B. WEST and A. GOLDBERGER explain: "His function was a superposition of harmonic terms: a fundamental with frequency Ωo and unit amplitude, a second periodic term of frequency bΩo with amplitude 1/a, a third periodic term of frequency bΩo with amplitude 1/a, and so on. The resulting function is an infinite series of periodic terms, each term of which has a frequency that is a factor b larger than the preceeding and an amplitude that is a factor of 1/a smaller. Thus, in giving a functional form to CANTOR's ideas, WEIERSTRASS was the first scientist to construct a fractal function" (1987, p.360).
The WEIERSTRASS function offers some singular properties: "Because of the infinite layers of detail, one cannot draw a tangent to a fractal curve, which means that the function, although continuous, is not differentiable" (Ibid).
Its mode of construction implies that the curve is self-similar at any level. It also can be properly interpolated in a continuous way, however acquiring more and more closely defined values at microscopic levels.
At any chosen level the curve is a fluctuation of the more macroscopic level and the median of the immediately inferior level of fluctuations.
Moreover, any WEIERSTRASS function has its proper measure of self-similarity, in terms of frequency and amplitude, "which is precisely the fractal, or HAUSDORFF dimension: logna/lognb "(Ibid).
The WEIERSTRASS function is a "scaling relation, often called the renormalization group transformation" (Ibid).
Its analogy – unfortunately rather imperfect – with embedded complex cyclical processes is striking.
See: "equilibria (levels of)"; "cyclical or periodic".
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Bertalanffy Center for the Study of Systems Science (2020). Title of the entry. In Charles François (Ed.), International Encyclopedia of Systems and Cybernetics (2). Retrieved from www.systemspedia.org/[full/url]
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