Cantor Set
Start with the interval [0, 1]. Remove the middle third. Then remove the middle third of each remaining segment. Repeat forever. What remains is the Cantor set — a fractal dust of points that is uncountably infinite yet has total length zero. Watch the Devil’s Staircase emerge as the Cantor function climbs from 0 to 1 while being constant almost everywhere.
Measure = (2/3)n → 0 · Fractal dimension = log 2 / log 3 ≈ 0.6309
The middle-thirds construction
Georg Cantor introduced this set in 1883 as an example of a perfect set (closed, with every point a limit point) that is nowhere dense — it contains no intervals at all. The construction is deceptively simple: begin with the closed interval [0, 1], remove the open middle third (1/3, 2/3), then remove the middle third of each remaining interval, and continue forever. At step n, you have 2n intervals each of length 3−n, so the total remaining length is (2/3)n, which converges to zero. Yet the set that remains is uncountably infinite — it has the same cardinality as the entire real line.
Uncountable yet measure zero
The Cantor set provides one of the most elegant paradoxes in mathematics: it is simultaneously very large (uncountable, like the reals) and very small (Lebesgue measure zero, like the rationals). The uncountability can be proved by noting that every point in the Cantor set has a ternary (base-3) expansion using only the digits 0 and 2 (never 1). These expansions are in bijection with binary sequences (replace each 2 with 1), and the set of all binary sequences is uncountable by Cantor’s diagonal argument. Meanwhile, the total length removed is 1/3 + 2/9 + 4/27 + … = 1, confirming the measure is zero.
The Devil’s Staircase
The Cantor function (or Devil’s Staircase) is a continuous function from [0, 1] to [0, 1] that rises from 0 to 1 but has derivative zero almost everywhere. On each removed interval, the function is constant (a “flat step”). All its increase happens on the Cantor set itself — a set of measure zero. This makes it a famous counterexample in real analysis: a continuous, non-decreasing, surjective function that is not absolutely continuous. The function is also self-similar: the left half of the graph (scaled up) looks identical to the whole, reflecting the fractal structure of the underlying set.
Fractal dimension
The Cantor set has Hausdorff dimension log 2 / log 3 ≈ 0.6309. This is strictly between 0 (the dimension of a countable set of points) and 1 (the dimension of a line segment), capturing the intuition that the Cantor set is “more than points but less than a line.” At each step of the construction, the set breaks into 2 copies of itself, each scaled by a factor of 1/3. The dimension d satisfies 2 · (1/3)d = 1, giving d = log 2 / log 3. This self-similarity dimension equals the Hausdorff dimension for the Cantor set, which is not always the case for more complicated fractals.