Projecting Points on Spheres to Lower Dimensions

Take points at random inside the sphere (or hypersphere) in dimensions, and project them to that sphere. Now project that set of points to either the interval , the unit disk, or the unit 3D ball (the interior of the 3D unit sphere) by dropping all but one, two, or three coordinates.

When the original dimension is low, the second projection will be to the endpoints of the interval, the unit circle, or the unit 3D sphere. For higher dimensions the projected points lie in the interval, the disk, or the ball.

The points cluster on the outside for low dimensions and toward the center for higher dimensions—this is simply due to dropping coordinates when projecting. For example, if , it is most likely that . Still, it looks mysterious.