Curse of Dimensionality: Distance Concentration

Every distance-based technique in data science — k-nearest neighbors, clustering, PCA-style variance capture — quietly assumes that "near" and "far" mean something. This simulator generates a real N-point cloud with d independent coordinates and computes its actual full-dimensional statistics: what fraction of the points fall inside the hypersphere inscribed in their bounding cube, and how much the nearest and farthest pairwise distances still differ (the relative contrast). Drag the dimension slider from 2 up to 150 and watch both numbers collapse — almost every point ends up in a "corner" outside the sphere, and every pair of points ends up nearly the same distance apart. A random 3D projection (the same Johnson–Lindenstrauss idea dimensionality-reduction tools rely on) renders the cloud so you can see the shape of the collapse, while the numbers underneath are computed honestly in the full d-dimensional space.