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 cuts 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.

It is represented by an extended Schläfli symbol t_0,2{p,q,...}.

For polyhedra, a cantellation operation offers a direct sequence from a regular polyhedron and its dual.

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.

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.

See also

• Uniform polyhedron
• Uniform polychoron
• Rectification (geometry)

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" from MathWorld.
• Olshevsky, George, Truncation at Glossary for Hyperspace.