Manifold learning (nonlinear dimensionality reduction) assumes high-dimensional data lies on a low-dimensional manifold embedded in the ambient space. Methods like Isomap, LLE, and UMAP attempt to find a low-dimensional embedding that preserves the manifold's intrinsic geometry. The key insight: geodesic distances (distances along the manifold) can differ drastically from Euclidean distances — two points close in Euclidean space may be far apart on the manifold. The left panel shows the 3D manifold rotating; the right shows its intrinsic 2D parameterization, colored by position on the manifold. This unrolling reveals the true latent structure hidden in the high-dimensional space.