The Hodge star ★ is an isomorphism from k-forms to (n−k)-forms on an n-dimensional oriented Riemannian manifold. In 3D: ★(dx) = dy∧dz, ★(dy∧dz) = dx — it maps vectors to planes and vice versa. Maxwell's equations take an elegant form: dF = 0 and d★F = ★J, where F is the electromagnetic 2-form. The Laplace-Beltrami operator is Δ = d★d★ + ★d★d, unifying div, grad, and curl. Hodge decomposition states every form uniquely splits into exact + co-exact + harmonic parts — the harmonic part counts topological holes (Betti numbers).