Cantellation is a non-symmetric relationship between some pairs of regular maps of the same genus. Any self-dual regular map can be cantellated.If a self-dual regular map is described by
This relationship is never symmetric: the cantellated regular map has twice as many edges as the original.
For example, if we cantellate the tetrahedron we get the octahedron.
If you have a regular map and want to cantellate 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 cantellate the cube, we get the cuboctahedron.
If a regular map has Petrie polygons of size r, and we cantellate it, the result has holes of size r.
ARM denotes halving by η.
Other relationships between regular maps