Frustration on Fractals

Abstract

MANDELBROT [1] has observed that shapes occurring in nature often look highly fragmented. Such shapes, known as fractals, can be attributed to coastlines, the mammalian brains, and so on. The length or volume of such objects depends on the length scale used. Consider, for instance, the Sierpinski gasket. To construct the 2-dimensional gasket we begin with a triangle. The midpoints of its edges are connected, creating 4 triangles. The central triangle is removed and the same procedure is continued for each of the new triangles. In the n-th step of the construction the length scale is ε_n = 2⁻ⁿ and the area within the gasket is proportional to ε ^2−D_n where D = ln3/ln2. If no triangles were removed, the area would be scale invariant and D = 2. Note that the number of triangles in the gasket grows as ε ^−D_n . On going from one length scale to the other the gasket looks the same, i.e. it is self-similar. Other fractals are usually self-similar in a statistical sense, but still they can be characterized by a fractal dimensionality D. In order to determine D of an object in a d-dimensional space, one may pick a length scale, a, and find the corresponding measure of the object within the volume (aL)^d. For large L’s the measure will go as L^D.