Sunday, 20 June 1999

How long?

Four months ago, after spending a week's holiday walking the Pembrokeshire Coast Path, I decided to walk the coastline of Britain. A question I commonly think about — both of my own accord, and when asked by others — is just how far it is I'll walk. My hunch is that it'll be between two and three thousand miles.

As of this week I've completed about 200 miles of the walk - half of that in West Wales, the other half in East Anglia. Yet, looking at my progress map, it appears that the distance I've walked in East Anglia is about twice that which I've walked in Wales.

In his 1983 book The Fractal Geometry of Nature, Benoît Mandelbrot considers the same question. Mandelbrot does not give a definitive answer. Instead, he gives the cryptic answer, "it depends how long your ruler is"; his thesis is that a definitive answer will remain elusive.

Coastline length turns out to be an elusive notion that slips between the fingers of one who wants to grasp it. All measurement methods ultimately lead to the conclusion that the typical coastline's length is very large and so ill determined that it is best considered infinite. Hence, if one wishes to compare different coastlines from the viewpoint of their "extend," length is an inadequate concept.

I can explain my observations by saying that the Pembrokeshire coastline has a much higher fractal dimension than the East Anglian coast; any quick glance at a map will confirm this. Accordingly, the relatively long ruler length of the road atlas flattens out the kinks in the Pembrokeshire coast while on foot there are plenty of bays and headlands to walk around thus elongating the distance. Conversely the lower fractal dimension of the Anglian coast results in fewer such detours and very little extension of distance.

All great stuff, but it still doesn't answer the question. Maybe I'm walking in order to find out. Best estimates are that the coastline of England and Wales is three to four thousand miles long, with Scotland doubling that. There's a lot of walking left to do.

Posted by pab at 22:57