Politics should play no role in mathematics, but mathematics is ubiquitous, even in politics. Take the security fence that Israel is building on the West Bank. Its legality was examined by the International Court of Justice, but not only is the course of the construction in dispute, the two sides cannot even agree on a simple fact: its length.

An Israeli Army spokesman declared that the barrier around Jerusalem would be 54 kilometers long, but Khalil Toufakji, a geographer at the Center for Palestinan Studies in Jerusalem, said he had checked the Army’s data and reached the conclusion that it would be 72 kilometers long.

For once both sides could be right—or wrong. The reason lies in the mathematical theory of fractals, which describes geometric patterns that are repeated at ever smaller scales. The length of the wall depends on the scale of the map used: Where 1 centimeter on paper corresponds to 4 kilometers in nature (scale 1:400,000), the barrier is only about 40 kilometers long. On a more detailed map, on which 1 centimeter corresponds to 500 meters (scale: 1:50,000), more details of the barrier’s meandering course can be discerned, and its length jumps to 50 kilometers. On a scale of 1:10,000 even more details become visible and the wall seems to become longer still.

Now take a walk along the concrete blocks that make up the barrier. One soon notices that the wall often curves around single houses, snakes between two fields, or avoids topographical obstacles. Those details are too small to be included even on large-scale maps. Hence the physical length of the barrier is longer than can be indicated even on the most exact maps. All of a sudden, the differences

in the declarations of the Israelis and the Palestinians start to add up.

It all started with an article by the French mathematician Benoit Mandelbrot. In his 1967 paper “How Long Is the Coast of Great Britain?,” Mandelbrot did not even try to answer his question. Instead he established that it has no meaning. On large-scale maps of Great Britain, bays and inlets are visible that cannot be discerned on less detailed maps. And if one checks out the cliffs and beaches on foot, a longer coastline emerges, its exact length being determined by the water level at the moment of measurement.

The observation also holds for land-based frontiers. Except for geographical borders defined as straight lines, as for example between North and South Dakota, there is no “correct” length of a border. In Spanish and Portuguese textbooks, for example, the lengths of the common border deviate by up to 20 percent. This is because the smaller state uses larger-scale maps to depict the fatherland, which results in longer borders.

The only quantitative statement that can be made, according to Mandelbrot, is about the line’s “fractal dimension.” This is a number that, in a way, describes the jaggedness of a geometrical object. For all coastlines and borders the fractal dimension is between 1 and 2. The more a line winds and meanders, the higher its fractal dimension. The border between Utah and Nevada has the fractal dimension 1, as we expect from regular lines. The British coastline has a fractal dimension of 1.24, and the even more jagged Norwegian coast has a fractal dimension of 1.52.

Fractal theory applies not only to lines on surfaces but also to surfaces in space. If the alpine landscape of Switzerland were ironed flat, for example, this country could well be as large as the Gobi Desert. A few years ago two physicists computed that the surface of Switzerland has the dimension 2.43. With this value it falls about halfway between the flat desert, which has a dimension of 2.0, and three-dimensional space.

With his bewildering article Mandelbrot heralded the age of fractals. Soon the strange shapes were discovered everywhere in nature: in trees and ferns, blood vessels and bronchia, broccoli and cauliflower, lightning, clouds and snow crystals, even in the movements of stock markets.

As for the West Bank fence, maybe it’s just as well it’s convoluted. If it were completely straight, it could theoretically reach all the way from Beirut to Mecca. Then the International Court of Justice would really have something to ponder.