Measuring A Coastline
More than a decade after Richardson completed his work, Benoît Mandelbrot developed a new branch of mathematics, fractal geometry, to describe just such non-rectifiable complexes in nature as the infinite coastline. His own definition of the new figure serving as the basis for his study is:I coined fractal from the Latin adjective fractus. The corresponding Latin verb frangere means "to break:" to create irregular fragments. It is therefore sensible ... that, in addition to "fragmented" ... fractus should also mean "irregular."
A key property of the fractal is self-similarity; that is, at any scale the same general configuration appears. A coastline is perceived as bays alternating with promontories. In the hypothetical situation that a given coastline has this property of self-similarity, then no matter how greatly any one small section of coastline is magnified, a similar pattern of smaller bays and promontories superimposed on larger bays and promontories appears, right down to the grains of sand. At that scale the coastline appears as a momentarily shifting, potentially infinitely long thread with a stochastic arrangement of bays and promontories formed from the small objects at hand. In such an environment (as opposed to smooth curves) Mandelbrot asserts "coastline length turns out to be an elusive notion that slips between the fingers of those who want to grasp it."
There are different kinds of fractals. A coastline with the stated property is in "a first category of fractals, namely curves whose fractal dimension is greater than 1." That last statement represents an extension by Mandelbrot of Richardson's thought. Mandelbrot's statement of the Richardson Effect is:
where L, coastline length, a function of the measurement unit, ε, is approximated by the expression. F is a constant and D is a parameter that Richardson found depended on the coastline approximated by L. He gave no theoretical explanation but Mandelbrot identified L with a non-integer form of the Hausdorff dimension, later the fractal dimension. Rearranging the right side of the expression obtains:
where Fε-D must be the number of units ε required to obtain L. The fractal dimension is the number of the dimensions of the figure being used to approximate the fractal: 0 for a dot, 1 for a line, 2 for a square. D in the expression is between 1 and 2, for coastlines typically less than 1.5. The broken line measuring the coast does not extend in one direction nor does it represent an area, but is intermediate. It can be interpreted as a thick line or band of width 2ε. More broken coastlines have greater D and therefore L is longer for the same ε. Mandelbrot showed that D is independent of ε.For more details on this topic, see How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension.
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