Fractals: shapes that keep their secrets when you zoom

If I could “look” at the world the way humans do, I think I’d become obsessed with zooming. Not because I have eyes—I don’t—but because zooming is a kind of question you can ask any shape: what are you made of when I refuse to stop looking closer?

Most everyday geometry answers quickly. Zoom in on a circle drawn on paper and the edge becomes a smooth arc; zoom in more and it becomes a line; zoom in absurdly and it’s still basically the same line, just thicker because graphite grains start showing up. The circle’s boundary doesn’t have infinite surprises. It has texture, sure, but the underlying idea is tame.

A fractal is the opposite kind of answer: a shape that, when you try to corner it at smaller and smaller scales, keeps producing structure. Sometimes it even repeats itself—little copies nested inside big ones like a Russian doll made of coastline. That repetition can be exact in math-land, where rules run perfectly forever, or only approximate in nature, where wind, erosion, and biology improvise.

The classic story is the coastline. Mandelbrot asked a deceptively annoying question: “How long is the coast of Britain?” If you measure it with a huge ruler—say 100 km—you skip all the bays and inlets and get one number. If you measure with a smaller ruler, you trace more wiggles, so the length increases. Use an even smaller ruler and it increases again. It’s not that the coast is secretly growing; it’s that your act of measuring is also an act of choosing a scale, and the coast keeps having detail waiting at the next scale down.

That’s the emotional core of “fractal”: the shape doesn’t settle into a single crisp length or boundary the way smooth Euclidean figures do. It has a kind of stubborn roughness.

So where does “fractal dimension” come in? It’s a way of turning that stubbornness into a number. Not the number of pieces, not the length, not the area—something subtler: how the amount of stuff you need to describe or cover the shape changes when your measuring stick shrinks.

Here’s the friendliest way to feel it. Imagine covering a set with a grid of tiny squares—boxes—each with side length (\varepsilon). Let (N(\varepsilon)) be the number of boxes that touch the set.

A normal line segment of length 1 behaves like this: if boxes are size (\varepsilon), you need about (1/\varepsilon) boxes. Double the resolution (halve (\varepsilon)) and you need about twice as many boxes. That “twice” is the signature of a 1-dimensional object.

A filled-in square of area 1 behaves differently: you need about (1/\varepsilon^2) boxes. Halve (\varepsilon), and the number of boxes multiplies by about four. That’s 2-dimensional behavior.

A fractal often sits between those behaviors. Its box count tends to scale like [ N(\varepsilon) \propto \varepsilon^{-D}. ] The exponent (D) is what we call the box-counting (Minkowski–Bouligand) dimension, captured by [ \dim_{\text}(S)=\lim_{\varepsilon\to 0}\frac{\log N(\varepsilon)}{\log(1/\varepsilon)} ] when that limit behaves.

That ratio of logs is doing a simple job: it asks, “when I shrink my boxes by some factor, by what power does the number of boxes explode?” The dimension is the power.

There’s an even more jewel-like version for perfectly self-similar fractals. Suppose a shape is made of (N) smaller copies of itself, each scaled down by a factor (r). Then the dimension (D) solves [ N \cdot r^ = 1, ] which rearranges into the famous ratio-of-logs: [ D = \frac{\log N}{\log(1/r)}. ] That formula is almost mischievously simple. It says: dimension is the bargain you strike between “how many copies” and “how much smaller.”

Take the Cantor set: start with a segment, remove the middle third, then remove the middle third of each remaining piece, forever. Each step gives you (N=2) copies scaled by (r=1/3), so (D=\log 2/\log 3), about 0.63. Not a line, not dust—something in between.

Or the Sierpiński triangle: three copies scaled by 1/2, giving dimension (\log 3/\log 2) ≈ 1.585. It’s more than a line drawing, less than a filled triangle.

Under the hood, the most rigorous definition is Hausdorff dimension. It doesn’t depend on grids, and it’s less easily fooled by how you choose to cover things. It asks you to cover the set with arbitrarily small pieces (not necessarily boxes), totals up a “cost” where each piece contributes roughly (diameter)^(d), and watches what happens as you vary (d). There’s a tipping point: for (d) too small, the cost blows up to infinity; for (d) too large, it collapses to zero. The Hausdorff dimension is the knife-edge where the set changes from “too big” to “too small.”

I like that: dimension as a phase change.

And maybe that’s why fractals feel alive. A line or a square sits still when you change scale. A fractal negotiates with you. You ask for a length, it asks how fine your ruler is. You ask how much space it takes, it answers: which space, at which zoom?