A fractal is, by definition, a curve whose perceived complexity changes with measurement scale. Not all curves can be measured in this way. The following animation illustrates how a smooth curve can be meaningfully assigned a precise length: A precise value for this length can be found using calculus, the branch of mathematics enabling the calculation of infinitesimally small distances. Using a few straight lines to approximate the length of a curve will produce an estimate lower than the true length when increasingly short (and thus more numerous) lines are used, the sum approaches the curve's true length. Measuring with rulers, one can approximate the length of a curve by adding the sum of the straight lines which connect the points: The length of basic curves is more complicated but can also be calculated. 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. In Euclidean geometry, a straight line represents the shortest distance between two points. The basic concept of length originates from Euclidean distance. ( February 2015) ( Learn how and when to remove this template message)Īn animation showing the increasing length of the coastline with decreasing measuring units (coarse-graining length) Unsourced material may be challenged and removed. Please help improve this article by adding citations to reliable sources in this section. This section needs additional citations for verification. 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. When measuring a coastline, however, the closer measurement does not result in an increase in accuracy-the measurement only increases in length unlike with the metal bar, there is no way to obtain a maximum value for the length of the coastline. The more accurate the measurement device, the closer results will be to the true length of the edge. It is possible, for example, to accurately measure the length of a straight, idealized metal bar by using a measurement device to determine that the length is less than a certain amount and greater than another amount-that is, to measure it within a certain degree of uncertainty. The problem is fundamentally different from the measurement of other, simpler edges. Various approximations exist when specific assumptions are made about minimum feature size. ![]() Since a landmass has features 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. ![]() The measured length of the coastline depends on the method used to measure it and the degree of cartographic generalization. Although the "paradox of length" was previously noted by Hugo Steinhaus, the first systematic study of this phenomenon was by Lewis Fry Richardson, and it was expanded upon by Benoit Mandelbrot. This results from the fractal curve–like properties of coastlines i.e., the fact that a coastline typically has a fractal dimension. ![]() The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. With 50 km (31 mi) units, the total length is approximately 3,400 km (2,100 mi), approximately 600 km (370 mi) longer. If the coastline of Great Britain is measured using units 100 km (62 mi) long, then the length of the coastline is approximately 2,800 km (1,700 mi).
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