Mathematics of scale: Big, small and everything in between

Breathe. As your lungs expand, air fills 500 million tiny alveoli, each a fraction of a millimeter across. As you exhale, these millions of tiny breaths merge effortlessly through larger and larger airways into one ultimate breath.

These airways are fractal.

The branches within lungs are an example of self-similarity. Brockhaus and Efron Encyclopedic Dictionary/Wikimedia
Fractals are a mathematical tool for describing objects with detail at every scale. Mathematicians and physicists like me use fractals and related concepts to understand how things change going from small to big.

You and I translate between vastly different scales when we think about how our choices affect the world. Is this latte contributing to climate change? Should I vote in this election?

These conceptual tools apply to the body as well as landscapes, natural disasters and society.

Fractals everywhere

In 1967, mathematician Benoit Mandelbrot asked, “How long is the coast of Britain?”

It’s a trick question. The answer depends on how you measure it. If you trace the outline on a map, you get one answer, but if you walk the coastline with a meter stick, the result is quite different. Anyone who has tried to estimate the length of a rugged hiking trail from a map knows the treachery of the large-scale picture.

Satellite image of Great Britain and Northern Ireland. NASA
That’s because lungs, the British coastline and hiking trails all have fractality: their length, number of branches or some other quantity depends on the scale or resolution you use to measure them.

The coastline is also self-similar – it’s made out of smaller copies of itself. Fern fronds, trees, snail shells, landscapes, the silhouettes of mountains and river networks all look like smaller versions of themselves.

That’s why, when you’re looking at an aerial photograph of a landscape, it’s often hard to tell whether the scale bar should be 50 km or 500 m.

Your lungs are self-similar, because the body finely calibrates each branch in exact proportions, making each branch a smaller replica of the previous. This modular design makes lungs efficient at any size. Think of a child and an adult, or a mouse, a whale. The only difference between small and large is in how many times the airways branch.

Self-similarity and fractality appear in art and architecture, in the arches within arches of Roman aqueducts and the spires of Gothic cathedrals that mirror the forest canopy. Even ancient Chinese calligraphers Huai Su and Yan Zhenqing prized the fractality of summer clouds, cracks in a wall and water stains in a leaking house in 722.