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 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.


It is represented by an extended Schläfli symbol t0,2{p,q,...} or 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

Cube cantellation sequence.svg

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[edit]

Uniform polyhedral and tilings
Form Polyhedra Tilings
Coxeter rTT rCO rID rQQ rHΔ
eT eC = eO eI = eD eQ eH = eΔ
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
2-uniform polyhedra
Coxeter rrt{2,3} rrs{2,6} rrCO rrID
eP3 eA4 eaO = eaC eaI = eaD
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]