# 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.

Other relationships between regular maps

General Index