22. Boundaries and Coastlines:

The Fractals Paradox

The Question:

Measurements should be precise. As long as the measuring rods are standardized it does not matter whether one uses a ten-centimeter ruler or a twenty-centimeter rod to measure the outline of a shape. And to determine the length of a coastline or a border on a map, for example using a compass, it should make no difference whether one consults a pocket-size guidebook or a full-size atlas.

Correct?

The Paradox:

Sometimes. And sometimes not!

No matter what map one uses, the borders of Colorado with its neighboring states add to 2100 kilometers. The coastline of South Africa according to the CIA World Factbook and most other sources is 2798 kilometers long. And the border between Portugal and Spain has been determined in 1926 by the “Convention of Limits” to be 1213 kilometers long.

But the latter in particular presents a problem. When measuring the Portuguese-Spanish border with a compass on a map in a certain Spanish encyclopedia, the length turns out to be only 987 kilometers. A similar discrepancy occurs with the border between Belgium and the Netherlands. It is 449 kilometers long according to one map, and 380 kilometers according to another.

So, the length of Colorado’s borders and of South Africa’s coastline do not change when consulting different atlases. But the borders between Spain and Portugal and between Belgium and the Netherlands vary, depending on which geographical map one uses.

And then there’s the British coastline. Measuring it by using an extremely rough scale interval of 200 km, the length of the coastline is about 2400 km. Reducing the interval to 50 km, the length of the coastline becomes about 3400 kms. And it keeps growing and growing, the smaller the interval. The CIA World Factbook lists the British coastline at 12,429 kilometers, Britain’s mapping authority records it as 17,820 km.

As a final example, let me cite the security fence that snakes along the border between Israel and the Palestinian territory. Within the municipal area of Jerusalem, this fence is actually a concrete wall. According to the Israeli army’s spokesperson, it is 54 kilometers long; according to a geographer at the Center for Palestinian Studies in Jerusalem, it is 72 kilometers long.

Why the differences? What’s going on?

Background:

The English scientist Lewis Fry Richardson, (1881–1953) was among the first people to apply modern mathematical techniques to meteorology. He proposed, for example, to forecast weather by solving differential equations. As a confirmed pacifist he also sought to use mathematics in order to prevent wars. One parameter that he believed would be useful in his models was the length of the borders between two countries. But when he tried to determine their lengths by measuring them on maps with a compass, he discovered the curious discrepancies mentioned above. The smaller the opening of the compass’s legs, the longer many borders turned out to be.

In the 1960s, the French mathematician Benoît Mandelbrot (1924-2010) stumbled upon Richardson’s work while investigating the behavior of commodities markets. He did the measurements for the length of the coast of Britain and noted the differences. He also noticed similarities with his research on the wiggliness of commodities prices, when depicted on graph paper. His paper “How Long is the Coast of Britain”, published in 1967 in the journal Science, made the general public aware of a new mathematical discipline that would turn out to be of great importance to geography, economics, and many other disparate fields: fractal geometry.

Dénouement:

Whenever borders are defined as straight lines, as for example the borders of Colorado which were originally defined to be lines of latitude and longitude, or the 49th parallel North which is the border between Canada and the United States, or the straight line through the Sahara Desert which forms the border between Libya and Egypt, their lengths are indeed precisely measured at maps of any scale.

But when a border follows topographical conditions – a river, a valley, a mountain range – or when a coastline snakes along the waterfront, things become more complicated. Since larger scale maps show more details, what seemed a straight-line coast on a small scale may turn out to be a collection of small bays and inlets at a larger scale. And a linear segment of a border may reveal recesses and protuberances on a larger scale.

This may explain the discrepancy in the length of the border on the Iberian Peninsula. Portugal, being about one fifth the size of Spain, can depict its maps at a larger scale on the page of an atlas than its neighbour. For the map of Spain to fit onto the page, the scale must be much smaller. Hence, the portuguese map displays many more details, and the length of the border seems much longer.

This is not so for the coastline of South Africa. Its coastline is relatively smooth, virtually an arc of a circle, with very few natural harbours. So, no matter which scale is used, its length is constant. Britain’s coastline, on the other hand, is jagged and wiggly. And as one zooms in, more and more jags and wiggles appear and the coastline becomes longer and longer.

The Israeli-Palestinian border wall is a man-made demarcation line. As disruptive as it is, it does snake around homes and gardens and tries to fit into the topographical surroundings. The wiggles and curves are visible, if at all, only on a very large-scale map which may explain the different lengths specified by the Israeli and the Palestinian spokespersons.

Technical supplement:

In effect, there is no meaning to the notion of a border’s or a coastline’s length. To make some sense nevertheless, Mandelbaum defined the ‘fractal dimension.’ For borders and coastlines, the fractal dimension is between 1.0 (a straight line) and 2.0 (a line that is so wiggly that is densely covers a two-dimensional surface). Hence, this dimension measures the line’s wiggliness. The fractal dimension for the British coastline is computed to be 1.26, for the South African coastline it is 1.02. The fractal dimension of the Spanish/Portuguese border is 1.18.

Fractal dimensions are also defined for higher-dimensions. A jagged surface has a higher dimension than a level surface. Mandelbau defined the fractal dimension of a flat surface as 2.0, and of a surface kneaded so tightly that it fills three-dimensional space as 3.0. Switzerland, for example, has an area of about 41,000 square kilometers. But if the alps were ironed out flat, it would possibly be as large as the Gobi Desert. The nation’s fractal dimension has been calculated to be 2.43, nearly half-way between a flat desert and three-dimensional space.

Comments, corrections, observations: