Fractals and Dimension

The Lorenz attractor was "strange" partly because its dimension came out at roughly 2.06 — not a whole number. That should feel impossible: a line is 1-dimensional, a surface 2-dimensional, a solid 3. How can anything have two-point-oh-six dimensions? The answer is that our schoolroom notion of dimension is too crude for the objects chaos produces. Those objects are fractals — shapes with detail at every scale, self-similar under magnification — and they demand a richer, quantitative idea of dimension. This page builds that idea, and it is the natural companion to the iterated maps whose attractors turn out to be fractal.

Self-similarity, made precise

A shape is self-similar if it is built of scaled-down copies of itself. Watch the Koch curve assemble: take a segment, replace its middle third with two sides of a triangle, and repeat on every segment forever. Each stage is four copies of the previous one, each shrunk to a third the size.

Here is the paradox that forces a new definition. The Koch curve has infinite length (each step multiplies the length by 4/3, forever) yet bounds a finite area and never fills any patch of the plane. It is more than a line but less than a surface. To pin down "how much more," count copies against scale.

If a self-similar set is made of N copies of itself, each scaled down by a factor r < 1, its dimension is the exponent D for which N \cdot r^{D} = 1, i.e.

D = \frac{\ln N}{\ln (1/r)}.

This reproduces ordinary dimension — a segment splits into N = 2 halves (r = 1/2), giving D = \ln 2/\ln 2 = 1; a square into N = 4 quarter-squares (r = 1/2), giving D = \ln 4/\ln 2 = 2 — but it also returns non-integers for fractals.

For the Koch curve N = 4 and r = 1/3, so

D = \frac{\ln 4}{\ln 3} \approx 1.2619.

Between a line and a surface, exactly as the picture promised.

Box-counting: dimension for shapes that aren't perfectly self-similar

Real attractors — the Lorenz butterfly, the logistic attractor — are only approximately self-similar, so the copy-counting formula won't apply directly. The fix is the box-counting dimension. Cover the object with a grid of boxes of side \varepsilon and count how many boxes, N(\varepsilon), contain a piece of it. As the boxes shrink,

N(\varepsilon) \sim \left(\frac{1}{\varepsilon}\right)^{D}, \qquad D = \lim_{\varepsilon \to 0} \frac{\ln N(\varepsilon)}{\ln(1/\varepsilon)}.

Take logarithms and this says: plot \ln N(\varepsilon) against \ln(1/\varepsilon) and the points fall on a line whose slope is the dimension. That is the whole method — no self-similarity required, just counting boxes at finer and finer resolution. The chart shows the tell-tale slopes: a smooth curve rides slope 1, the Koch curve slope 1.26, the Sierpiński triangle slope 1.58, a filled region slope 2.

The idea generalises the metrics you met when studying metric spaces: box-counting works in any space where you can measure distance and lay down a grid, which is exactly why it applies equally to a coastline, a fern, and a strange attractor.

Worked example: the Sierpiński triangle and the Cantor set

Sierpiński triangle. Take a triangle, cut it into four half-size copies, and throw away the middle one — leaving N = 3 copies each scaled by r = 1/2. Then

D = \frac{\ln 3}{\ln 2} \approx 1.585.

More than the Koch curve (it is "bushier"), still short of filling the plane.

Cantor set. Take [0,1], delete the open middle third, and repeat on each remaining piece: N = 2 copies scaled by r = 1/3, so

D = \frac{\ln 2}{\ln 3} \approx 0.631.

A dimension less than one — the Cantor set is more than a scatter of isolated points (it is uncountable) but less than any curve. These non-integer dimensions are not artefacts; they are honest measurements of how densely a set fills space, and they classify strange attractors just as sharply as they classify these hand-built fractals.

In a famous 1967 paper Benoît Mandelbrot asked exactly this and answered: it depends on your ruler. Measure a coastline with a 100 km ruler and you miss every bay; with a 1 km ruler you catch the bays but miss the coves; with a 1 m ruler, the rocks. The measured length keeps growing as the ruler shrinks — precisely the Koch curve's infinite-length paradox in the wild. What is stable is the box-counting dimension: Britain's coast comes out around D \approx 1.25, Norway's craggier fjords nearer 1.5. Mandelbrot coined the word "fractal" (from Latin fractus, broken) to name this whole family, and turned a schoolboy's unanswerable question into a measurable number.

The scaling factor in D = \ln N / \ln(1/r) is the linear shrink r, not an area or volume ratio — a very common slip. When a square is halved in side you get N = 4 pieces with r = 1/2 (not r = 1/4); using 1/4 would wrongly give D = 1. And mind the reciprocal: it is \ln(1/r) in the denominator, so a 1/3 scaling contributes \ln 3, a positive number. Finally, remember that infinite length and finite dimension coexist happily: "infinitely long" does not make the Koch curve two-dimensional — its dimension is a firmly finite 1.26.