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 a polyhedron 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 t_0,2 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.

Coxeter	rrt	rrs	rrCO	rrID
Conway notation	eP3	eA4	eaO = eaC	eaI = 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
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Animation