Self-Similarity or Scale-Invariance

For a limited discussion of fractal geometry, some simple descriptive definitions should suffice. Self-similarity is a characteristic of basic fractal objects. As described by Mandelbrot 58 When each piece of a shape is geometrically similar to the whole, both the shape and the cascade that generate it are called self-similar. Another term that is synonymous with self-similarity is scale-invariance, which also describes shapes that remain constant regardless of the scale of observation. Thus, the self-similar or scale-invariant macromolecular assembly possesses the same topology, or pattern of atomic connectivity, 62 in small as well as large segments. Self-similar objects are thus said to be invariant under dilation. 

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Power law relaxation is no guarantee for a gel point. It should be noted that, besides materials near LST, there exist materials which show the very simple power law relaxation behavior over quite extended time windows. Such behavior has been termed self-similar or scale invariant since it is the same at any time scale of observation (within the given time window). Self-similar relaxation has been associated with self-similar structures on the molecular and super-molecular level and, for suspensions and emulsions, on particulate level. Such self-similar relaxation is only found over a finite range of relaxation times, i.e. between a lower and an upper cut-off, and 2U. The exponent may adopt negative or positive values, however, with different consequences and... 

For surface fractals with uniform density, D = 3. The self-similarity or scale invariance of the fractal aggregate impUes that the structure of the aggregate, on the average, remains unchanged over any extended interval and any its portion is similar to the aggregate as a whole. When refering to a surface, the Z) = 2 corresponds to a smooth surface, and D = 3 corresponds to the maximum roughness. 

It is pertinent to point out here that, in the scale interval between tte lattice ( mtant, lo, and the correlation length, x> all properties of InC are similar to those at the critical point. More precisely, in the interval, Iq L x> acquires the property of self-similarity (or scale invariance) which is a characteristic feature of fractals . 

Fractals are defined in general as objects made of similar parts to the whole in some way either exactly the same except for scale or statistically the same. In short, fractals are self-similar or scaling, that is, invariance against changes in scale or size (scale-invariance). 

The word fractal was coined by Mandelbrot in his fundamental book.1 It is from the Latin adjective fractus which means broken and it is used to describe objects that are too irregular to fit into a traditional geometrical setting. The most representative property of fractal is its invariant shape under self-similar or self-affine scaling. In other words, fractal is a shape made of parts similar to the whole in some way.61 If the objects are invariant under isotropic scale transformations, they are self-similar fractals. In contrast, the real objects in nature are generally invariant under anisotropic transformations. In this case, they are self-affine fractals. Self-affine fractals have a broader sense than self-similar fractals. The distinction between the self-similarity and the selfaffinity is important to characterize the real surface in terms of the surface fractal dimension. 

Fractals are geometric structures of fractional dimension their theoretical concepts and physical applications were early studied by Mandelbrot [Mandelbrot, 1982]. By definition, any structure possessing a self-similarity or a repeating motif invariant under a transformation of scale is caWcd fractal and may be represented by a fractal dimension. Mathematically, the fractal dimension Df of a set is defined through the relation ... 

Certain structures, when examined on different scales from small to large, always appear exactly the same. Such structures are said to be self-similar or to be endowed with the symmetry of self-similarity. Self-similar structures are found to be invariably associated with the geometrical relationship known as the golden ratio, or what Johannes Kepler (1571 - 1630) referred to as the "divine proportion", adding ... 

Scale invariance, whether self-similar or self-affine, has a formidable consequence it makes us lose our bearings and brings the value of our measurements into question. The example of the screwed up paper balls demonstrates... 

But power-law distributions may lack any characteristic scale. This property prevented the use of power-law distributions in the natural sciences until mathematical introduction of Levy s new probabilistic concepts and the physical introduction of new scaling concepts for thermodynamic functions and correlation functions (see ref. [40]). In financial markets, invariance of time scales means that even a stock expert cannot distinguish in a time series analysis if the charts are, for example, daily, weekly, or monthly. These charts are statistically self-similar or fractal. 

The simplest fractals are mathematical constructs that replicate a given structure at all scales, thus forming a scale-invariant structure which is self-similar. Most natural phenomena, such as colloidal aggregates, however, form a statistical self-similarity over a reduced scale of applicability. For example, a colloidal aggregate would not be expected to contain (statistical) self-similarity at a scale smaller than the primary particle size or larger than the size of the aggregate. 

Conversely, the relationship (7.2) expresses a time-scale invariance (selfsimilarity or fractal scaling property) of the power-law function. Mathematically, it has the same structure as (1.7), defining the capacity dimension dc of a fractal object. Thus, a is the capacity dimension of the profiles following the power-law form that obeys the fundamental property of a fractal self-similarity. A fractal decay process is therefore one for which the rate of decay decreases by some exact proportion for some chosen proportional increase in time the self-similarity requirement is fulfilled whenever the exact proportion, a, remains unchanged, independent of the moment of the segment of the data set selected to measure the proportionality constant. 

The scaled structure factor F(x) or F(x) characterizes morphological aspects of the growing pattern such as those shown in Fig. 2, while the size of the pattern at t is characterized by qm(t) or Am(t) (eq.2). If the pattern grows with the dynamical self-similarity at a given T and (t)> is invariant with t, F(x) is universal with t as shown schematically in Fig.3(b). Obtaining this universal scaling function F(x) confirms the validity of the dynamical scaling hypothesis. The... 

Mathematical or nonrandom fractals are scale invariant, i.e. the pattern is the same at all scales (self-similar). Natural, real or random fractals are quasi or statistically self-similar over a finite length scale that is most often determined by the characterization technique that is employed. An object or process can be classified as fractal when the length scale of the property being measured covers at least one order of magnitude. Fractal structures obey a power law, allowing the fractal dimension D to be determined from experimental data ... 

Figure 2.9 Trace of the Brownian motion of a particle in a plane. The boxed detail of the trace (magnified in the upper left portion of the figure) suggests an invariance of scale or self-similarity the detail looks like the whole. Modified from [14].

In all the cases examined so far, it is the matter distribution of the object that has exhibited the property of self-similarity. These objects are called mass fractals. Other situations are encountered, where it is not the matter distribution which has self-similarity, but rather the pore distribution in these cases, we speak of pore volume fractals. Some structures are found in which only the contour or surface manifests scale invariance these are called boundary or surface fractals, and the exponent we need to know is the boundary fractal dimension. To obtain the corresponding exponents, we calculate the autocorrelation function, the mass distribution or the number of boxes, restricting ourselves to the relevant subsets (the points occupied by matter, the points in the pore volume, or the points lying in the interface). 

Quantitatively, a self-afRne fractal is defined by the fact that a change Ax XAx (and possibly Ay —> XAy) transforms Az into X Az, where H lies between 0 and 1. The case H — 1 corresponds to a self-similar fractal. Self-affine fractal structures are no longer characterised by just one (mass or boundary) fractal dimension they require two. The first is local and can be determined by the box-counting method, for example it describes the local scale invariance and its value lies between 1 and 2. The second is global and its value is a simple whole number describing the asymptotic behaviour of the fractal. In the case of a mountain, this global dimension is simply 2. When viewed from a satellite, even the Himalayas blend into the surface of the Earth. 

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