Schlegel diagram

Summary

A Schlegel diagram is a picture of a $d$-dimensional convex polytope inside one of its facets, one dimension lower. It is obtained by looking at the polytope from a point just outside that facet and projecting the rest of the boundary onto it. The result is a subdivision of the facet into convex polytopes, one for each other facet, that keeps the full face structure. Schlegel diagrams make $4$-dimensional polytopes visible in space and give a quick proof that graphs of $3$-polytopes are planar.