Saturday, 8 February 2014

From Trivial to Noise via Fractals

Fractals are really interesting objects. They’re interesting because they bust the myth that mathematics and its objects are always boring, predictable, and certainly not beautiful or intricate. Certainly not like objects we see in nature; clouds, mountains, flowers, water streams, snowflakes.

Let’s look some of these so-called boring, predictable, flat objects. A square, a circle, and even a 3-dimensional cube or cylinder.

These objects are certainly simple compared to what we see in nature and find intricate or beautiful. Actually some people do find these objects really fascinating, but I suspect most people wouldn’t want to stare at these objects for a long time, exploring their nooks and crannies. This last bit is important - these objects don’t have nooks and crannies, they don’t reveal new details for us to explore the longer we look at them. If we looked at them with a magnifying glass we wouldn’t find anything new or surprising. In that sense, we can call them boring.

Now lets look at something totally unordered, noisy, following no rule or regulation, other than being totally random.

The picture is of the familiar white noise we used to see on old television sets when they weren’t tuned quite right to a broadcasting channel. It’s not that interesting either. Actually we humans have a tendency to try to project recognisable things onto noise, which is why we love finding faces and dragons in the clouds! If we set that human tendency aside, we’d find purely random objects boring.

Now look at the following objects; a snowflake, some clouds, river systems and a piece of cauliflower.


There is something about these objects that leads us to explore them a little longer. It’s not just that they are very detailed, but that there is some semi-regularity about them. There appears to be a theme that pervades them, a repeated pattern, but repeated not in that boring easily predictable way, but in a way that is somewhere between the two extremes of boring regularity, and total messy randomness. This is an important idea. The idea that objects that appeal to our human senses and minds are somewhere between total boring regularity and total randomness.

We’ll be coming back to this idea later, carefully treading a path between regularity and randomness where mathematics produces some really intricate beautiful objects.

If we zoomed into a bit of the above images with a microscope or a telescope, it would be difficult to guess at what scale we were looking at the object. Each little floret of the cauliflower looks like a collection of smaller ones, and you can’t guess whether the bit you’re looking at is 5 centimetres wide or 5 millimetres. Each bit of cloud fluff looks like a collections of smaller clouds fluffs - is it a metre wide or a kilometre? Each river network looks like it could have been taken from a height of 10 metres or equally at 1000 metres. This is called self-similarity at different scales.

The following objects are less organic and natural than clouds and plants, but they do have that self-similarity we talked about above. Their construction is much more regular. We include them here because when people talk about fractals they often think of these objects too. Personally I think they’re interesting to look at but the interest wears off quickly because they’re closer to the “regular” world than the more organically less predictable natural objects.

The first one is a Koch snowflake, and the second is a Sierpinski gasket. You can easily find out how to make them, but the basic idea is to take a simple regular shape such as a hexagon or triangle, and repeatedly add smaller versions of this shape at ever greater levels of detail.

The following summarises this spectrum of objects, from totally random to totally ordered, with the interesting objects between the two. It’s as if the tension, the constant battle, between total order and total randomness, produces the most natural organic intricate beautiful objects.