Elegance

Elegance - the beauty of an idea characterised by minimalism and intuitiveness while preserving exactness and precision.

Complex - not simple, easy or straightforward. A Complex system is one that has multiple paths to multiple answers.

So now I've learnt how to make Julia sets.  Using the same colour palette as for my Mandelbrots here are a couple of my home made images. Of course there are far better ones on the internet, but I'm pretty pleased that I have even got this far. 

The Julia and Mandelbrot sets are mathematically very similar, so my program can generate either. The maths is so very simple - elegant indeed. But the resulting fractal images are complex, anything but simple. And there's the rub - the conjunction, the juxtaposition, the synergy of elegance and complexity. We see it again and again - the human body for one - such complexity from relative simplicity of DNA. The 90 or so naturally occurring chemical elements that are each made up of so few elementary particles (the periodic table and all that) giving rise to such diversity of materials and, indeed, life. How such a vast assortment of vividly coloured flowers are created from dirt, water and sunlight. Water gushing over a precipice: any student can calculate its maximum rise in temperature as it falls, but how to predict its impact as you swim beneath its fall?. And of course the much maligned "butterfly effect".

For those interested but too idle to google for themselves, the Mandelbrot set is the set of points c on the complex plane (aka Argand diagram) for which the iteration formula is bounded (z does not fly off to infinity), starting with z = 0.  There is a Julia set for every point c in the Mandelbrot set and it is the set of points z on the complex plane for which the same iteration formula is bounded, starting with z = that point. It can be proved (so I am told) that if, during the iteration, the modulus of  z exceeds 2, then z is bound to be unbounded: we need iterate no further. For cases that never or are slow to reach that stage, the iteration ought to go on ad infinitum - in practice such a requirement is onerous, and instead one sets an arbitrary limit of the number of iterations, for example 500.

Here's two of my favourite Mandelbrot zooms:

A trip to infinity - 2 hours 

The deepest hard zoom at 100,000,000 iterations - zoom depth 10^2126

The guys who create such videos (God bless them) have access to better software than mine but even so take perhaps a month or more of computer time to do the job. Not for the faint hearted or those of us with other more pressing calls on our time!

My brain still has problems understanding how so much complexity is bound up in so simple an equation.

This, my interest in Mandelbrot et al, started when reading an article about deep learning. Deep learning is AI (artificial intelligence) on steroids. The mechanics is based on a neural network (i.e. that tries to mimic how the human brain works). Faced with a complex task, for example recognising a particular face in a picture of a crowd, rather than trying to figure out software code that analyses the face you "simply" show the deep-learning-thing (DLT) zillions of pictures of the face in question from many angles and in many circumstances and with varying expressions.  Then when faced with the picture to be analysed, the DLT does a sort of mathematical correlation with all the pictures it has been shown and comes up with a score, a likelihood of each face in the picture being the one. Etc. A very different way of approaching software and clearly akin to what goes on in the brain (for those of use lucky enough to have one). Deep learning is the in-thing at the moment.  If you are young and mentally supple enough to get a grasp of all this, I'd imagine that designing these DLT's is the way to go. Or if interested in investing your fortunes, investing in DLT's might prove profitable. 

Deeping learning, complex systems, AI, chaotic systems, fractals (e.g. the Koch snowflake which has infinite perimeter and yet a finite area.  Plenty here to keep me amused for many a let's-avoid-Covid evening.