Is there a formula?
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- ribit0
In the idealized case of equally sized clusters (the Hierarchical Model) on a square 2d lattice. Ramified fractals only result when the end of one aggregate sticks to the side of the other, where end and side are the shortest and longest dimensions of a rectangle circumscribing the cluster. A series of side-to-end aggregations yields an invariant cluster shape with an aspect ratio (side/end) of phi_2=1.618.... This number is the Divine Proportion of the ancients. It occurs in clusters because during binary aggregation in the Hierarchical Model, the end and side both grow in accord with the Fibonacci series, 1, 1, 2, 3, 5, 8, 13..., for which the ratio of any two consecutive series members limits to the Divine Proportion. Moreover, since during binary aggregation the number of monomers per cluster doubles while the cluster's dimensions increase by phi_2, the fractal dimension can be calculated simply as D_2=log2/log phi_2=1.44, a value in excellent agreement with simulation. Remarkably, these concepts can be generalized to any spatial dimension. We define a d-dimensional Fibonacci series and a d-dimensional Divine Proportion from which the d-dimensional fractal dimension can be calculated with excellent numerical agreement with simulation... so the gutter could be any size you like really.