elb's hovel of thoughts

Thursday, February 10, 2005

Fractal geometry

Allow to introduce a brance of science which may or may not be familiar to most - fractal geometry; where patterns present on a small scale are also found replicated on a large scale.

Is this possible, that what is observable on a scale of a few millimetres can be found present on a scale of centimetres, metres, decametres, even kilometres and megametres? Yes.

This sense of scale is present to us in nature, and can also be replicated via simple geometric progession, which many of you would have learnt doing mathematics at a grade 10 or 11 level.

Let us begin with a basic geometric progression (here called a Cantor set) involving a single dimension - a straight line

Step 1: We have a line -------------------------------
Step 2: We now remove the line in between the 1/3 rd and 2/3 rd marks.
This leaves us ----------- --------- -------- ---------
Step 3: We now remove the 'open middle third' of each remaining set
This leaves us --------------- ----- -------- --- -- ---
Step 4: Repeat, giving ----- - -- - - -------- - - -- - -

Get the idea?

The same can be done with two dimensions; i.e. Koch curves and Sierpinski triangle; in fact, a full blown Koch island has similarities with a snowflake! The most famous of them all though, is probably the Mandelbrot Set (notice how the very miniature patterns still look like the parent shape!)

Mandelbrot set

This also can be done with three dimensional objects; i.e. Menger sponge, and even objects with higher dimensions (don't ask how!).

In nature, probably a good example is the fern leaf

Image taken from http://www.stenslandsystems.com/

Notice how the individual leaves take on the same pattern as the overall shape of the leaf!

Other good examples include a river. Why a river? Because, overall, a real river is really in the shape of a gigantic fan! That's right, a river isn't just a solo meandering body of water, when you take into account its tributaries and the tributaries' tributaries and the tributaries' tributaries' tributaries, it becomes the shape of a delta. That's right, a river's delta near the sea can be scaled up to reflect the true overall shape of the river. And it can be scaled down to show deltas forming within the delta, and even more and more sub-deltas being formed.

Another good example is the human lung, the delta scale applies just as well for this case. And so it is for lightning, for volcanoes, clouds, snowflakes, and so on.

If you still cannot follow me, let me present to you a question: How long is the coastline of say, the Malaysian Peninsula. Looking up a map would not give you the correct answer. Why? If say, we are to take a straight ruler that is ten kilometres long, then we will receive an accuracy limited to that ruler; i.e. only huge geographical changes in coastline would get accounted for. Smaller peninsulas and bays and cracks than 10km would NOT get recorded! If we change to a ruler that is a kilometer long, then geographical changes smaller than a kilometre would not be recorded! And so on. Thus; the TRUE coastline would be needed to be measured using a ruler that is probably several molecules across!

This concept of course, can also mutate itself into questions such as: how small can an iceberg get? If the concept of fractals were to be applied, then theoretically the smallest iceberg is undefinable while the largest occurs very rarely.

Anyway, I hope I haven't managed to get any of you too confused just yet :)

-Copyright Patrick 2005


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