Zoom in, zoom out4 min read . Updated: 30 Aug 2012, 05:35 PM IST
The fascinating thing about fractals is: you don’t need advanced mathematics to generate them. The equations involved are simple
Your daughter’s geography teacher asks her to find out how long India’s (mainland) coastline is. She comes to you, the all-knowing parent, to ask for help. “Easy!" you say. You pull out your Random House Mini World Atlas that’s gathering dust on your top shelf and open to the page marked “South Asia", scale 1cm to 500km. The Indian peninsula is like two sides of a triangle, you note, and lay your ruler along each side in turn to get an approximate answer. Close to 7cm, thus about 3,500km.
Your daughter runs off happily. Idly, you flip the pages. On the “India and West China" page, you note the larger scale of the map and also that India’s east coast is better represented by two straight lines (one from Kanyakumari to Vijayawada, the other from there to the Sundarbans). Measuring this way, you estimate the length at 4,200km.
Hmm, you think. 3,500 as a rough estimate, but suddenly it’s up to 4,200? Daughter, come back!
If you’ve been following my columns— it’s ok, you can admit it—you’ll remember that I touched on this coastline question in “As Many Holes As You Want". When you move to ever-larger scales, as I wrote there, you find “straight lines are no longer enough: you start noticing gulfs (Kutch) and estuaries (Sundarbans) and bays (Marine Drive), each of which add to the length."
As you move up map scales this way— that is, as you “zoom in"—you get ever-larger estimates of the length. At some point you will reach the generally accepted figure for India’s coastline: it’s about 5,700km long. But you needn’t stop there. Wherever a cartographer has assumed a straight line, or a smooth curve, you will find distortions as you zoom in, each adding to the measure. Take the causeway out to the Haji Ali Dargah, about 500m long. That little spur alone adds a kilometre.
Or take rocks along shorelines. If we measure them all, with all their nooks and indentations, our number will keep rising. And you can extend this reasoning to mangrove roots, pebbles, grains of sand, on and on. Longer estimates every time, dependent only on what level of detail you want to stop at.
If you think about it, there’s no limit to detail, to how much of it you can find to measure. When you magnify the coast, you don’t eventually find straight lines and smooth curves. You find more detail, that’s all.
The astonishing implication: the coastline is as long as you want it to be. In effect, it is infinitely long.
Our idea of length itself goes for a toss here. You can reasonably say your daughter’s skipping rope, for example, is 2m long. But generally accepted though it is, you can’t quite as reasonably say India’s coastline is 5,700km long.
To me, there’s something intensely appealing about this idea of zooming ever closer to see ever-finer detail. And where things are intensely appealing, it’s a good bet there are mathematicians trying to understand them as well. Case in point: Benoit Mandelbrot. Through the 1960s and 1970s, this French mathematician was fascinated with forms and shapes, like coastlines, that retain their complexity no matter how far you zoom in. He found the same theme—and it helps to visualize each of these as you read—in ferns, in crooked bolts of lightning, in pieces of broccoli or cauliflower, in river estuaries.
Even in the structure of the human lung, with its air passages that branch indefinitely.
In trying to describe such shapes, Mandelbrot invented a name for them in 1975: “fractals". With good reason, Mandelbrot called them the “geometry of nature". If architects use straight lines and circles, nature abounds in fractal shapes. “Clouds are not spheres," said Mandelbrot, “mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."
Fuelled by that thinking, fractals are invaluable in computer or film simulations of natural phenomena. Want a mountainous landscape as a backdrop for your car chase? Fractals can help. Want snowflakes, clouds? Fractals, again. All because each of these shapes looks fundamentally the same at whatever magnification you choose: yes, think fractals. And so fractals are a natural fit for the way computers work. This is probably why the study of fractals parallels the explosion in the use of computers over the last 35 years.
But the most fascinating thing about fractals is this: you don’t need advanced mathematics to understand or generate them. The equations involved are remarkably simple. Every computer science student, at some point early in her career, sets out to produce fractals on her computer screen. With even minor tweaking of her original equations, she can produce an array of beautiful fractals. Simple equations, but they produce objects of stunning complexity that retain that complexity even when magnified.
Stunning complexity, from remarkable simplicity. Think of how you first approximated the coast of India, that infinitely intricate shape, with two sides of a triangle.
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. A Matter of Numbers will explore the joy of mathematics, with occasional forays into other sciences. Comments are welcome at email@example.com
To read Dilip D’Souza’s previous columns, go to www.livemint.com/dilipdsouza