roughness

=Roughness=

toc This article defines a word and clarifies its application and usefulness in scale studies.

=Definition= Roughness is the general, vernacular term for the quality that Mandelbrot described in fractals--in his words, "the basic idea behind fractals is measuring roughness." Euclidean objects (points, lines, rectangles, spheres) are “smooth” as their slope changes gradually, everywhere differentiable, and they have integer dimensions. Real objects are rough: their slope changes abruptly, everywhere discontinuous, and they have fractional dimensions. Real objects include real data such as stock exchange data, mountain height, and river flow rates.

Roughness is also an important measurement standard of the Bureau International des Poids et Mesures (BIPM), which divides scientific metrology into nine technical subject fields. In the subject field of length, subfield of surface quality, BIPM lists as important "roughness standards" and "roughness measurement equipment."

=Fractal= Below is a sample fractal pattern. Its scale invariance is expressed as self-similarity: a closeup of a branch at one scale is similar to the pattern at other scales.



The roughness of most fractals is described with a single number specifying the dimension of the fractal, e.g. 1.2.

There are also some rare multifractal systems in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is used.

=Usage= Roughness is a measure of how many scales or orders of magnitude are needed to fully characterize a phenomenon.

=Scale Invariance of Fractals= It is sometimes said that fractals are scale-invariant, although more precisely, one should say that they are self-similar. A fractal is equal to itself typically for only a discrete set of values λ, and even then a translation and rotation must be applied to match up to the fractal to itself. Thus, for example the Koch curve scales with Δ = 1, but the scaling holds only for values of λ = 1 / 3n for integer n. In addition, the Koch curve scales not only at the origin, but, in a certain sense, "everywhere": miniature copies of itself can be found all along the curve.

Some fractals may have multiple scaling factors at play at once; such scaling is studied with multi-fractal analysis.