Rectification (geometry)

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.

A rectification operator is sometimes denoted by the letter r with a Schläfli symbol.

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.

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.

The rectified polyhedron turns out to be expressible as the intersection of the original platonic solid with an appropriately scaled concentric version of its dual.

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.

If a polytope is regular, this form is represented by an extended Schläfli symbol notation t1{p,q,...} or r{p,q,...}.

If an n-polytope is (n-1)-rectified, its facets are reduced to points and the polytope becomes its dual.