In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a cantellation is a 2nd-order
truncation
In mathematics and computer science, truncation is limiting the number of digits right of the decimal point.
Truncation and floor function
Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb ...
in any dimension that
bevels a
regular polytope
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
at its
edges and at its
vertices, creating a new
facet
Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cu ...
in place of each edge and of each vertex. Cantellation also applies to
regular tilings and
honeycomb
A honeycomb is a mass of hexagonal prismatic wax cells built by honey bees in their nests to contain their larvae and stores of honey and pollen.
Beekeepers may remove the entire honeycomb to harvest honey. Honey bees consume about of honey t ...
s. Cantellating a polyhedron is also rectifying its
rectification.
Cantellation (for polyhedra and tilings) is also called ''
expansion
Expansion may refer to:
Arts, entertainment and media
* ''L'Expansion'', a French monthly business magazine
* ''Expansion'' (album), by American jazz pianist Dave Burrell, released in 2004
* ''Expansions'' (McCoy Tyner album), 1970
* ''Expansio ...
'' by
Alicia Boole Stott
Alicia Boole Stott (8 June 1860 – 17 December 1940) was an Irish mathematician. Despite never holding an academic position, she made a number of valuable contributions to the field, receiving an honorary doctorate from the University of Gron ...
: 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
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mor ...
''t''
0,2 or ''r''
or ''rr''.
For
polyhedra
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on t ...
, a cantellation offers a direct sequence from a
regular polyhedron
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equiv ...
to its
dual.
Example: cantellation sequence between cube and octahedron:
Example: a
cuboctahedron
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...
is a cantellated
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...
.
For higher-dimensional polytopes, a cantellation offers a direct sequence from a regular polytope to its
birectified form.
Examples: cantellating polyhedra, tilings
See also
*
Uniform polyhedron
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent.
Uniform polyhedra may be regular (if also ...
*
Uniform 4-polytope
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.
There are 47 non-prismatic convex uniform 4-polytopes. There ...
*
Chamfer (geometry)
References
*
Coxeter, H.S.M. ''
Regular Polytopes'', (3rd edition, 1973), Dover edition, (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
*
{{polyhedron-stub
Polyhedra
4-polytopes