# Cantellation

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
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 cantellated. This yields a regular map described by
M:{p,4}   E   2V   2E.

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,

• 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 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.

### The name "cantellation"

The term "cantellation" was coined by Coxeter. It is further defined in the Wikipedia entry cantellation.

### Halving

The book "Abstract Regular Polytopes"ARM, page 197, uses the term "halving" for an operation closely related to what we call "cantellation". 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-cantellating it.

ARM denotes halving by η.