{{Short description|Geometric operation on a regular polytope}} [[Image:Small rhombicuboctahedron.png|thumb|A cantellated cube - Red faces are reduced. Edges are bevelled, forming new yellow square faces. Vertices are truncated, forming new blue triangle faces.]] [[Image:Cantellated_cubic_honeycomb.png|thumb|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 a polyhedron 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.

== Notation == A cantellated polytope is represented by an extended Schläfli symbol '''''t'''''<sub>0,2</sub>{''p'',''q'',...} or '''''r'''''<math>\begin{Bmatrix}p\\q\\...\end{Bmatrix}</math> 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:'''

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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 ==

{| class=wikitable |+ Regular polyhedra, regular tilings |- !Form !colspan=3|Polyhedra !colspan=2|Tilings |- !Coxeter !rTT !rCO !rID !rQQ !rH&Delta; |- !Conway<BR>notation !eT !eC = eO !eI = eD !eQ !eH = e&Delta; |- align=center !rowspan=2|Polyhedra to<BR>be expanded |Tetrahedron |Cube or<BR>octahedron |Icosahedron or<BR>dodecahedron |Square tiling |Hexagonal tiling<BR>Triangular tiling |- align=center |40px40px |40px40px |40px40px |40px40px |40px40px |- !Image !100px !100px !100px !100px !100px |- !Animation !100px !100px !100px ! ! |}

{| class=wikitable |+ Uniform polyhedra or their duals |- !Coxeter !rrt{2,3} !rrs{2,6} !rrCO !rrID |- !Conway<BR>notation !eP3 !eA4 !eaO = eaC !eaI = eaD |- align=center !rowspan=2|Polyhedra to<BR>be expanded |Triangular prism or<BR>triangular bipyramid |Square antiprism or<BR>tetragonal trapezohedron |Cuboctahedron or<BR>rhombic dodecahedron |Icosidodecahedron or<BR>rhombic triacontahedron |- align=center |40px40px |40px40px |40px40px |40px40px |- !Image !100px !100px !100px !100px |- !Animation ! ! !100px !100px |}

== See also == * Chamfer (geometry) * Conway polyhedron notation * Uniform 4-polytope * Uniform polyhedron

== 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 == * {{mathworld | urlname = Expansion | title = Expansion}}

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Category:Polyhedra Category:4-polytopes