A 4D Voronoi foam: seeds scattered through four dimensions, each owning the points nearer to it than to any other seed. Each owned region is a cell, a convex 4D polytope, and the cells fill 4D with no gaps.

You see one 3D slice. Every slice is itself a foam of convex polyhedra, but not a Voronoi one: a seed a distance h off the slice competes as if handicapped by h². Seeds near the slice own big cells; far ones own small cells or none. Mathematicians call this a power diagram, and every power diagram is a slice of a Voronoi foam one dimension up.

Sweep w and nothing moves sideways (each seed lands in the same place); only the handicaps change. Cells grow and shrink, and the foam changes shape in exactly three ways:

• A cell is born as a tiny tetrahedron where four cells met, when the slice first touches that 4D cell's corner.
• A cell dies, shrinking to a point.
• A face flips: it closes to an edge shared by three cells, or an edge opens into a face.

Real 3D foams coarsen by the last two. A 4D foam is, in a sense, a 3D foam's whole history at once.

The counts are exact. Every slice is computed by clipping each cell with its bisector planes, and the events are counted between successive slices for cells clear of the box's walls.