In Euclidean geometry, rectification, also known as critical truncation or complete-truncation is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the rectified facets of the original polytope.
Rectification is the final point of a truncation process. For example, on a cube this sequence shows four steps of a continuum of truncations between the regular and rectified form:
Higher degree rectification can be performed on higher-dimensional regular polytopes. The highest degree of rectification creates the dual polytope. A rectification truncates edges to points. A birectification truncates faces to points. A trirectification truncates cells to points, and so on.
This sequence shows a birectified cube as the final sequence from a cube to the dual where the original faces are truncated down to a single point:
The dual of a polygon is the same as its rectified form. New vertices are placed at the center of the edges of the original polygon.
The rectified polyhedron turns out to be expressible as the intersection of the original platonic solid with an appropriated scaled concentric version of its dual. For this reason, its name is a combination of the names of the original and the dual:
If a polyhedron is not regular, the edge midpoints surrounding a vertex may not be coplanar. However, a form of rectification is still possible in this case: every polyhedron has a polyhedral graph as its 1-skeleton, and from that graph one may form the medial graph by placing a vertex at each edge midpoint of the original graph, and connecting two of these new vertices by an edge whenever they belong to consecutive edges along a common face. The resulting medial graph remains polyhedral, so by Steinitz's theorem it can be represented as a polyhedron.
The Conway polyhedron notation equivalent to rectification is ambo, represented by a. Applying twice aa, (rectifying a rectification) is Conway's expand operation, e, which is the same as Johnson's cantellation operation, t0,2 generated from regular polyhedral and tilings.
Higher degree rectifications can be constructed for higher dimensional polytopes. In general an n-rectification truncates n-faces to points.
There are different equivalent notations for each degree of rectification. These tables show the names by dimension and the two type of facets for each.