SELF-SIMILARITY in the WEIERSTRAS FUNCTION
| Collection | International Encyclopedia of Systems and Cybernetics |
|---|---|
| Year | 2004 |
| Vol. (num.) | 2(2) |
| ID | ◀ 2987 ▶ |
| Object type | Methodology or model |
B. WEST and A. GOLDBERGER explain: “His function was a superposition of harmonic terms: a fundamental with frequency $_{o}$ and unit amplitude, a second periodic term of frequency b $_{o}$ with amplitude 1/a, a third periodic term of frequency b2 $_{o}$ with amplitude 1/a2 , 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: log$_{n}$a/log$_{n}$b ”(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: \term“{equilibria} (levels of)”; “cyclical or periodic”.
See also
See also Resonance