Fractal Dimension, Between a Line and an Area

Definition A fractal dimension is a number $d$ that describes how a set fills space across scales. Every kind of fractal dimension makes precise, in its own way, the idea that covering the set with pieces of a small size $\varepsilon$ takes about $(1/\varepsilon)^d$ pieces as $\varepsilon$ shrinks toward $0$. The main kinds are the similarity dimension, which counts exact smaller copies of the set, the box-counting dimension, which counts the boxes of a grid that the set meets, and the Hausdorff dimension, which allows covers by sets of any shape. For a segment, a filled square and a solid cube all three give the familiar values 1, 2 and 3, but for other sets they need not be whole numbers. The surprise is that Sierpinski’s triangle, which is made of three half-size copies of itself, has dimension

in all three senses. Such a value records how fast the number of pieces grows as the pieces shrink. It does not mean that the set has one and a half directions to move in. For some other sets the different kinds of fractal dimension disagree, so a value should always be given together with its kind.

Definition A similarity with ratio $r$ is a map of the plane or of space that multiplies every distance by the same number $r$, possibly combined with turning, reflecting and shifting. Suppose a set $F$ is the union of $N \ge 2$ pieces, each the image of $F$ under a similarity with the same ratio $r$, where $0 < r < 1$, and let $s = 1/r$ be the magnification that brings a piece back to full size. The similarity dimension of this description of $F$ is the number $d$ with $N = s^d$, which is

When the ratios $r_1, \ldots, r_N$ differ, it is the one number $d \ge 0$ with

The similarity dimension belongs to the list of maps, not directly to the set. It equals the Hausdorff dimension and the box-counting dimension of $F$ when the pieces satisfy the open set condition, which asks for a bounded open set $V$, not empty, that each map sends into $V$, with the images of $V$ under different maps not overlapping. The condition holds for the Cantor set, the Koch curve, Sierpinski’s triangle and the other examples on this page whose pieces meet at most along edges or at single points. Without the condition, the similarity dimension is still an upper bound for the Hausdorff dimension, but it can be too large.

Definition For a base $b > 0$ with $b \ne 1$ and a number $x > 0$, the logarithm $\log_b x$ is the exponent $y$ for which $b^y = x$. The base 10 and the base $e \approx 2.718$, whose logarithm is written $\ln$, are the most common. For any two allowed bases $b$ and $c$,

So for $s > 0$ with $s \ne 1$ and $N > 0$, the equation $s^d = N$ has exactly one solution, $d = \log N / \log s$, where both logarithms use the same base and any base gives the same quotient.

Definition The diameter of a set $U$, written $\operatorname{diam} U$, is the least upper bound of the distances between pairs of its points. Let $F$ be a set on a line, in the plane or in space, and let $d \ge 0$ and $\delta > 0$. Cover $F$ by finitely or countably many sets $U_1, U_2, \ldots$, each of diameter at most $\delta$, and form the sum

The greatest lower bound of these sums over all such covers is written $\mathcal{H}^d_\delta(F)$. As $\delta$ shrinks, fewer covers are allowed and this number can only grow, and its limit $\mathcal{H}^d(F)$, possibly infinite, is the $d$-dimensional Hausdorff measure of $F$. For every set $F$ there is one number $D \ge 0$ such that $\mathcal{H}^d(F)$ is infinite for every $d < D$ and $0$ for every $d > D$. This number $D$ is the Hausdorff dimension of $F$. At $d = D$ the measure may be $0$, a positive number or infinite. For a segment the measure $\mathcal{H}^1$ is its length, and for a region in the plane $\mathcal{H}^2$ is a fixed multiple of its area, so segments and regions get the dimensions 1 and 2. The Hausdorff dimension is defined for every set, with no exact copies needed. It is named after Felix Hausdorff, who introduced it.

Definition The topological dimension of a set is a whole number defined step by step. The empty set has topological dimension $-1$. A set has topological dimension at most $n$ if every point of it has arbitrarily small neighborhoods within the set whose boundaries, taken within the set, have topological dimension at most $n - 1$. The topological dimension is the smallest such $n$. It does not change when a set is bent or stretched by a one-to-one continuous map whose inverse is also continuous, so it describes how a set is connected up rather than how its detail scales. For every set in the plane or in space, the topological dimension is at most the Hausdorff dimension. Benoit Mandelbrot, who coined the word fractal, proposed calling a set a fractal when its Hausdorff dimension is larger than its topological dimension, although no single definition of fractal is used by everyone.