Menger sponges

Figure 1.2 Generation of the (A) Sierpinski triangle (gasket) (the first three iterations are shown), (B) Menger sponge (the first two iterations are shown) from their Euclidean counterparts.

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... 

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. 

Considering first the theoretical issue, diagrammed in Figure 4.15 is the N — 12 first-generation Menger sponge (top of figure), and several... 

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.

Menger Sponge Torus Square Planar and Simple Cubic Lattice (Hypothetical) ... 

Hollander-Kasteleyn expressions or from polynomial representations of numerical data on n) for lattices of various [A. d, v]. It was noted previously that when calculations are performed in which all symmetry-distinct sites, both internal and external in the geometries represented are considered, values of n) for the Menger sponge lie between the corresponding values for (n) ind — 2 and J = 3 for all N > 72. The data in Table III.l 1 show that this correlation holds even if one restricts consideration to surface-only locations of the reaction center, except for a few special site locations (viz., the corner sites in all cases, and the additional site 2 in I). 

Figure 2.6 First three stages in the construction of the Menger sponge.

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