# 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${\displaystyle {\begin{Bmatrix}p\\q\\...\end{Bmatrix}}}$ 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.

## Examples: cantellating polyhedra, tilings

Regular polyhedra, regular tilings
Form Polyhedra Tilings
Coxeter rTT rCO rID rQQ rHΔ
Conway
notation
eT eC = eO eI = eD eQ eH = eΔ
Polyhedra to
be expanded
Tetrahedron Cube or
octahedron
Icosahedron or
dodecahedron
Square tiling Hexagonal tiling
Triangular tiling
Image
Animation
Uniform polyhedra or their duals
Coxeter rrt{2,3} rrs{2,6} 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
Image
Animation