Box counting
Named by 2 essays across one field — each of them below, with the objects they name alongside it.
A cluster grown by random walkers
Release a particle far away, let it wander until it touches a growing cluster, and freeze it there. Twenty thousand particles later the cluster is a branching tree whose mass grows like the radius to the power 1.7. Every simulation agrees on the number; the only theorem says it is at least 1.5.
![A dust plus itself is a line. C level 6: 64 intervals, length 0.0878; C+C: 1 interval(s) [0, 2].](/f/a-dust-plus-itself-is-a-line-89387b47.svg)
Two dusts that add up to a line
The middle-thirds Cantor set has no length at all, and yet adding it to itself — every sum x + y of two of its points — fills the whole interval from 0 to 2 without a gap. The reason is a single number, the thickness, and it is exactly 1 for this set. Make the dust a little thinner and its sum with itself collapses to a dust again; make two different dusts and whether they add to a line depends on their dimensions, unless arithmetic gets in the way.
Named alongside it
The objects these essays reach for when they reach for this one.
Fractal dimensionSelf-similarityCantor setGrowth modelHarmonic measureIterated function systemMeasure zeroPower lawRandom walkSimulationTernary expansion