In high dimensions, our intuitions completely break down. Spheres become hollow, distances become meaningless, and almost all space becomes unreachable corners. Explore the bizarre geometry of high-dimensional spaces.
The volume of a unit ball (radius 1) rapidly approaches zero as dimensions increase. By 20 dimensions, it's smaller than a grain of sand!
What fraction of a cube contains its inscribed sphere? In 2D it's 78%. In high dimensions, it's essentially 0% — all volume migrates to the corners!
Almost all volume of a high-dimensional sphere is concentrated in a thin shell near the surface. The interior becomes empty!
Random points in high dimensions all become approximately equidistant. Nearest neighbors become as far as the farthest neighbors!
Many algorithms rely on distance metrics. In high dimensions, all points appear equidistant, breaking clustering, k-NN, and kernel methods.
To uniformly sample a 100D hypercube, you'd need more samples than atoms in the universe to achieve the same coverage as 100 points in 2D.
Searching for optimal solutions becomes exponentially harder. The vast majority of space is far from any sampled point.
Protein folding involves hundreds of dimensions. How does nature navigate this curse? (Levinthal's Paradox)