An Ode to the Curves of Sir Benoit Mandelbrot

Celadon Surf by ~mandelbrat on deviantART

If you've been around the internet for any length of time, you're bound to have stumbled onto trippy stuff like this. Well it's based on a branch of mathematics called fractal geometry and it's... What's that... you're saying you've heard about fractals and have had questions about it ever since high school? And what, you were just thinking how great it'd be if someone were to explain the basics and their relevance to everyday living? Well today is clearly your lucky day! Pioneered in 1975 by Sir Benoit Mandelbrot, fractal curves (also called Mandelbrotian curves) are rarely given more than a passing mention in school. Thus denied, the average barely-pubescent male must then seek his curves elsewhere... *wink wink*

But I digress. Traditionally, what we're taught in school is the regular ol' Cartesian geometry. We graduate from the doodling of our early years to learn about straight lines and how to construct them. Then it's on to squares, circles, rectangles and triangles. Then it advances to three-dimensional shapes and all their possibilities; not just spheres, cubes and cones but then conical-sections as well. Conical-sections incidentally, are the different 2-D curves (the ellipse and others) you would get if you were to take a katana and go to town on a traffic cone.

So with all of these shapes available, it seems like we're pretty much set to go out with a pen and capture pretty much anything on paper and so we did! We built bridges and buildings and skyscrapers and even sent a man to the moon! But over time, it started to become apparent that Cartesian geometry had some serious limitations when it came to describing the geometry of naturally occurring objects. So for example if you had to design a car you were golden. But what if you had to describe the way a tree was growing, or the way that lightning strikes the ground. Sure, you can draw it out but the point is that you don't really know any more about it after you're done.

Enter Benoit Mandelbrot. *Cue Superman theme*

"Why is geometry often described as 'cold' and 'dry?' One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line... Nature exhibits not simply a higher degree but an altogether different level of complexity."

~Benoit Mandelbrot

The Fractal Geometry of Nature (1977), Introduction, xiii.

So when he looked at a picture of lightning, he noticed several properties like self-similarity (that a zoomed-in version looks exactly like the full picture) and scale invariance (the zoomed-in version has the same mathematical description as the overall picture). Basically there's no difference between the branching that happens at any scale level, it's the same rule applied recursively over and over again until you get the entire branching structure.

Self-Similarity illustrated in a lightning strike

This was a huge breakthrough because it suddenly allowed people to study things that were earlier dismissed as just being "too chaotic". Medical science made massive headway in terms of our understanding of the lungs (because of the way bronchi branch into bronchioles and then subdivide until it finally reaches the alveoli) and in the circulatory system (again, because of how chaotically it seems the arteries branch into capillaries) among others. Fractal geometry also led to a greater understanding of seemingly chaotic events and heavily influenced modern Chaos Theory. Fractals can also be seen pretty much everywhere because the mathematics allows you to automatically generate say, a landscape or a sea of clouds rather than having to actually model them by hand.

A landscape from Minecraft

(Side note : Check out Minecraft if you have some time to kill and want to do something that feels constructive. Or watch the minecraft-yogscast if you just want to waste time. ^_^)

I'm hoping some poor soul didn't have to sit and draw this all out. :p

But here's where it gets really interesting for me. What Sir Benoit Mandlelbrot essentially accomplished was to actually measure the seeming randomness of things, what he called 'roughness', and in doing so managed to create a new geometry. A geometry for thing that didn't have a geometry! Over the course of his work he expressed all of his various insights in what is called The Mandelbrot Set where as you continue to zoom in you keep seeing the same patterns getting generated over and over to infinity. The video below is a zoom down to the 6th-level mini-set, set to Jonathan Coulton's "Mandelbrot Set". Check out the lyrics btw, highly amusing. :)

What's cool about the Mandelbrot set is that it demonstrates that an infinitely complicated structure can arise from the recursive application of simple rules. This is super important, so I'm going to say it again: "...an infinitely complicated structure can arise from the recursive application of simple rules."

And here, i think, we may have been too timid with the way we apply fractals. We've applied the learnings of fractal geometry to a whole bunch of things but when we've applied them to ourselves we seem to stop short at a "hardware" level. So we're quite comfortable talking about how our lungs and our blood-vessels and even the rhythm of a heartbeat have a fractal construction (Check out the Music of the Heart project btw. Music from ECG recordings = super cool tunes). But at a "software" level it seems we haven't considered that we might ourselves be somewhat fractal in nature in terms of even our personalities. The truth is that it's a worrying thought that we might not be completely in control. But as much as we might protest that we are the masters of our own fate or destiny or whatever, evidence would suggest that the course of every life has a strong element of the haphazard to it. At the very least we can admit that we don't control every single aspect of what goes on in each and every day. But rather than let that be a worrying thought, maybe fractals could offer another way to examine our lives. Like if you want to figure out what your life might be like in the next twenty or forty years, take a day and just kind of walk around your own life as it is right now. Patterns might start to emerge that might let you glimpse the future. Or maybe you're not particularly happy about something in your life right now. Rather than something drastic where you're booking a trip to the Himalayas to meet with the Rishis, maybe the answers you need are right in this moment.

As an approach to problem-solving as well, fractal geometry might have something to offer. Say there's a particular situation you feel anxious about, it might not be practical to try to fix the entire big picture all in one massive burst of change. Maybe all it would take is to handle the littlest thing right now, and then again in the next moment and again over and over and to just stay watchful as each moment compounds on itself. Eventually a new picture will have emerged out all of the individual decisions. I read a quote once that neatly summarised it:

"Watch your thoughts, they become words.

Watch your words, they become actions.

Watch your actions, they become habits.

Habits become character, and character becomes destiny."

~Lao Tze

And again,

"We are what we repeatedly do. Excellence then, is not an act, but a habit."

~Aristotle

And at a larger scale, if a city is composed of individuals then maybe whole societies could be analysed as having this sort of fractal, recursive element to it. I'm not quite sure how one would do so but if you could model a city in this manner then solutions might present themselves to problems like large-scale waste management, or poverty or health care or racism even, who knows! Anyway I'm not really going anywhere with this line of thought, it just continues to fascinate me how much Philosophy is present in Math and Science and how they kind of just blur together at the edges. In closing, I leave you in the capable hands of the legend himself: