The calculation of the spatial spectrum of multidimensional fractals using the fast Fourier transform

Abstract

The Fast Fourier Transform was applied to spatial spectrum modeling of a one-dimensional fractal (Cantor set), a two-dimensional fractal (Sierpinski carpet), and a three-dimensional fractal (Menger sponge). A spectrum is developed for different levels. The spatial spectrum was also obtained and modeled for various filling parameters. The ParaView software package was used for 3D modeling. Many natural phenomena have distinctive features, which are often associated with fractal structures. Visually, fractals represent a geometric figure, replication of which is exactly the same at every scale [1]. This ability is called self-similarity. Fractals are interesting because of widespread presence in natural formations [1-3]. In this case, natural fractals are called "statistical", and artificial "exact". Statistical fractals can be observed in various polymers, biological structures, electrical circuits, galactic clusters and fluctuations in exchange prices [4]. Exact fractals are generated from mathematical approach [5]. Can these precise mathematical abstractions be found in physical reality? Yes, it is optical fractals [3]. This concept includes "diffractals" (diffraction pattern on fractal lattice) [6, 7], eigen modes of unstable resonators [8], distributions in nonlinear optics [3, 9]. Particularly interesting can be the coincidence of certain properties of "accurate" and "statistical" fractals [10], such as aerosols, smoke, moire [11-13], which is very important applied to optical signal transmission through a heterogeneous or random medium [14-17]. Examination of diffraction on fractal lattice [6, 7, 18-20] can solve other important problems - the formation of periodically self-reproducing fields [21-26], the creation of multi focus [27-30] or specified longitudinal distributions [31-33], and in achromatic depicting systems [34-37]. One of the most important characteristics of fractals is the spatial spectrum [38-41], which are also important in the analysis of crystal structures [42-44]. Taking into account possible multidimensionality of fractals, the calculation of the spatial spectrum can lead to problems associated with computational complexity, which depends on the technical capabilities of modern computers. The solution to the problem can be the usage of the fast calculation algorithm. Within this paper, the fast transformation is used to develop the spatial spectrum of multidimensional fractals with different characteristics.

Keywords

• cantor set
• Sierpinski carpet
• Sierpinski carpet
• fast Fourier transform
• 3D modeling

1. Introduction

will be a simulated fractal.

n E 

Ei , parameters specified in the range of 0,1 , whereby a1  b1 , a2  b2 and a1  b1  1 , a2  b2  1 . The simulated fractal can be found by the previously applied for the one-dimensional case formula ( 1 ). If we set the parameters a1  , a2 

, b1  b2  2 3 and

The three-dimensional case is implemented reciprocally to the two-dimensional case. The unit cubes E0  0,1 0,10,1

E1  0, a1   b1 ,1  0, a2   b2 ,1  0, a3   b3 ,1 was taken, whereby a1 , a2 , a3 , b1 , b2 and b3 are fractal parameters specified in the range of 0,1 , whereby a1  b1 , a2  b2 , a3  b3 and a1  b1  1 , a2  b2  1 , a3  b3  1 . If we set the parameters a1 

If we set the parameters a1 

, a2  3 3 3 3 3 (Fig. 2 a), the boundary section of which is a Sierpinsky carpet.

1 3 1 2 1

1 , a2  , a3  1

2 , b2  2

and b3  3 8 , a3 

we get a scalable three-dimensional fractal (Fig. а)

b)

F (u)   f (x) (u)   f (x) exp  2 ixu d n x,

Rn whereby f (x) is the input function specified as a vector, which is a binary representation of the fractal, F (u) is the output function, [] is the Fourier transform operator.

As can be seen from the Table 1, with the number of iteration increasing, the spatial spectrum from the fractal structure becomes more complex and the energy at higher frequencies increases. However, the pattern of the spectrum maintains a regular structure, which is also characteristic of crystalline structures [42-44].

3. Conclusion

As a result of the work, the spatial spectrum was calculated and visualized from a two-dimensional (Sierpinski carpet) and a three-dimensional (Menger sponge) fractal structure using the Fast Fourier Transform algorithm.

Acknowledgements References

The work was supported by the Ministry of Education and Science of the Russian Federation.

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