The Coastline Paradox
On first reflection, if someone asked you the question: “How long the coastline of Great Britain?” (or any other landmass for that matter) You’d probably give it your best guess and return an integer in miles or kilometres.
The true answer however, is that it depends! Determining the length of a countries coastline is nowhere near as simple as it would first appear.
In 1951, Lewis Fry Richardson, a mathematician and pacifist was studying armed conflict. More specifically, he was trying to answer the hypothesis of whether the length of border shared by two countries had any bearing upon the likelihood of them going to war.
At the time, the prevailing method of estimating the length of a border (or coastline) was to use a large scale map or aerial photograph and a set of chart dividers. The dividers were then used to divide the border into n equal length, straight line segments of length l where the end of each line bisected the border or coastline being measured (effectively down-sampling the level of detail).
Richardson discovered that the sum of the segments (total border length) is inversely proportional to the common length of the segments l. The result astounded Richardson as in other words, the finer the resolution you use to measure a coastline, the longer the total measurement will be — as l approaches zero, the length of the coastline will tend towards infinity.
The measured length of a coastline directly depends on the resolution of measurement used.
Whilst conducting his research into border disputes, Richardson noticed that the Spanish/Portuguese border was stated to be 987 km by Spain and 1,214km by Portugal, a difference of 227 km, which could now be explained purely by the fact that the two countries had measured their border using different scales. This was the beginning of the coastline paradox.
If the coastline of Britain is measured using units 100 km long, the total length of the coastline is approximately 2,800 km long whereas if units of 50km are used, the total length is approximately 3,400 km, a difference of 600 km.
The beginnings of Fractal Geometry
This coastline paradox, first observed by Richardson provided the inspiration for one of Benoit Mandelbrot’s initial publications on the topic of fractals. Mandlebrot’s paper “How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension” was first published in Science in 1967, although it wasn’t until 1975 that he coined the term fractal.
In the paper, Mandelbrot discusses an empircal law discovered by Richardson which could be approximated by a function of the form:
L(G) = MG¹⁻ᴰ
Where:
- L = length
- G = Measurement scale
- M = positive constant
- D = constant called the dimension, ≥ 1.
Intuitively, if a coasline looks smooth, the dimension should be closer to 1, and the more irregular the coastline, the closer D should be to 2.
Mandlebrot goes on to describe various mathematical curves related to the Koch Snowflake, the earliest described Fractal. These curves are defined as being strictly self similar whereas coastlines are less definite in their construction as they are formed by natural events that create patterns in statistically random ways. Idealised fractals on the other hand are formed by the repeated iteration of simple sequences.
Whilst Mandlebrot does not claim that any coastline or border actually has fractional dimension he shows that Richardsons law is compatible with the idea that geographic shapes such as coastlines can be modelled by random self-similar figures of fractional dimension i.e. Coastlines are inherently fractal in nature.
This is how the simple question of “How long is the Coast of Britain?” and the Coastline Paradox informed some of Mandlebrot’s earliest thinking on fractals and formed the basis of his linking of mathematical objects to natural forms, an overriding theme of the rest of his life’s work.
Reference:
How long is the Coast of Britain? Statistical Self-Similarity and the Fractal Dimension, Benoit Mandlebrot. https://science.sciencemag.org/content/156/3775/636