Rectification

Rectification (also called complete truncation) is an operation on polytopes which creates a new polytope by cutting away area around each vertex to create new facets with new vertices at the midpoints of the original edges. It can be thought of a form of truncation with deeper cuts.

Polygons
Polygons are the lowest dimension for which rectification is possible. Truncating a polygon replaces each vertex with an edge and each edge with a vertex. Thus the truncation of a polygon is the same as its dual. Since regular polygons are isomorphic to their duals, the truncation of a regular polygon is the polygon itself. In terms of Coxeter-Dynkin diagrams this can be written:

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Polyhedra
A cube (left) and a rectified cube (right). The newly exposed faces are colored yellow.

Rectifying a polyhedron results in a new polyhedron. For every face $F$ in the original polyhedron, there is a corresponding face in the rectified polyhedron $F'$ which is the rectification of the original $F$. Furthermore, if two faces were adjacent by an edge in the original polyhedron, their rectifications are adjacent in the rectified polyhedron. Recfication additionally adds more faces corresponding to vertices in the original polyhedron. These faces are similarly adjacent by a vertex iff the vertexes the correspond to in the initial polyhedron were adjacent by an edge. For an initial polyhedron with vertex, edge and face counts $$V(\mathcal{P})$$, $$E(\mathcal{P})$$ and $$F(\mathcal{P})$$, its rectification $$a(\mathcal{P})$$ has counts $$V(a(\mathcal{P})) = E(\mathcal{P})$$, $$E(a(\mathcal{P}))=E(\mathcal{P})$$ and $$F(a(\mathcal{P}))=F(\mathcal{P})+V(\mathcal{P})$$.

The rectification of a regular polyhedron and its dual are isomorphic. This is not true in general for polytopes of higher rank.

Maps
The idea of rectification can be generalized to an operation on maps. Rank 3 maps correspond to graph embeddings and their rectification corresponds is the medial graph.