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Circular fractal

Circular fractal after the third iteration

A circular fractal is a class of geometric (constructive) fractals (see, for example, [1] [2] ) constructed by repeatedly inscribing other circles of smaller radius into the circle . (see Fig. 1a, 1b, 1c).

Application

Constructive circular fractals can be used as models of various natural structures in chemistry, biology, material technology, etc. Fractals of this type were proposed in [3] [4] as models of clusters of magnetic flux tubes in the upper layers of the solar convection zone. More complex constructions of this kind were also considered, for example, circular fractals with overlapping elements simulating twisted magnet flux tubes [5] , see also [6] [7] [8] . It is also possible to construct multifractal structures of a similar type for modeling more complex structures. Unlike Sierpinski carpets , such fractals are not built from rectangular or triangular, but from circular elements.

The first three of a potentially infinite sequence of such circular fractals are shown in Fig. 1a, 1b, and 1c.

Fig. 1a
Fig. 1b
Fig. 1c

To calculate the Hausdorff dimensions ( d ) of these objects, you can use the well-known formula for structural fractals:d=log⁡(n)/log⁡(one/a) {\ displaystyle d = \ log (n) / \ log (1 / a)} {\displaystyle d=\log(n)/\log(1/a)} . In the case of Fig. 1a, the value n = 3. Parameter a is the ratio of the characteristic lengths of neighboring scales. In this case, it isri+one/ri {\ displaystyle r_ {i + 1} / r_ {i}} {\displaystyle r_{i+1}/r_{i}} ; Whereri {\ displaystyle r_ {i}} r_{i} - radius of a larger circle,ri+one {\ displaystyle r_ {i + 1}} {\displaystyle r_{i+1}} - radius of the circle of the neighboring smaller scale. From simple geometric considerations, we find: a = 0.4641. Substituting these values ​​in the formula, we obtain d≈1.43. For the variant in fig. 1b, respectively, n = 4, a = 0.4142 ..., d≈1.57 ... For the variant shown in Fig. 1c, we have: n = 7, a = 1/3 and, dimension d≈1.77 ... Increasing the number of inscribed circles, we get an infinite sequence of fractal objects, with Hausdorff dimensions d → 2.

Example

Seven circles of radius R / 3 are inscribed in a circle of radius R so that they all touch but do not intersect each other. In each of these seven circles, seven circles R / 9 fit in, etc.

 
Circular fractal after the fourth iteration
 
Circular fractal after the fifth iteration

Notes

  1. ↑ Morozov A. D. Introduction to the theory of fractals. - Moscow-Izhevsk. Institute for Computer Research, 2002, 160 pp.
  2. ↑ Bozhokin S.V., Parshin D.A. Fractals and multifractals. - Izhevsk. SIC “Regular and chaotic dynamics”, 2001, 128 p.
  3. ↑ Chumak O. V. Fractal dimensions of ITF associations. - Astronomical Circular, No. 1546, 1990
  4. ↑ Chumak O. V. Entropies and fractals in data analysis. - M.-Izhevsk: Research Center “Regular and chaotic dynamics”, 2011, 164 p.
  5. ↑ Chumak O. V. Self-similar fractals with overlapping elements as a model of photospheric magnetic structures. - Astronomical Circular, No. 1546, 1990
  6. ↑ Chumak OV, Zhang H. - Size-flux relation in active regions. - Chinese Journal Astron. and Astroph., Vol. 3, No. 2, 2003, pp. 175-182
  7. ↑ Chumak O. V. Fractal dimensions and ratios “area - flux” for local magnetic fields on the Sun. - Astronomical Circular No. 1545, 1990.
  8. ↑ Chumak O. - Self-similar and self-affine structures in observational data on solar activity - Asrton & Astroph. Trans. V. 24, No. 2, 2005, pp. 93-99

Literature

  • Chumak O. V. Entropies and fractals in data analysis . - RHD , 2011 .-- 164 p. - ISBN 978-5-93972-852-2 . (inaccessible link)
Source - https://ru.wikipedia.org/w/index.php?title=Krugovoy_fractal&oldid=95478302


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Clever Geek | 2019