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 (for polyhedra and tilings) 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{p,q,...} or r$$\begin{Bmatrix}p\\q\\...\end{Bmatrix}$$ or rr{p,q,...}.

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.