## Coastline Length of Britain

I always assumed that most people had heard of the famous 1967 Mandelbrot paper on this. Apparently though, not the Ordnance Survey (*or*the CIA). Reproduced below is my correspondence so far on this matter....

From: Me To : Ordnance Survey Customer Services Date : 2006-02-16

Dear Sir / Madam,

Nearly 40 years after the publication of Benoit Madelbrot's famous paper, "How Long is the Coastline of Great Britain", in which he demonstrates that the coastline is actually a fractal (and therefore of infinite length), I notice that at http://www.ordnancesurvey.co.uk/oswebsite/freefun/didyouknow/ the Ordnance Survey states quite definitively that "The coastline length around mainland Great Britain is 11,072.76 miles". Elsewhere, in places such as http://www.ordnancesurvey.co.uk/media/news/2004/jan/iczmap.html, you repeat this figure of 11,000 miles, or 18,000km.

I am curious to know as to how this figure was reached. Is it a 'piece of string' measurement taken from the largest scale map that you publish? Or did somebody actually walk the coastline path with a pedometer? Or is a 'standard granularity' applied, turning the coastline into a series of straight lines?

( Funnily enough, the CIA Factbook, possibly the most widely consulted geopolitical text ever, states that the length is 12,429 km, or 7723 miles. Are their pieces of string slightly thicker than yours, do you think - or are they simply working off a less detailed map? )

Eagerly looking forward to your reply,

Ben Daglish

From: Ordnance Survey Customer Services To : Me Date : 2006-02-20

Dear Mr Daglish,

Thank you for your email dated 16th February 2006 regarding the above.
Please see below the information Ordnance Survey holds regarding the
coastline of Britain:

The Coastline length of mainland Great Britain is 11072.76 miles
or 17819.88 km as measured along Mean High Water.

*[SNIP - Table of Lengths of England, Scotland, Wales etc. ]*

The Coastline is measured from Mean High Water which extends into estuaries
until the normal tidal limit (NTL) is reached. The islands included in the figures are: *[SNIP - Island list (more fractals!)]*

*Then the killer line...*

Source: Ordnance Survey Boundary Line data at 1:10000 scale. Revised: October 2001

*And an added note, presumably from the mapping dept.*

You will notice that the information was captured from the 2001 Boundary-Line dataset. Because the coastline length is constantly changing, I have recalculated the value from the latest version of the Boundary-Line dataset. This has provided a GB coastline length of 31442km (mainland including islands).

I hope this information is satisfactory for your requirements and thank you for your interest in Ordnance Survey.

With regards

[Name Spared ]

General Enquiries Helpdesk Advisor | Ordnance Survey | Customer Service Centre

From: Me To : Ordnance Survey Customer Services Date : 2006-02-20

Dear [Name]

Thank you for your answer, and the comprehensive table of figures. Unfortunately,
I suspect that my original question was maybe misunderstood, as shown by the line
at the bottom of the table...

Source: Ordnance Survey Boundary Line data at 1:10000 scale.

All this means is that the coastline length is 11,072 miles *when measured in straight lines at 1:10000 scale*. If a 1:5000 scale had been used, the figure would be completely different - possibly twice as much. If measured at a 1:1000 scale, it would be completely different again. And, as I said, if you walked around the coastline with a pedometer, providing a '1:1' scale, it would be completely different again (depending on how small the pedometer was, and how far the walker was prepared to 'wiggle in and out of the cracks').

Basically, as stated in my previous mail, the coastline is of infinite length. All the Ordnance Survey have done in this case is taken a completely arbitrary decision to settle on a 1:10000 granularity in order to provide a figure that is in fact pointless (other than to possibly serve as a comparison to other coastlines measured at the same granularity). This is surely obvious, given that the CIA obviously decided upon a completely different granularity when providing their figure, which is some 5,000 km less than yours.

If you Google for "coastline of Britain", you will find (near-exclusively, with the exception of a few mentions of the CIA figure) many references to the original Mandelbrot paper, which will explain this more clearly.

So my question is: why provide a figure for the coastline length at all? If it is ever given, it should *always* be pointed out that this was measured at a scale of 1:10000, otherwise it becomes a completely useless measurement. In fact, stating a figure in the first place seems rather futile - it's information for information's sake, providing no useful function at all. Is this figure ever related to *anything else* in the 'real world'? If it is to provide an approximation for (say) walkers of the coastline path, then it should be measured at one scale, along one particular set of straight lines. If it is to provide sailors an idea of how long it would take to go around the coast, it should be measured with a completely different set of straight lines. Is it possible to provide a single case where the figure of 11,072 miles has actually been used? (If so, then I suspect subsequent calculations would be severely flawed).

Again, I look forward to your reply.

Yours,

Ben Daglish

From: Me To : Ordnance Survey Customer Services Date : 2006-02-20

Dear [Name]

A quick correction - although I state a number of times that the coastline is infinite, of course this is a 'pure maths' infinity - quantum physics unfortunately limits the measurement to multiples of the Planck Length...the Wikipedia quote below is taken from the article about Lewis Fry Richardson (who actually worked on this problem some 20 years before Mandelbrot).

"Note that Richardson's results do not mean that the coastline of Britain is actually infinitely long. This would require the ability to measure with infinitesimally small rulers, something which quantum physics says cannot be done, as there is a lower limit to the smallness of a measurement, the Planck length. What Richardson's results do show is that natural geographic features, when considered over a wide range of scales, do not behave in the same way as the objects of Euclidean geometry."

So, in fact, it *would* be possible for the OS to provide an 'accurate' figure in terms of quantam physics - though this would be (at the very least) in the realm of billions, rather than thousands, of miles.

Cheers,

Ben

From: Ordnance Survey Customer Services To : Me Date : 2006-03-03

Dear Mr Daglish

Thank you for your email dated 20th February 2006 regarding the above.

Everything that you say is correct (with the possible exception of the coastline length being infinite). The reason that we supply this information is all to do with the NIMSA agreement, i.e. we are required to supply this information (or any other information) by ODPM as and when it is required. The NIMSA activity requires Ordnance Survey to update mapping and address data in rural and moorland areas is one of the eight monitors on which our performance is measured. NIMSA is a not-for-profit agreement between the Office of the Deputy Prime Minister (ODPM) and Ordnance Survey.

*So It's not our fault - it's John Prescott's!*

The fact that these facts and figures seem to cause more trouble than they are worth (i.e. someone will always dispute them) puts us in catch-22 position. The coastline length (as with all our Geofacts) can vary depending on what products are used for the calculation. This is why we state that the length has been calculated from Boundary-Line at a scale of 1:10 000.

*Except they don't, unless hassled*

The information is primarily intended as an indication of the coastline length (e.g. for pub quiz type situations). It is not intended to be a 'true' length - as

i) we couldn't calculate a 1:1 value; and

ii) the length will vary from day-to-day (due to geographical factors, e.g. coastal erosion).

*So where're the qualifiers? It's a bit like children's armbands that say "This product is *not* meant to be a life-saving device." Where, when they publish this figure, do they ever say "This is not intended to be a 'true' length" ?*

The fact is that I could calculate a variety of lengths based on different datasets (Boundary-Line, Meridian 2, Strategi), coordinate systems (Spherical or Cartesian), projections (British National Grid or WGS84), etc.

*Yes, which was my point.*

Out of interest...We are currently generating a 'complete' coastline from the OS MasterMap Topography Layer - so will be able to provide the most accurate figure we are currently capable of.

*??? How accurate?*

I hope this information is satisfactory for your requirements and thank you for your interest in Ordnance Survey.

With regards

[Name Spared ]

General Enquiries Helpdesk Advisor | Ordnance Survey | Customer Service Centre