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Minimum rank2
Extended Schläfli symbol
Conway polyhedron notation

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.

Truncation by dimension[edit | edit source]

Polygons[edit | edit source]

Polygons are the lowest dimension for which rectification is possible. Rectifying an n-gon produces another n-gon.

Polyhedra[edit | edit source]


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 𝓟 with vertex, edge and face counts , and , its rectification has counts , and .

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

Maps[edit | edit source]

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

See also[edit | edit source]

External Resources[edit | edit source]