Rectification is a non-symmetric relationship between some pairs of regular maps of the same genus. Any regular map can be rectified. If and only if a regular map is self-dual, it can be rectified to give another regular map.If a self-dual regular map is described by
This relationship is never symmetric: the rectified regular map has twice as many edges as the original.
For example, if we rectify the tetrahedron we get the octahedron.
If you have a regular map and want to rectify it,
The same procedure can be applied to a regular map which is not self-dual. However the result is not a regular map, it is semiregular. For example, if we rectify the cube, we get the cuboctahedron.
If a regular map has Petrie polygons of size r, and we rectify it, the result has holes of size r.
ARM denotes halving by η.
Other relationships between regular maps