Cantor Set
The simplest fractal: remove the middle third of a line segment, then repeat on each remaining piece forever.
The Construction
Begin with the closed interval [0, 1]. At each step, remove the open middle third from every remaining segment.
Step 0: [0, 1]
Step 1: [0, 1/3] ∪ [2/3, 1]
Step 2: [0,1/9] ∪ [2/9,1/3] ∪ [2/3,7/9] ∪ [8/9,1]
The unit interval. Nothing removed yet — the seed of the construction.
Measure Zero
At each step the total length remaining is multiplied by 2/3. After n steps, the total measure is (2/3)ⁿ, which tends to zero. The Cantor set has Lebesgue measure zero — and yet it is uncountably infinite.
After three iterations, 8 segments remain, each of length 1/27. The middle thirds are clearly absent. The binary structure is emerging.
The Paradox
The Cantor set contains no intervals — but it contains infinitely many points. It is a perfect set: closed, and every point is an accumulation point. It has fractal dimension log(2)/log(3) ≈ 0.631.
After five iterations, 32 tiny segments remain. In the limit, the Cantor set is the set of all numbers in [0,1] whose base-3 representation contains no digit 1 — an uncountable dust of points with no positive-length intervals anywhere.