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.
A cantellated polytope is represented by an extended Schläfli symbol t0,2{p,q,...} or r 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.
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 |
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 |