If we check out official Government figures we find that the mainland coast of Australia has a length of 35877 km, but this is a somewhat contentious matter, as is the measurement of all coastlines around the world.

The problem is that the figure we come up with is very much dependent of HOW we measure, and this is an important issue in arriving at an estimate.

If we measure the coastline with a ruler 1400 km long (specifying that both ends of our ruler must touch the coast with each measurement) we come up with a total of around 10800 km.

(Click on image to enlarge)

However we see that large parts of the coastline are omitted in this way and obviously a more accurate result will follow from using a smaller ruler.

If we use one about 700 km long, our new result is about 11300 km – an increase of 500 km or 4.6% with respect to our first measurement. (Click on image to enlarge)

But we are still missing a large amount of coastline including all the bays and inlets, so a smaller ruler should be used.

But proceeding in this way, we realise that by using smaller and smaller rulers, so that we can even measure around mangroves and individual rocks on beaches, our measured coastline will appear to become ever longer.

So we are left with the apparent paradox - the smaller our ruler the longer the coastline.

However, instinctively we feel that our estimates, although increasing with smaller rulers, will eventually converge to a limit that will be the true figure. But this may not be the case.

In the 1980’s the mathematician Benoit Mandlebrot pioneered a new type of geometry that was composed of figures called fractals. These much more closely resembled nature than the cubes, spheres, straight lines and triangles of traditional geometry.

Fractals have the interesting property of being “self-similar” – if you zoom in for a closer look at the side of a fractal the same detail of the full figure is maintained. No matter how many times you zoom in the same picture emerges without limit. You can see this effect here:

http://en.wikipedia.org/wiki/File:Phoenix%28Julia%29.gif

This produces the rather amazing fact that the “coastline” of a fractal is infinitely long – the smaller your measurement ruler the longer the coastline. (Click on image to enlarge)

The obvious question is therefore “is a continental coastline a fractal?” Here academic opinion is divided and to a certain extent we enter the realm of philosophy.

If we consider the Australian coastline is a fractal it is therefore infinitely long, just as are the coastlines of Tasmania, Europe and the United States. If not, we can stick with the official Government figure of 35877 km as given above.

A similar, but much simpler situation arises in many Australian backyards. We can ask “How long is the back fence” and if this consists of a paling structure, the answer is probably longer than you think.

(Click on image to enlarge)

Lets say the fence consists of 200 palings 10 cm wide and 1 cm deep. If we measure it with a string we will come up with a figure of 20 metres. But if we measure around the palings as shown, our back fence will become 22 metres long.

If we want to be really precise and measure carefully around each wood splinter it will be longer again and further, if we consider that the palings are fractals then we will have an infinitely long back fence. But don’t tell the Council – they might increase your rates.

## Saturday, August 8, 2009

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