What is Fractal Dimension?

A point has dimension 0. A line has dimension 1. A plane has dimension 2. A solid has dimension 3. This all seems obvious — but what dimension is the coastline of Britain? Or the boundary of the Mandelbrot set? The answer, it turns out, is not a whole number.

The Problem with Coastlines

In 1967, Benoit Mandelbrot published a famous paper: “How Long Is the Coast of Britain?” His answer: it depends on how closely you measure. Use a ruler 100 km long and you get one answer. Use a ruler 10 km long and you get a larger answer, because it fits into more of the bays and promontories. Use a ruler 1 km long and the measured length grows again.

As the ruler shrinks to zero, the measured length of a rough coastline approaches infinity. A smooth curve (like a circle) would converge to a finite length. The difference is encoded in the fractal dimension.

Familiar Dimensions

Dimension tells you how an object scales. If you scale a line segment by 3, its “size” (length) multiplies by $3^1 = 3$. If you scale a square by 3, its area multiplies by $3^2 = 9$. If you scale a cube by 3, its volume multiplies by $3^3 = 27$.

The exponent — 1, 2, or 3 — is the dimension. This is the self-similarity scaling law:

The Koch Snowflake: Dimension ≈ 1.26

The Koch snowflake is built by taking a triangle and replacing the middle third of each side with a smaller triangle, then repeating forever. Each iteration multiplies the number of sides by 4 ($N=4$) while scaling each piece by $r=3$. So its dimension is:

It's more than a line (1) but less than a plane (2). It has infinite length but zero area. This non-integer value is the fractal dimension.

Measuring with Box Counting

One practical way to estimate fractal dimension is the box counting method. Cover the fractal with a grid of boxes of side length $\varepsilon$ and count how many boxes $N(\varepsilon)$ are needed to cover the shape. Then shrink $\varepsilon$ and count again:

Box counting: as the grid gets finer, box count grows faster for rougher shapes.

The Mandelbrot Set: Dimension 2

Here is a remarkable theorem: the boundary of the Mandelbrot set has Hausdorff dimension exactly 2, despite being a 1-dimensional curve. This was proved by Mitsuhiro Shishikura in 1998 after years of work.

What does dimension 2 mean for a curve? It means the boundary is so infinitely wiggly that it effectively fills a 2D region, even though it's technically a 1D mathematical set. Zoom into any boundary point and you'll find more and more fine detail — no matter how deep you go.

This is what you're witnessing when you zoom into the gallery locations on FractalSet: the maximal possible roughness of a curve.