nothing of much merit. Quite a lot of them are just old photos of drunken orgies from my past - well, to be truthful there was never much in the way of 'orgy' but there was plenty of drunkeness! However, amongst them is a much treasured series of pictures, nicely framed, illustrating the Mandelbrot set - and, no, me neither until 1999, that is, when I directed Tom Stoppard's classic play Arcadia and I had to delve into the subject.
In writing this play Stoppard's endless curiosity takes him into the intricacies of chaos theory and the peculiarities of iterated algorithms - yeah, yeah, I know, but give me a chance and I'll try to explain it but don't expect to much in the way of accuracy! If you have a mathematical equation which you work out to a result and you then feed the result back into the equation and re-work it out, and then do it again with the new result, and then again, and again, and so onfor several zillion times that's called an iterated algorithm. So far, so eye-stabbingly tedious . . . but! Suddenly, as you plot the results on your graph with the line showing a steady, boring, predictable increase with each result, the line shoots off the graph for no apparent reason! You check your equation, check your sums, but there it is, an inexplicable change.
I will pause for a second because even from here I can sense the sound of crashing bodies as sundry swots reading this faint away at my crass explanation - sorry, chaps, but 'sums' was never my strong point - oh, you guessed!
Anyway, this mathematical discovery has profound implications. For example, the constant reproduction of biological cells in which sets of chemical equations are iterated normally with standard results, can sometimes, out of the blue, produce entirely unexpected results - either for good or ill! Even human reproduction can be looked at as another example of an iterated algorithm Here-in, perhaps, lies a very much better explanation for evolution than 'Archbishop' Dawkins' theory of tiny little incremental changes being forged, or destroyed, in the white heat of survival of the fittest. Big changes wrought every so often by the mathematics of iterated algorithms would provide a more believable explanation for complex and infinitely variable living beings, as Richard Bird suggests in his book Chaos and Life. In fact, human reproduction can be looked upon as an iterated equation as genetic codes are constantly melded together producing, so to speak, fairly standard people but occasionally throwing up quite exceptional human beings such as, for example . . . er, but modesty forbids!
Another area of iteration occurs in the natural world about us. Look close up at a broccoli flower and you will see the same shape 'iterated' in the tiny flowers that make it up. Put it under a microscope and you will see yet more repetitions.
This characteristic in nature of endless repetitions at different scales is what attracted the attention of the late Benoit Mandelbrot, a Polish Jew who spent the war in hiding from the Nazis in occupied France, according to an article in The WSJ by Stephan Wolfram:
Mandelbrot ended up doing a great piece of science and identifying a much
stronger and more fundamental idea—put simply, that there are some geometric
shapes, which he called "fractals," that are equally "rough" at all scales. No
matter how close you look, they never get simpler, much as the section of a
rocky coastline you can see at your feet looks just as jagged as the stretch you
can see from space. This insight formed the core of his breakout 1975 book,
There was no book like it. It was a new paradigm, both in presentation and in
the informal style of explanation it employed. Papers slowly started appearing
(quite often with Mandelbrot as co-author) that connected it to different
fields, such as biology and social science. Results were mixed and often
controversial. Did the paths traced by animals or graphs of stock prices really
have nested structures or follow exact power laws?
Below, courtesy of Wiki is an illustration of a Mandelbrot set in action and you can see how a simple pattern repeated can produce extraordinary complexity:
Anyway, that's enough bad science for today - my brain hurts and I suspect that my swot readers have suffered enough.