A runcinated cubic honeycomb (partial) - The original cells (purple cubes) are reduced in size. Faces become new blue cubic cells. Edges become new red cubic cells. Vertices become new cubic cells (hidden).
In geometry, runcination is an operation that cuts a regular polytope (or honeycomb) simultaneously along the faces, edges, and vertices, creating new facets in place of the original face, edge, and vertex centers.[citation needed]
For a regular {p,q,r} 4-polytope, the original {p,q} cells remain, but become separated. The gaps at the separated faces become p-gonal prisms. The gaps between the separated edges become r-gonal prisms. The gaps between the separated vertices become {r,q} cells. The vertex figure for a regular 4-polytope {p,q,r} is an q-gonal antiprism (called an antipodium if p and r are different).
For regular 4-polytopes/honeycombs, this operation is also called expansion by Alicia Boole Stott, as imagined by moving the cells of the regular form away from the center, and filling in new faces in the gaps for each opened vertex and edge.
