Cantellation (geometry)

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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 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.

Examples: cantellating polyhedra, tilings[edit]

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

See also[edit]


  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 (pp.145-154 Chapter 8: Truncation, p 210 Expansion)
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966

External links[edit]