Notes

Chapter 5: Two Dimensions and Beyond

Section 4: Substitution Systems and Fractals


Fractal dimensions

Certain features of nested patterns can be characterized by so-called fractal dimensions. The pictures below show five patterns with three successively finer grids superimposed. The dimension of a pattern can be computed by looking at how the number of grid squares that have any gray in them varies with the length a of the edge of each grid square. In the first case shown, this number varies like (1/a)^1 for small a, while in the last case, it varies like (1/a)^2. In general, if the number varies like (1/a)^d, one can take d to be the dimension of the pattern. And in the intermediate cases shown, it turns out that d has non-integer values.

The grid in the pictures above fits over the pattern in a very regular way. But even when this does not happen, the limiting behavior for small a is still (1/a)^d for any nested pattern. This form is inevitable if the underlying pattern effectively has the same structure on all scales. For some of the more complex patterns encountered in this book, however, there continues to be different structure on different scales, so that the effective value of d fluctuates as the scale changes, and may not converge to any definite value. (Precise definitions of dimension based for example on the maximum ever achieved by d will often in general imply formally non-computable values, as in the discussion of page 1138.)

Fractal dimensions characterize some aspects of nested patterns, but patterns with the same dimension can often look very different. One approach to getting better characterizations is to look at each grid square, and to ask not just whether there is any gray in it, but how much. Quantities derived from the mean, variance and other moments of the probability distribution can serve as generalizations of fractal dimension. (Compare page 959.)

From Stephen Wolfram: A New Kind of Science [citation]