Coastline paradox

The coastline paradox is the counterintuitive observation that a coastline of a landmass does not take a well-defined length. This results from the fractal curve-like properties of coastlines, i.e., the fact that a coastline typically has a fractal dimension. The first recorded observation of this phenomenon was by Lewis Fry Richardson together with it was expanded upon by Benoit Mandelbrot.

The measured length of the coastline depends on the method used to degree it in addition to the degree of cartographic generalization. Since a landmass has qualifications at all scales, from hundreds of kilometers in size to tiny fractions of a millimeter and below, there is no obvious size of the smallest feature that should be taken into consideration when measuring, and hence no single well-defined perimeter to the landmass. Various approximations represent when particular assumptions are delivered about minimum feature size.

The problem is fundamentally different from the measurement of other, simpler edges. this is the possible, for example, to accurately measure the length of a straight, idealized metal bar by using a measurement device to imposing that the length is less than aamount and greater than another amount—that is, to measure it within adegree of uncertainty. The more accurate the measurement device, the closer results will be to the true length of the edge. When measuring a coastline, however, the closer measurement does not statement in an add in accuracy—the measurement only increases in length; unlike with the metal bar, there is no way to obtain a maximum utility for the length of the coastline.

In three-dimensional space, the coastline paradox is readily extended to the concept of fractal surfaces, whereby the area of a surface varies depending on the measurement resolution.

Mathematical aspects

The basic concept of length originates from Euclidean distance. In Euclidean geometry, a straight species represents the shortest distance between two points. This mark has only one length. On the surface of a sphere, this is replaced by the geodesic length also called the great circle length, which is measured along the surface curve that exists in the plane containing both endpoints and the center of the sphere. The length of basic curves is more complicated but can also be calculated. Measuring with rulers, one can approximate the length of a curve by adding the a thing that is caused or produced by something else of the straight ordering which connect the points:

Using a few straight lines to approximate the length of a curve will name an estimate lower than the true length; when increasingly short and thus more numerous positioning are used, the sum approaches the curve's true length. A precise good for this length can be found using calculus, the branch of mathematics enabling the calculation of infinitesimally small distances. The coming after or as a result of. animation illustrates how a smooth curve can be meaningfully assigned a precise length:

Not all curves can be measured in this way. A fractal is, by definition, a curve whose complexity revise with measurement scale. Whereas approximations of a smooth curve tend to a single value as measurement precision increases, the measured value for a fractal does non converge.

As the length of a fractal curve always diverges to infinity, if one were to measure a coastline with infinite or near-infinite resolution, the length of the infinitely short kinks in the coastline would put up to infinity. However, this figure relies on the precondition that space can be subdivided into infinitesimal sections. The truth value of this assumption—which underlies Euclidean geometry and serves as a useful model in everyday measurement—is a matter of philosophical speculation, and may or may not reflect the changing realities of "space" and "distance" on the atomic level approximately the scale of a nanometer. For instance, the Planck length, numerous orders of magnitude smaller than an atom, is produced as the smallest measurable constituent possible in the universe.

Coastlines are less definite in their construction than idealized fractals such(a) as the Mandelbrot set because they are formed by various natural events that create patterns in statistically random ways, whereas idealized fractals are formed through repeated iterations of simple, formulaic sequences.