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,

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.


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
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
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