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The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal-like properties of coastlines. The first recorded observation of this phenomenon was by Lewis Fry Richardson and it was expanded by Benoit Mandelbrot. More concretely, the length of the coastline depends on the method used to measure it. Since a landmass has features at all scales, from hundreds of kilometres in size to tiny fractions of a millimetre and below, there is no obvious size of the smallest feature that should be measured around, and hence no single well-defined perimeter to the landmass. when specific assumptions are made about minimum feature size. == Mathematical aspects == The basic concept of length originates from Euclidean distance. In the familiar Euclidean geometry, a straight line represents the shortest distance between two points; this line has only one length. The geodesic length on the surface of a sphere, called the great circle length, is measured along the surface curve which exists in the plane containing both end points of the path and the centre 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 sum of the straight lines which connect the points: Using a few straight lines to approximate the length of a curve will produce a low estimate. Using shorter and shorter lines will produce sums that approach the curve's true length. A precise value for this length can be established using calculus, a branch of mathematics which enables calculation of infinitely small distances. The following animation illustrates how a smooth curve can be meaningfully assigned a precise length: However, not all curves can be measured in this way. A fractal is by definition a curve whose complexity changes with measurement scale. Whereas approximations of a smooth curve get closer and closer to a single value as measurement precision increases, the measured value of fractals may change wildly. The length of a "true fractal" always diverges to infinity, as if one were to measure a coastline with infinite, or near-infinite resolution, the length of the infinitely smaller bends of the coastline would add up to infinity.〔Post & Eisen, p. 550.〕 However, this figure relies on the assumption that space can be subdivided indefinitely. 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). The Planck length, many orders of magnitude smaller than an atom, is proposed as the smallest measurable unit possible in the universe. Coastlines differ from mathematical fractals because they are formed by numerous small events, which .〔Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, ''Chaos and Fractals: New Frontiers of Science''; Spring, 2004; p. (424 ).〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「coastline paradox」の詳細全文を読む スポンサード リンク
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