Ball Shape Lab
A ball in five or fifty dimensions can't be drawn directly, but three honest views of it can be, and they are all exact. Drag the dimension and watch the ball and its surface change together: what the ball looks like sliced through the middle, where its volume actually lives, and how much of the box around it it manages to fill.
Left — the ball, sliced. Cut the unit ball straight through its center and stack the slices: the width drawn at each height is how much of the ball's volume that slice holds. For n = 2 this is a circle, for n = 3 a rounder spindle, and as n climbs the shape sharpens into a needle — the ball's volume abandons the poles and crowds into the thin slab through the middle. This is the equator belt from the Dimension Lab, seen as a solid. Middle — where the volume lives. The disc is shaded by how much volume sits at each distance from the center; the rings are the shells at 10% steps of the radius. In two dimensions the disc is evenly filled; by n = 20 the inner 90% of the radius is nearly empty and the ball is all skin. That is the meaning of S(n)/V(n) = n: with every added dimension the surface takes a bigger share of the ball. Right — the ball in its box. The ball always touches the six (or 2n) faces of the cube that encloses it, yet the fraction of the cube it fills collapses: 79% in the plane, 52% in space, a quarter of a percent by n = 10. The corners of a high-dimensional cube are enormously far from its center, and the ball never reaches them.