Cantellation (geometry)


In geometry, a cantellation is a 2nd order truncation in any dimension that bevels a regular polytope at its edges and at its vertices, creating a new facet in place of each edge and of each vertex. Cantellation also applies to regular tilings and honeycombs. Cantellating is also rectifying its rectification.
Cantellation is also called expansion by Alicia Boole Stott: it corresponds to moving the faces of the regular form away from the center, and filling in a new face in the gap for each opened edge and for each opened vertex.

Notation

A cantellated polytope is represented by an extended Schläfli symbol t0,2 or r or rr.
For polyhedra, a cantellation offers a direct sequence from a regular polyhedron to its dual.
Example: cantellation sequence between cube and octahedron:
Example: a cuboctahedron is a cantellated tetrahedron.
For higher-dimensional polytopes, a cantellation offers a direct sequence from a regular polytope to its birectified form.

Examples: cantellating polyhedra, tilings

CoxeterrrtrrsrrCOrrID
Conway
notation		eP3eA4eaO = eaCeaI = eaD
Polyhedra to
be expanded		Triangular prism or
triangular bipyramid		Square antiprism or
tetragonal trapezohedron		Cuboctahedron or
rhombic dodecahedron		Icosidodecahedron or
rhombic triacontahedron
Polyhedra to
be expanded
Image
Animation