Where we learn that it’s the conditions,
not the fractal itself, that is complex.
I think it’s kind of weird that I didn’t have the whole world of fractals in a bear-hug of an embrace till I was…oh…probably in my 30’s? I guess I had just never stumbled upon them and didn’t know that the genesis, definitively, of a fractal is alls about the maths. I’m sure I had seen them and probably even paused / remarked / gazed in pure delight at a fractal or two or a zillion in my day-to-day life prior to actually reading in-depth about them. Being drawn to visual patterning, especially when there’s a mathematical basis for that patterning, is totally the norm for me.
I do recall the exact moment of my discovery + understanding of fractals. I was sitting at a bar, by myself, eating dinner, and reading a math book. It was a thin paperback book from the late ‘80s and it was entirely about fractals; the mathematics that brought them into existence, but also the how and why they come about, the different geometric shapes they can take on, and how you, too, can make a fractal with the help of a computer(though “it’s important to have a screen with a high-resolution” which is then specified as being “640x400pixels”)! And I remember that I dove so deeply into that fractal world that my dinner got cold before I could finish it yet I didn’t want to put the book aside at all because I knew that as the sun was setting outside, the lighting in the bar would be matching the evening’s progression into dimness and subsequently that bistro bar would start taking on a romantical feel and though reading by candlelight sounds like it could be a cozy good time, it also puts a hella strain on the eyes. So cold dinner be damned, I decided to keep reading because I just wanted to know more about these fractals.
Just in case the word ‘fractal’ brings absolutely nothing to mind for you, according to the book I was reading at the bar that day – Fractals by Hans Lauwerier – the definition of a fractal is:
Other resources for defining the word often include ‘self-similarity’ when describing what makes a fractal definitively A Fractal, as opposed to being just any old geometric shape, and it’s sometimes referred to as a never-ending pattern (which is true, though that doesn’t mean that the visual of that pattern needs to be shown into infinity in order to qualify as a fractal – though if it does do that, it definitely supports the definition further).
You could also just go with the definition as quoted from the memoir of the granddaddy of fractals, Benoit Mandelbrot, who is legit the one who coined the word back in 1975 – less than 50yrs ago! – of this wacky geometry he was witnessing.
The crux of a fractal is the visual geometric edges of a shape – like, a coastline is an edge of a shape, as is the edge of a snowflake, – and you notice that when you zoom in on that edge, that edgy pattern repeats itself.
Yet, it’s not always something as easily-digestible as this:
Sometimes it is (what I would call) deeply scalable, like this:
That first image showed the repeated geometric pattern at different scales, yes, but the second image showed that, in addition to the edges seeming to represent fractal characteristics, if you were to zoom-in on one section and repeatedly scale in closer and closer (as shown on the right part of the image), the mass area contained within the edges exemplified self-similarity as well. It’s like – if you measured the space between the twists and turns along an edge, and you also measured the mass area of space existing within the patterned edges of twists and turns, that scaling ratio would also be, the friggin same. It’d be self-similar.
And this happens in nature. See: the ferns on the homepage of this very site.
And though I know perfectly well that us humans may not be considered to be as natural an entity as a fern is, I was inspired to consider the things that we do appear to, well, naturally do, that appear to naturally come about.
I thought: Are there actions, as in human actions – and I do mean actions, not reactions – that’d one would consider as a natural action, perhaps instinctual, a considered movement that lead us to – nay, dictate us to- the timing (aka, in this case, the measurement) of the patterns of those actions that we then we repeat to do.
Can human actions show up as fractals?
It was when I read the late global health physician and medical anthropologist Paul Farmer’s To Repair the World and he referenced fractals when talking about “the broader world around us and the past upon which it’s built” that I, too, could see how it could overlap into daily life, especially when I thought about a repeating geometric shape scaling, getting bigger or getting smaller. He goes on to say,
“things that [can] look wildly different from one perspective but the same from another perspective, another scale…we need to think fractally, at several scales at once, in order to get our arms around the most vexing problems of our day and in order to innovate at all realms.”To Repair the World – pgs 164-165
At this point, I was deep into the fractal research universe and marveling that there was a whole methodology of using fractal analysis in every discipline from the natural sciences to technology to financial investment services. But the thing that I was hooked on was Farmer’s fine point that “thinking fractally” was more than just identifying and scaling a patterned process. He knew that recognizing the role that human behavior and all its quirks played was something to embrace in the creation, and innovation, of impactful and sustainable systems.
What could we do, as leaders or as a collective,
to recognize and harness these most complex of patterns within our own beings?
Consider this section from the afterword of Mandelbrot’s memoir, written by one of his Yale colleagues, Michael Frame.
To be sure, many scientists and artists had noticed this, and the examples of continuous, nowhere differentiable curves were familiar from basic real analysis courses. But Benoit saw much more, a way to quantify these recurring patterns so that complicated shapes might be easily understood dynamically, as processes, not just as objects.Michael Frame, from the afterword of “The Fractalist”
As processes! Well, then.
Bottomless wonders spring from simple rules…repeated without end.Benoit B. Mandelbrot
Thanks math, you’re the best.