Take a line segment. Remove the middle third. Now do that to each remaining piece. Repeat. What's left after infinity iterations is the Cantor set: uncountably many points, total length zero, fractal dimension log 2 / log 3 ≈ 0.631. The first fractal, and a counterexample to nearly every comfortable intuition about size and dimension.
why a set can have zero length but still be uncountable
The Cantor set is the limit of a recursive process: start with [0, 1], remove the open middle third (1/3, 2/3), leaving [0, 1/3] ∪ [2/3, 1]. Iterate on every remaining interval.
After n iterations, there are 2^n intervals, each of length (1/3)^n. Total length = (2/3)^n → 0. So the limit set has measure zero — it's negligible in the Lebesgue sense.
And yet: a point belongs to the Cantor set iff its ternary expansion uses only digits 0 and 2 (never 1). The set of all such sequences has the same cardinality as ℝ. The Cantor set is uncountable — same size as the real line.
Fractal dimension: at each step the set is 2 copies of itself scaled by 1/3. The Hausdorff dimension d satisfies 2 · (1/3)^d = 1, giving d = log 2 / log 3 ≈ 0.6309. Between a point (dim 0) and a line (dim 1) — a fractional dimension. Hausdorff and Besicovitch invented this notion specifically to capture sets like this.
The Cantor set is the foundation of fractal geometry: a topological dust, perfect (every point is a limit point), nowhere dense, self-similar at every scale. Mandelbrot named the field "fractal geometry" in 1975; Cantor (1883) was first.
It shows up everywhere: dynamical systems (the Smale horseshoe), the spectrum of the Fibonacci Hamiltonian, the boundary of Julia sets, sums of independent Bernoulli random variables, and the distribution of digits in unit-interval expansions.