>For instance, consider a unit cube in n dimensions. The volume of this cube (1 unit on each side) remains constant, but the distance from the center to a corne

>For instance, consider a unit cube in n dimensions. The volume of this cube (1 unit on each side) remains constant, but the distance from the center to a corner increases with the square root of the number of dimensions. This implies that in high dimensions, most of the volume of a hypercube is located at its corners. This is bizarre to comprehend because it's hard to stop imagining a "cube" in the 3D sense. This is incorrect. If you put a little box of small side length x at each corner, then each box has volume x^n and there are 2^n many of them, so the total volume they take up is 2^nx^n=(2x)^n, which is very small. Maybe what you meant is that most of the volume of a high-dimensional hypersphere is near its surface.