Hickey versus the fractals

A friend of mine, who shall remain Hickey, sent a group of us this email:

http://kottke.org/10/02/insanely-deep-fractal-zoom

This is probably one for you, Mairtin, but can anyone explain to me why I'm supposed to be impressed by this. I'm seeing these Mandelbrot set's everywhere on the tinterweb [sic] lately and I don't get it. Why isn't it just an animation that goes on for a long time (in theory can't it go on infinitely)? What exactly is it supposed to convey, mean, represent?

It has been a while since I deigned to put up a blog post and while writing my first [lengthy, oft-times kitsch and pontificating] response to Hickey's challenge, I decided that this fit the bill.

Mairtin McNamara for the defence, your Hickey.

The Mandelbrot set is the flagship for a group of mathematical oddities known as fractals.

Fractals look the same at any scale of magnification.(But, as you said, theoretically.)

This is their key characteristic. Well, that and the fact that they arise from a few simple steps or the solution to a few equations; You get something of potentially infinite vastness, but contiguous enough to shift smoothly from coarse to fine scales, all from a tiny package of information.

Evidence A for the defence:

This is bread and butter in biology. The construction of nerve fibers, the lungs, the folds in the brain, fingerprint patterns, blood vessels, etc. are all examples of the application of fractal behaviour by human genome/body. With a fractal expression it is tantamount to having a button called "create vein system."

It is a few years since I read it, but in Max Gleick's book, Chaos, he as an example of how flexible fractals can be, he writes about a guy who created a set of expressions that yielded the face of a dog, and more easily the leaf of a fern.

Fractal geometries also depict the behaviour of dynamic systems.* I say depict, since it is useless as a predictive tool- as evidenced [I haven't used that as a verb in a while...] by the lack ironclad predictions in the stock market, or the bloody weather.

Simply, it shows you the quality of things, what kind of thing to expect.

Evidence B for the defence:

A weather forecast is the most likely outcome based on numerous simulations performed on current available data. It is an educated guess, not a prophesy [particularly since Met Eireann uses arcane code from the 1960s that is to modern forecasting what an ass-and-cart is to a Ferrari Enzo...]

Fractals come into this because they form in the visualisations/plots of the data- you could probably interpret it as a piece of music, if you were so inclined, but I wouldn't recommend it though.

I digress... the fractals show as almost repeating and never overlapping phase trajectories. In the context of weather the shape of the fractals show in broad-stroke terms how many different types of day could unfold and whether the weather of one of these days is a strange attractor of sorts.

Evidence C for the defence:

Fractals hold an interest in game graphics and special effects, since both want simple little packages that use little energy to provide complex results: clouds of smoke, water splashes, explosions, river deltas, branches of trees, the way wind bends the blades of grass in a lawn, hair on the head of a back-flipping cheerleader, etc. can be represented by some kind of fractal and simulating contextually , or phenomenologically can hugely simplify things, when compared to modelling individual elements in painstaking detail.

Closing statement:

In summary, it is unlikely we could exist today without fractal phenomena, since our bloated DNA would be rife with lethal errors in their convoluted algorithm "how to construct vein system" ["error at vein-889-9-3-1, address not found"]

The defence rests, your Hickey.

I haven't read it in about 6 years, but there is a book called the Collapse of Chaos that does an amazing job of addressing the flaws of the deterministic view of the world adopted by most scientists since year dot, and how it compares with the contextual (i.e. qualitative view) that fractals, Poincaré plots, horseshoe plots, etc. represent. A caution though: it is densely written, I could only go through 10-20 pages at a time before needing a break to think about what they were saying. Max Gleick's book I mentioned earlier is a better one to cut your teeth on, or even to wet your toes with.

You'll notice the sparsity of references in this, it is more a personal essay, than a formal response. Nonetheless, it will be this week's blog post, for no other reason than my canookie uncle demanded one of me..

*Dynamic systems is a catchall phrase: populations and food supplies, the rate change of data failures in a transmission with respect to time, weather- as I note at length-, and almost anything else you can think of. Hell even how a day in your life progresses can be considered as a fractal... if you lived it over and over again like Mr Murray in Groundhog Day and assigned numerical values to the things you did in your day and the outcome thereof... You get the idea.

In writing this little essay on "what the Mandelbrot set means to me" I am reminded of a comment my seconday school English teacher made at the end of one of my essays:

Somewhat overwhelmed by its own verbosity

I imagine that it will be my epithet...¹

Feeling guilty for inundating my friends with a wall of text, left in their inboxes waiting to be sprung on a Sunday morn, I wrote a more brief one:

If that wall of text was too much to consider reading on a Sunday morning:

The set is the iconic representation of infinite recursion, something rich, complex and almost the same arising from a simple set of rules.

Poetically speaking it represents life.

It is seen all over nature- from trees to blood vessels.

By corollary, fractal shapes that we make can follow the spirit of the law of the world around us as opposed to the letter of it.

If I get the chance, I'll comb through the, currently, baseless, but generally reasonable, claims I made above and insert citations. I make no promises though.

*****

¹I could write another post on how much I enjoyed my teacher's sense of humour, but then it wouldn't be very funny, would it?