Geometrical Similarity as Topological

The GSTE Principle Geometrical Similarity as Topological Equivalence... 

The above treatment of similarity is the basis of the GSTE Principle treating Geometrical Similarity as Topological Equivalence [108]. 

The GSTE Principle Geometrical Similarity as Topological Equivalence is a general principle for similarity analysis [108]. 

ADMA = adjustable density matrix assembler AFDF = additive fiizzy density fragmentation GSTE = geometrical similarity as topological equivalence MEDLA = molecular electron density loge assembler MEP = molecular electrostatic potential RBSM = resolution-based similarity measures SGM = shape group methods VDWS = van der Waals surface ID, 2D, 3D = one, two, and three dimensions. 

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. 

These fascinating bicontinuous or sponge phases have attracted considerable theoretical interest. Percolation theory [112] is an important component of such models as it can be used to describe conductivity and other physical properties of microemulsions. Topological analysis [113] and geometric models [114] are useful, as are thermodynamic analyses [115-118] balancing curvature elasticity and entropy. Similar elastic modulus considerations enter into models of the properties and stability of droplet phases [119-121] and phase behavior of microemulsions in general [97, 122]. 

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