Cantellation (geometry)

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.

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.

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.

Notation

A cantellated polytope is represented by an extended Schläfli symbol t_0,2{p,q,...} or r${\displaystyle {\begin{Bmatrix}p\\q\\...\end{Bmatrix}}}$ 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

Regular polyhedra, regular tilings

Form	Polyhedra			Tilings
Coxeter	rTT	rCO	rID	rQQ	rHΔ
Conway notation	eT	eC = eO	eI = eD	eQ	eH = eΔ
Polyhedra to be expanded	Tetrahedron	Cube or octahedron	Icosahedron or dodecahedron	Square tiling	Hexagonal tiling Triangular tiling
Image
Animation

Uniform polyhedra or their duals

Coxeter	rrt{2,3}	rrs{2,6}	rrCO	rrID
Conway notation	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
Image
Animation

See also

• Uniform polyhedron
• Uniform 4-polytope
• Chamfer (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". MathWorld.