Cantor Dust

About: The Cantor set (Georg Cantor, 1883) is constructed by repeatedly removing the open middle third of each remaining interval. After infinitely many steps: no intervals remain, yet uncountably many points survive. Its Lebesgue measure is zero (total length removed = 1), but its Hausdorff dimension is log(2)/log(3) ≈ 0.631 — fractional, between a point and a line. The 2D Cantor dust takes the Cartesian product C × C, producing a fractal dust with dimension 2 × 0.631 ≈ 1.26. This construction underpins modern measure theory and set theory.