The Curse of Dimensionality

What to observe

1. Empty space. Cut every axis into 10 bins and the number of cells is 10^d. At d = 20 that is a hundred billion billion boxes, and your 600 points sit in 600 of them. The 2D picture still looks nicely filled, which is the trap: every low-dimensional projection lies about how crowded the real space is.
2. Watch the curve on the right. To gather 10% of the data you must span63% of every axis at d = 5 and 89% at d = 20. A "local" neighbourhood stops being local, so anything built on neighbours (k-NN, kernels, histograms) quietly stops being local too.
3. Volume in the crust. Drag d up and watch the dots evacuate the middle of the ball. The outer 5% shell holds under 10% of a 2D disc but over99% of a 100D ball. Nothing moved: it is just that volume scales like r^d, so almost all of it is near the surface.
4. All distances equal. The cloud around the query point collapses into a thin ring: by d = 60 the spread of the distances is only a few percent of their mean, and it keeps shrinking like 1/√d. Nearest and farthest stop being meaningfully different, which is why exact nearest-neighbour search loses its grip in high dimensions.
5. Everything orthogonal. Two random directions have cos θ with mean 0 and standard deviation 1/√d. By d = 100 nearly every pair sits within a few degrees of 90°. High-dimensional space is roomy enough for enormous numbers of almost-orthogonal directions, which is exactly what makes random projections and superposition in embeddings work.
6. Nothing but corners. The ball inscribed in a cube touches all 2d faces yet fills 1 part in 10²⁸ of it at d = 50. All the volume hides in the 2^d corners, and the cube's half-diagonal is √d while its inradius stays 1: the box is really a spiky sea urchin, not a box.