The Cantor set: remove the middle fraction of each interval, iterate. Explore generalized Cantor sets with variable removal fraction. Watch the devil's staircase (Cantor function) emerge.
The standard Cantor set (middle-third removal) has Hausdorff dimension d_H = log(2)/log(3) ≈ 0.631. It is uncountable, measure zero, nowhere dense, and totally disconnected — the prototype of fractal dust. The Cantor function (devil's staircase) is continuous, monotone, yet has zero derivative a.e.