n-sphere-optimal-embedding-r-n+1

Summary

This is a hard geometric limit: a closed shape simply cannot be laid down in its own dimension without tearing, so a circle needs a plane, a sphere needs 3D space, and in general an n-dimensional closed surface needs at least n+1 dimensions to sit in cleanly. The practical upshot is that any system trying to represent closed manifolds is guaranteed to need one extra dimension of ambient space, and the n-sphere is the tightest, most efficient shape that fills that minimum requirement.