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 t_0,2{p,q,...} or $r\begin{Bmatrix} p \\ q \end{Bmatrix}$ 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

See also

• Uniform polyhedron
• Uniform 4-polytope

References

• 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

• Weisstein, Eric W., "Expansion", MathWorld.
• Olshevsky, George, Truncation at Glossary for Hyperspace.