STRUCTURE FACTOR OF DETERMINISTIC FRACTALS WITH ROTATIONS

Abstract

We derive a recursion relation for the Fourier transform of any self-similar multifractal mass distribution. This is then used to find sufficient conditions under which S(k)↛0 as |k|→∞. Among two-dimensional distributions for which the similarity transformations contain 2π/n rotations, it is found that for values of n equal to 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18 and 30, distributions may be constructed satisfying the above condition. The possible scaling factors in the similarity transformations are strongly constrained by the value of n. In three dimensions, the equivalent condition is that all rotations/reflections are elements of a finite group, together with similar constraints on the scaling factors.