Frequency fractal

Frequency Fractal is a Fractal wherein there is a distribution of frequency over an infinite frequency range. Therein one can observe self-similarity of a set of frequency over all frequency scales under consideration. Frequency fractal conceived by Ghosh et al is different from conventional Fractal in the sense that it is not about spatial self-similarity. Frequency Fractal has been a subject of research for a long time, in music, .

Humans can visually perceive self-similarity in a spatial Fractal, which is observed in nature as Fractal landscape, Fractal Flame, and widely used in constructing several technologies like Fractal Antenna. In contrast, Frequency Fractal may or may not be self-similar over the spatial scale, but its vibrational response could have self-similarity, it could be mechanical vibrational frequency like sound, or could be electromagnetic vibrational frequency like radio or microwave. Therefore, Frequency fractal is about frequency of vibration of a material and when the vibration or resonance frequency is measured in a material over a large frequency range. Frequency is related to wavelength and time, hence, Frequency Fractal could be termed as Clock Fractal or Wavelength Fractal.

Definition of self-similarity in frequency space

A compact frequency space F is self-similar if there exists a finite set S indexing a set of non-surjective homeomorphisms $\{f_{s}\}_{s\in S}$ for which

$F=\cup _{s\in S}f_{s}(X)$

If $F\subset Y$, we call F self-similar if it is the only non-empty subset of Y such that the equation above holds for $\{f_{s}\}_{s\in S}$. We call

${\mathfrak {L}}=(F,S,\{f_{s}\}_{s\in S})$

a self-similar structure.

Examples of Frequency Fractal

Frequency fractals in biology

Frequency Fractals are everywhere in nature, from biological systems to musical instruments. However, its potential is not harnessed and even the term Frequency Fractal did not find any importance in literature until very recently by Ghosh et al (Reference 2). Biological rhythms are known. For example, human motor activity has a robust, intrinsic fractal structure with similar patterns from minutes to hours. This is an example of time or Frequency Fractal. Cognitive neuroscience of music has extensively studied how brain hardware is designed to process frequency fractals The perception of past present and future is created in the brain via Frequency Fractal, even the hierarchical network of hardware are arranged in a fractal manner in order to process frequency fractals of music. The frequency glides of a continuous tone evokes magnetic field, the distribution of frequency has self-similarity hence fractal in nature.

Frequency fractals in music

A normal vibrating spring starts vibration from fundamental frequency and then its integral multiples. Both integers (0, 1, 2, 3, 4, ....) and integral frequencies are self-similar, hence fractal. Instead of integral numbers, one can take infinite series of self-similar relationships and then play the vibrations using musical instruments, it will be a frequency fractal. Per Nørgård a Danish composer played several such infinite series. The trick of creating such music is robust. Shepard tone is one example of harmonic overtones, it is a frequency fractal. Tritone paradox is a nice example how a single frequency fractal is perceived differently by different groups.

Typical features of a Frequency Fractal

Frequency fractals and platonic geometry

Ghosh et al (Reference 2) has constructed their computing model by replacing the frequency fractals with the platonic geometries, however, such a relationship exists in atomic orbital physics as spherical harmonics, though the correlation between frequency and geometry did not exist. Spherical harmonics exhibits a special kind of self-similarity where a new frequency/geometry appears with each new harmonics Geometry should replace algorithm was first proposed by A R Forest in 1971, this conceptual research field is named as Computational geometry.

Power scaling law in Frequency Fractals

Since frequency is directly related to the energy quanta, therefore scale invariant feature of Frequency Fractal ensures similar energy expense over entire system The scale invariance of frequency is a scale invariance of energy. If we consider a relation $f(x)=ax^{k}$, then, scaling the argument $x$ by a constant factor $c$ causes only a proportionate scaling of the function itself. That is,

$f(cx)=a(cx)^{k}=c^{k}f(x)\propto f(x).\!$

That is, scaling by a constant $c$ simply multiplies the original power-law relation noted above by the constant $c^{k}$. Thus, all power laws with a particular scaling exponent are equivalent and changes by constant factors, since each is simply a scaled version of the others. As a result, when logarithms are taken of both $f(x)$ and $x$, and the straight-line on the log-log plot is often called the signature of a power law. Such straightness is a necessary, but not sufficient, condition for a power-law relation, it could be multiple fractals then the scaling factor would change at different frequency range.

References

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