Self-similarity square lattices

Figure 22. Schematic for constructing a self-similar lattice via iterative growth of a square generation cell.

In order to get more experience with the newly proposed index ( ) we will consider the leading eigenvalue X of D/D matrices for several well-defined mathematical curves. We should emphasize that this approach is neither restricted to curves (chains) embedded on regular lattices, nor restricted to lattices on a plane. However, the examples that we will consider correspond to mathematical curves embedded on the simple square lattice associated with the Cartesian coordinates system in the plane, or a trigonal lattice. The selected curves show visibly distinct spatial properties. Some of the curves considered apparently are more and more folded as they grow. They illustrate the self-similarity that characterizes fractals. " A small portion of such curve has the appearance of the same curve in an earlier stage of the evolution. For illustration, we selected the Koch curve, the Hubert curve, the Sierpinski curve and a portion of another Sierpinski curve, and the Dragon curve. These are compared to a spiral, a double spiral, and a worm-curve. 

Let us consider the case of deterministic fractals first, i.e. self-similar substrates which can be constructed according to deterministic rules. Prominent examples are Sierpinski triangular or square lattices, also called gasket or carpet (in d = 2) and sponge (in d = 3), respectively, Mandelbrot-Given fractals, which are models for the backbone of the incipient percolation cluster, and hierachical lattices (see for instance the overview in Ref. [21]). In this chapter, however, we restrict the discussion to the Sierpinski triangular and square lattice for brevity. 

This generalized des Cloizeaux relation is in very good agreement with numerically ob-tmned values [74], The second term in Eq. (43) has its origin in the (self-similar) disordered nature of the backbone of critical percolation clusters and is expected to be absent on deterministic fractals such as the Sierpinski lattice. EE results support this conclusion as shown in Section 4 for both triangular and square Sierpinski lattices. 

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