Rectification (geometry)

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:

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.