High-dimensional spheres are "spikey"
Another way of looking at this phenomenon is to inscribe a high dimensional sphere in a hypercube. As the dimension grows, the sphere/cube volume ratio gets arbitrarily small, although sphere touches the cube at every side. It seems that most of the mass of the cube was focused near vertices. This is because the side of a cube is a whole lot closer than the vertex from the center of the cube -- for instance, if we take a million-dimensional cube with a side of length 2, then the distance from the center of the cube to the side is 1, but the distance from the center to the vertex is 1000. It's not the spheres that are "spikey", the cubes are.