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

M:{p,p} V V E

and is the same as taking the dual and un-rectifying it.

ARM denotes halving by η.

Other relationships between regular maps
General Index