# Rectification

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
M:{p,p}   V   V   E
(meaning, it is in manifold M, each face has p edges, each vertex has p edges, it has V vertices, V faces and E edges), then it can be rectified. This yields a regular map described by
M:{p,4}   E   2V   2E.

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,

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

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 a and holes of size b, and we rectify it, the result has Petrie polygons of size 2b and holes of size a.

### The name "rectification"

The term "rectification" is defined in the Wikipedia entry rectification.

### Halving

The book "Abstract Regular Polytopes"ARM, page 197, uses the term "halving" for an operation closely related to what we call "rectification". Halving converts
M:{4,p}   2V   E   2E
to
M:{p,p}   V   V   E
and is the same as taking the dual and un-rectifying it.

ARM denotes halving by η.