### Ireland’s Fractal Coastline

Reports of the length of Ireland’s coastline vary widely. The World Factbook of the Central Intelligence Agency gives a length of 1448 km. The Ordnance Survey of Ireland has a value of 3,171 km (http://www.osi.ie). The World Resources Institute, using data from the United States Defense Mapping Agency, gives 6,347km (see Wikipedia article [3]).

Fractals

How come the values differ so much? It is because the coastline is fractal in nature and the measured length depends strongly on the “ruler” or unit of length used. A straight line – like a road – has dimension 1 and a plane surface – like a field – has dimension 2. But some curves – like coastlines – are so wiggly that they have a dimension between 1 and 2. Such curves, with fractional dimensions, are called fractals.

An early example of a fractal curve is the Koch snowflake, which first appeared in a 1904 paper titled “On a continuous curve without tangents, constructible from elementary geometry” by the Swedish mathematician Helge von Koch. It is built up by repeatedly subdividing the sides of an equilateral triangle into three parts and erecting a smaller triangle on the central segment. Here are the first four stages:

The Koch snowflake (image from Wikimedia)

The length of the snowflake grows without limit as the subdivision process continues: each step adds one third to the length, so at stage N it is (4/3)N times the length of the original triangle, growing exponentially with N. The fractal dimension of the snowflake is log 4 / log 3 ≈ 1.26.

Richardson’s Scaling Law

Fractals were considered by the English Quaker mathematician Lewis Fry Richardson. When he measured the length of the west coast of Britain, his estimates varied widely with the scale of the map he was using. Larger scale maps include finer detail, resulting in a longer coastal length. As we may choose to include variations corresponding to every rock, pebble or grain of sand, it is impossible to assign a length unequivocally. All we can do is to describe how the length varies with our unit of measure.

Richardson measured the coastal length by walking dividers with fixed distance Δ along a map. He found that the dependence of measured length L on the distance Δ was

L = C x Δ (1-D)

For a very smooth coast like that of South Africa, D 1. For more convoluted coastlines, D > 1 and so L increases as Δ decreases. Richardson found that the west coast of Britain yielded a value D = 1.25, close to the dimension of the snowflake.

Ireland’s Coastline

Recently a group of students in the School of Physics, Trinity College Dublin, under the supervision of Professor Stefan Hutzler, studied Ireland’s coastline using Google Maps and a measure called the box dimension: the length of a curve is estimated by superimposing a square grid on it and counting the number N of grid boxes that contain a segment of the curve. For a smooth curve, the product of N and the width W of a box is insensitive to the width of the box, and gives an estimate L = N x W of the length of the curve. For fractal curves, the length increases as the box width is reduced. The figure below, taken from the Science Spin article [1] shows the idea.

The Kerry coast with superimposed coarse grid (left) and fine grid (right). Only boxes intersecting the coast are shown (from [1]).

Hutzler obtained a value D = 1.20 for the fractal dimension. Not surprisingly, the ragged Atlantic shore has a higher fractal dimension (D = 1.26) than the relatively smooth east coast (D = 1.10). We can see this visually by comparing the coasts of Mayo in the North-west and Wexford in the South-east:

Contrasting character of the NW and SE coasts (from Google Maps).

Independently, and using a different method, a group in the School of Computing and Mathematics, University of Ulster, used printed maps and dividers of varying length (see McCartney, Abernethy, and Gault [2]). This yields what they call the divider dimension. The graph below shows the measured length as a function of divider length.

Log-log plot of length versus step-size (from [2]).

The points lie close to a line on this log-log plot, and the slope m of the line gives the fractal dimension: D = 1 – m. McCartney et al. obtained a value D = 1.23 for the overall coastline. They found that the relationship

L = 15,550 X Δ -0.23

held for step sizes over three orders of magnitude, between 100m and 100 km (note that L is in km and Δ is in metres). This implies values in the following table:

 Stepsize Δ Length L 1 metre 15,550 km 10 metres 9,157 km 100 metres 5,392 km 1 km 3,175 km 10 km 1,870 km 100 km 1,100 km

We see from this table that the value of 3,171 km given by the Ordnance Survey of Ireland corresponds to a step size of 1km.

The close agreement between the two independent studies [1] and [2] confirms our ideas about the fractal nature of the Irish coastline and gives us confidence in the reported values of D.

Sources

[1] Hutzler, S. (2013). Fractal Ireland. Science Spin, 58, 19-20

[2] McCartney M., Abernethy G., and Gault L. (2010). The Divider Dimension of the Irish Coast. Irish Geography, 43, 277-284.

[3] Wikipedia article: List of countries by length of coastline.

Peter Lynch’s book about walking around the coastal counties of Ireland is now available as an ebook (at a very low price!).

### Population Projections

The Population Division of the United Nations marked 31 October 2011 as the “Day of Seven Billion”. While that was a publicity gambit, world population is now above this figure and climbing. The global ecosystem is seriously stressed, and climate change is greatly aggravated by the expanding population. Accurate estimates of growth are essential for assessing our future well-being. This week, That’s Maths in The Irish Times ( TM034  ) is about population growth over this century. Continue reading ‘Population Projections’

### A Simple Growth Function

Three Styles of Growth

Early models of population growth represented the number of people as an exponential function of time,

$\displaystyle N(t) = N_0 \exp(t/\tau)$

where ${\tau}$ is the e-folding time. For every period of length ${\tau}$, the population increases by a factor ${e\approx 2.7}$. Exponential growth was assumed by Thomas Malthus (1798), and he predicted that the population would exhaust the food supply within a half-century. Continue reading ‘A Simple Growth Function’

### The Antikythera Mechanism

The article in this week’s That’s Maths column in the Irish Times ( TM033 ) is about the Antikythera Mechanism, which might be called the First Computer.

Two Storms

Two storms, separated by 2000 years, resulted in the loss and the recovery of one of the most amazing mechanical devices made in the ancient world.  The first storm, around 65 BC, wrecked a Roman vessel bringing home loot from Asia Minor. The ship went down near the island of Antikythera, between the Greek mainland and Crete. Continue reading ‘The Antikythera Mechanism’

### The Watermelon Puzzle

An amusing puzzle appears in a recent book by John A. Adam (2013). The answer is very surprising. The book argues in terms of simultaneous equations. A simpler argument, using the diagram below, should make all clear. Continue reading ‘The Watermelon Puzzle’

### Euler’s Gem

This week, That’s Maths in The Irish Times ( TM032  ) is about Euler’s Polyhedron Formula and its consequences.

Euler’s Polyhedron Formula

The highlight of the thirteenth and final book of Euclid’s Elements was the proof that there are just five “Platonic solids”. Recall that a regular polygon is a plane figure with all sides and angles equal, for example a square. By joining identical polygons together, we can form solid bodies called regular polyhedra. Continue reading ‘Euler’s Gem’

### Hyperbolic Triangles and the Gauss-Bonnet Theorem

Poincaré’s half-plane model for hyperbolic geometry comprises the upper half plane ${\mathbf{H} = \{(x,y): y>0\}}$ together with a metric

$\displaystyle d s^2 = \frac { d x^2 + d y^2 } { y^2 } \,.$

It is remarkable that the entire structure of the space ${(\mathbf{H},ds)}$ follows from the metric.
Continue reading ‘Hyperbolic Triangles and the Gauss-Bonnet Theorem’