# Cantellation (geometry)

A cantellated cube - Red faces are reduced. Edges are bevelled, forming new yellow square faces. Vertices are truncated, forming new blue triangle faces.
A cantellated cubic honeycomb - Purple cubes are cantellated. Edges are bevelled, forming new blue cubic cells. Vertices are truncated, forming new red rectified cube cells.

In geometry, a cantellation is an operation in any dimension that bevels a regular polytope at its edges and vertices, creating a new facet in place of each edge and vertex. The operation also applies to regular tilings and honeycombs. This is also rectifying its rectification.

This operation (for polyhedra and tilings) is also called expansion by Alicia Boole Stott, as imagined by taking the faces of the regular form moving them away from the center and filling in new faces in the gaps for each opened vertex and edge.

## Notation

It is represented by an extended Schläfli symbol t0,2{p,q,...} or ${\displaystyle r{\begin{Bmatrix}p\\q\end{Bmatrix}}}$ or rr{p,q,...}.

For polyhedra, a cantellation operation offers a direct sequence from a regular polyhedron and its dual.

Example cantellation sequence between a cube and octahedron

For higher-dimensional polytopes, a cantellation offers a direct sequence from a regular polytope and its birectified form. A cuboctahedron would be a cantellated tetrahedron, as another example.

## Example polyhedral and tilings

Uniform polyhedral and tilings
Form Polyhedra Tilings
Coxeter rTT rCO rID rQQ rHΔ
Conway
notation
eT eC = eO eI = eD eQ eH = eΔ
Expanded
polyhedra
Tetrahedron Cube or
octahedron
Icosahedron or
dodecahedron
Square tiling Hexagonal tiling
Triangular tiling
Image
Animation
2-uniform polyhedra
Coxeter rrt{2,3} rrs{2,6} rrCO rrID
Conway
notation
eP3 eA4 eaO = eaC eaI = eaD
Expanded
polyhedra
Triangular prism or
triangular bipyramid
Square antiprism or
tetragonal trapezohedron
Cuboctahedron or
rhombic dodecahedron
Icosidodecahedron or
rhombic triacontahedron
Image
Animation