Scale invariant patterning

Patterns of this third class in fact demonstrate a complex form of scale-invariance by their self-similarity, in the infinite time limit, different magnifications observed at the same resolution are indistinguishable. The pattern generated by rule R90, for example, matches that of the successive lines in Pascal s triangle ai t) is given by the coefficient of in the expansion of (1 - - xY modulo-tv/o (see figure 3.2). 

For a limited discussion of fractal geometry, some simple descriptive definitions should suffice. Self-similarity is a characteristic of basic fractal objects. As described by Mandelbrot 58 When each piece of a shape is geometrically similar to the whole, both the shape and the cascade that generate it are called self-similar. Another term that is synonymous with self-similarity is scale-invariance, which also describes shapes that remain constant regardless of the scale of observation. Thus, the self-similar or scale-invariant macromolecular assembly possesses the same topology, or pattern of atomic connectivity, 62 in small as well as large segments. Self-similar objects are thus said to be invariant under dilation. 

Preliminary Dislocation Dynamics (DD) simulations using the model developed by Verdier et al. provide a plausible scenario for the dislocation patterning occuring during the deformation of ice single crystals based on cross-slip mechanism. The simulated dislocation multiplication mechanism is consistent with the scale invariant pattemings observed experimentally. 

Mathematical or nonrandom fractals are scale invariant, i.e. the pattern is the same at all scales (self-similar). Natural, real or random fractals are quasi or statistically self-similar over a finite length scale that is most often determined by the characterization technique that is employed. An object or process can be classified as fractal when the length scale of the property being measured covers at least one order of magnitude. Fractal structures obey a power law, allowing the fractal dimension D to be determined from experimental data ... 

Mandelbrot [2, 3] systematized and organized mathematical ideas concerning complex structures such as trees, coastlines and non-equilibrium growth processes. He pointed out that such patterns share a central property and symmetry which may be called scale invariance. These objects are invariant under a transformation, which replaces a small part with bigger part that is under a change in a scale of the picture. Scale-invariant structures are called fractals [7]. More recently the relevance of natural and mathematical structure has become clearer with the help of computer simulation. Self-similarity turns out to be a general invariance principle of these structures. 

Mahalanobis distance is based on correlations between variables by which different patterns can be identified and analyzed. It differs from Euclidean distance in that it takes into account the correlations of the data set and is scale-invariant, i.e., not dependent on the scale of measurements. 

Fergus, R., Perona, P. and Zisserman, A. (2003) Object class recognition by unsupervised scale-invariant learning. Proceedings of Computer Vision and Pattern Recognition (CVPR), 2 264—271. 

For the most part, calculating a fractal dimension for a material is only interesting if the value obtained is constant over a range of length scales. In other words, the pattern of organization must repeat itself sufficiently for the material to be structurally scale invariant. Still, calculation of D... 

Scale-invariant structures originating from growth processes have been found to be extremely widespread in nature. This observation have led to a number of careful experiments, and various growth models have been suggested to describe the fractal outcome but why did they become fractal in the first place To answer this question we must understand the spatio-temporal evolution. Dynamically, the interface is observed to be unstable, and the system eventually reaches a statistically stationary state where a rich ramified pattern is created. A major observation is that this state can be described by power laws - the pattern becomes scale invariant. 

The group could be a geometrical group such as the group of all translations, rotations, scalings, etc. The desired classification of pattern is invariant under the action of the group, i.e. 

In addition, all complexes display a reversible, one-electron reduction at a very negative potential Em —1.70 to -1.90 V vs Fc+/Fc, which is metal centered and nearly invariant with respect to the substitution pattern of the coordinated pheno-lates. It demonstrates the enormous stabilization of the high-spin ferric state by three phenolato ligands. The electrochemistry also nicely shows that unprotected ortho- or para positions of these phenolates lead to irreversible electron-transfer waves on the time scale of a cyclic voltammogram and that methyl substituents are inefficient protecting groups. 

The pattern can be obtained from the polymer temperature or concentration variations in addition to the change of G°N. The relaxation function may be too complicated a mathematical expression ever to be calculated, nonetheless, it obeys a property of invariance which allows the superposition of all normalised relaxation curves to one another by adjusting a suitable factor to the time scale of each curve. The time shift factor is found to obey the equation... 

The numerical value of the reference curvature b can be specified in absolute units or in units scaled relative to the size of the object G(a). If absolute units are used, then a relative convexity characterization of G(a) involves size information if an object G(a) is scaled twofold, then its shape remains the same, but with respect to a fixed, nonzero b value a different relative convexity characterization is obtained. That is, the pattern of relative shape domains Do(b)> D (b), and D2(b) defined with respect to some fixed, nonzero reference curvature value b (b K)) is size-dependent. On the other hand, if the reference curvature b is specified with respect to units proportional to the size of G(a), then a simple. scaling of the object does not alter the pattern of relative shape domains with respect to the scaled reference curvature b. In this case, the shape characterization is size-invariant, that is, a "pure" shape characterization is obtained. 

A fundamental characteristic of spatially periodic systems is the existence of a group of translational symmetry operations, by means of which the repeating pattern may be brought into self-coincidence. The translational symmetry of the array, expressing its invariance with respect to parallel displacements in different directions is represented by a lattice. This lattice consists of an array of evenly spaced points (Fig. 3-13), such that the structural elements appear the same and in the same orientation when viewed from each and every one of the lattice points. Another important property of spatially periodic arrays is the existence of two characteristic length scales, corresponding to the average microscopic distance between lattice... 

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