Fractal Menger sponge

Garrison, J.R., Jr., W.C. Peam, and D.U. Rosenberg. 1992. The fractal Menger sponge and Sierpinski carpet as models for reservoir rock/pore systems I. Theory and image analysis of Sierpinski carpets. In Situ 16 351-406. 

Fractals are self-similar objects, e.g., Koch curve, Menger sponge, or Devil s staircase. The self-similarity of fractal objects is exact at every spatial scale of their construction (e.g., Avnir, 1989). Mathematically constructed fractal porous media, e.g., the Devil s staircase, can approximate the structures of metallic catalysts, which are considered to be disordered compact aggregates composed of imperfect crystallites with broken faces, steps, and kinks (Mougin et al., 1996). 

Fig. 7. Two examples of fractal suspensions (a) The spheres are arranged in a Leibnitz packing with the construction process illustrated to n = 2 sphere 4 is created during the generation n = 1, while spheres 5, 6 and 7 are created during step n = 2. (b) The spheres are arranged according to a modified Menger sponge, again the contruction stage is shown to n = 2.

Another fractal structure of interest is considered by Adler (1986). A three-dimensional fractal suspension may be constructed from a modified Menger sponge, as shown in Fig. 7(b). A scaling argument permitted calculating the effective viscosity of such a suspension however, this viscosity should be compared with numerical results for the solution of Stokes equations in such a geometry before this rheological result is accepted unequivocally. 

Figure 9. The menger sponge a potential fractal representation of pore space within FCC particles. ( fractal dimension = 2.7268) [15]...

Calculate the fractal dimension of a Menger sponge (see Fig. 1.29), a three-dimensional version of the Sierpinski carpet. A solid cube is divided into 3x3x3 cubes and the body-center cube along with the six face-center cubes... 

Figure 4.15. The N = 12 first generation Menger sponge, a symmetric fractal set in three dimensions of Hausdorff (fractal) dimension tn 20/fn 3 = 2.7268. is configuration (1). Calculations based on (1) and the additional configurations (2-7) are discussed in the text.

Figure 7.7. The Menger sponge, a geometric fractal made of a cube from which three central parts (cube stacks) have been removed. This leaves 20 of the original 27 subcubes in a cubic arrangement. All the remaining subcubes get the same treatment. If this is iterated an infinite number of times the Menger sponge of no weight and infinite surface area is formed. Its dimension is 2.73. From B. B. Mandelbrot. The Fractal Geometry of Nature. W. H. Freeman Co., New York (1983). With kind permission from B. B. Mandelbrot.

This kind of fractals sometimes is identified with deterministic constructions like Cantor set (in P), Sierpinski triangle or Sierpinski square (in P ), with Sierpinski pyramid or Menger sponge (in P ), and so on. In our case of electric impedance property. 

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