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 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:

Cube cantellation sequence.svg

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 polyhedron-33-t0.pngUniform polyhedron-33-t2.png Uniform polyhedron-43-t0.svgUniform polyhedron-43-t2.svg Uniform polyhedron-53-t0.svgUniform polyhedron-53-t2.svg Uniform tiling 44-t0.svgUniform tiling 44-t2.svg Uniform tiling 63-t0.svgUniform tiling 63-t2.svg
Image Uniform polyhedron-33-t02.png Uniform polyhedron-43-t02.png Uniform polyhedron-53-t02.png Uniform tiling 44-t02.svg Uniform tiling 63-t02.svg
Animation P1-A3-P1.gif P2-A5-P3.gif P4-A11-P5.gif
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
Triangular prism.pngTriangular bipyramid2.png Square antiprism.pngSquare trapezohedron.png Uniform polyhedron-43-t1.svgDual cuboctahedron.png Uniform polyhedron-53-t1.svgDual icosidodecahedron.png
Image Expanded triangular prism.png Expanded square antiprism.png Expanded dual cuboctahedron.png Expanded dual icosidodecahedron.png
Animation R1-R3.gif R2-R4.gif

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]