# Truncation

Rectification is a non-symmetric relationship between some pairs of regular maps of the same genus.

If a regular map is described by
M:{n,2n}   an   2an   an2
(meaning, it is in manifold M, each face has n edges, each vertex has 2n edges, it has an vertices, 2an faces and an2 edges), then it can be truncated. This yields a map described by
M:{2n,3}   2an2   3an   3an2.

The truncated map may or may not be regular; for example members of series z all have the form {n,2n}, but only those in genera 0 and 1 yield regular maps when truncated. The relationship is never symmetric: the truncated regular map has more faces than the original.

If the original regular map has second-order Petrie polygons with p edges, then the truncated map has Petrie polygons with 3p edges.

For example, if we rectify the 4-hosohedron we get the cube. The second-order Petrie polygons of the 4-hosohedron have two edges; those of the cube have six edges.

Every regular map of the form {3,6} can be truncated to give a regular map {6,3}.

If you have a regular map and want to truncate it,

• replace each vertex by a face
• retain each face as a face