Thursday, December 01, 2005

Of Snowflakes and Fractals

Its December - the season of Christmas and what's more, lots and lots of snow. I have to confess, I am one of those strange creatures who actually embrace the cold. Low temperatures faze me not a bit; aestial, blue skies and bright sunshine drive me to distraction though. It's no surprise that I'm eagerly awaiting some snow in NY.

While on the topic of snow, I just have to dwell on the fascinating concept of fractal dimensions briefly. In addition to the more conventional notion of Euclidean geometry (where we deal with one, two and even three-dimensional objects), there exist many objects in nature that cannot be described using integer dimensions. For instance, a coastline is neither a straight line, nor a shape that can be described as an area - its dimension lies somewhere between 1 (a straight line) and 2 (an area).

The fractal dimension of an object, speaking intuitively, is roughly an estimate of the extent to which it fills the Euclidean shape in which it is contained. For instance, a coastline with a fractal dimension of 1.2 is closer to a straight line than a rectangle, while one with a fractal dimension of 1.8 has more curves. Nature is full of objects with fractal dimensions - coastlines, trees, mountains, and - you guessed it - snowflakes.

Koch's snowflake
is actually one of the more famous fractals. Its construction is pretty simple - consider an equilateral triangle. Every side of the equilateral triangle is then divided into thirds, and a new triangle is created on each of the middle thirds. Every successive iteration would increase the complexity of the figure, but more importantly, every new triangle in the figure would increasingly look like its predecessor. Use this link to see a demo (uses applets).

Self-similarity is one of the more important properties of fractals. For instance, in the abovementioned example I said that 'every new triangle...would increasingly look like its predecessor'. To put it another way, if you were to take a microscope and take a closer look at the triangles in the Koch snowflake, every triangle would look the same (assuming an infinite number of iterations, of course), irrespective of scale. This is true of a lot of natural phenomena, such as coastlines, mountains, snowflakes, pine-leaves etc - all of them have fractal dimensions. This phenomenon is a characteristic of fractals, and is known as self-similarity or scale-invariance.

Fractals are being analyzed more and more today, as they are being extensively used in fields as diverse as image compression algorithms, graphics & special effects (Star Wars uses it extensively for all those fantasy-landscapes), music compression algorithms, stock-market analysis and biology.

No comments: