Cantor dust

Fig. 14a-c. Mathematical models of dendritic/fractal systems a) Bifurcation of a line segment with connectivity paths to define Cantor dust, b) Bifurcation of a triangle (two-dimensions) to define Koch snowflakes, c) Fitting of a sphere (three-dimensional) to define a series of infinite nesting spheres... 

The function shown in Fig. 6 is connected with the Cantor set (Cantor dust ). This function has been called the devil s stairs. ... 

Each of the remaining segments are also divided into three equal parts. The function y = fix) over the middle parts of the segments is assumed to equal 1/8, 3/8, 5/8, 7/8, respectively that is, the increment in the function is equal to (1/2)3. Continuing this process, as n — oo we obtain a function that is defined at all points of segment [0,1], except at the points belonging to the Cantor set (Cantor dust, Fig. 6). 

This means that the function is continuous as n > oo. Thus, the graph of the function y f(x) looks like a staircase with an infinite number of steps whose total length is 1 because the length of Cantor dust (where there are no steps) is zero. This function goes up by 1, although it only increases over the set of zero length and does not make any jumps. 

Therefore, if we take df = In 2/In 3, then we obtain the finite dimension of the Cantor dust summed length. Thus, the measure (measurement) of set fIf depends on the dimensions of the objects covering it. Measuring a set gives a smart result if the ruler used to measure the set corresponds to the geometry (dimensions) of the set. 

Different relaxation functions are derived assuming that the actual (physical) ensemble of relaxation times is confined between the upper and lower limits of self-similarity. In this respect, the temporal fractal differs from a geometrical fractal (e.g., Cantor dust) for which only an upper limit (i.e., the initial segment before its subdivision) is assumed to exist. It is predicted that at times, shorter than the relaxation time at the lowest (primitive) self-similarity level the... 

What would happen when an infinite number of line segments were removed from an initial line interval. Cantor devised an example, which portrayed classical fractal made by iteratively taking away something. This operation created a dust of points-Cantor dust. The operation is shown in Figure 13.19 [3]. 

Typical examples of these fractals are the Cantor set ( dust ), the Koch curve, the Sierpinski gasket, the Vicsek snowflake, etc. Two properties of deterministic fractals are most important, namely, the possibility of exact calculation of the fractal dimension and the infinite range of self-similarity -°° +°°). Since a line, a plane, or a volume can be divided into an infinite number of fragments in different ways, it is possible to construct an infinite number of deterministic fractals with different fractal dimensions. Therefore, deterministic fractals cannot be classified without introducing other parameters, apart from the fractal dimension. 

The Cantor set is simply the dust of points that remain. The number of these points is infinite, but their total length is zero. Mandelbrot recognized the Cantor set as a model for the occurrences of... 

In Ref [51], the Halsey multifractal formalism [39], modified by Williford [58], application for particulate-filled pol5mier composites structure and properties description was considered. The Cantor set ( dust ) was used as mathematical model. It is assumed, that section of length /j and probabilistic... 

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