According to
Coxeter, this multidimensional term was defined by
Alicia Boole Stott for creating new polytopes, specifically starting from
regular polytopes to construct new
uniform polytopes. The
expansion operation is symmetric with respect to a regular polytope and its
dual. The resulting figure contains the
facets of both the regular and its dual, along with various prismatic facets filling the gaps created between intermediate dimensional elements. It has somewhat different meanings by
dimension. In a
Wythoff construction, an expansion is generated by reflections from the first and last mirrors. In higher dimensions, lower dimensional expansions can be written with a subscript, so e2 is the same as t0,2 in any dimension. By dimension: • A regular {p}
polygon expands into a regular 2n-gon. • The operation is identical to
truncation for polygons, e{p} = e1{p} = t0,1{p} = t{p} and has
Coxeter-Dynkin diagram . • A regular {p,q}
polyhedron (3-polytope) expands into a polyhedron with
vertex configuration p.4.q.4. • This operation for polyhedra is also called
cantellation, e{p,q} = e2{p,q} = t0,2{p,q} = rr{p,q}, and has Coxeter diagram . • : • : For example, a rhombicuboctahedron can be called an
expanded cube,
expanded octahedron, as well as a
cantellated cube or
cantellated octahedron. • A regular {p,q,r}
4-polytope (4-polytope) expands into a new 4-polytope with the original {p,q} cells, new cells {r,q} in place of the old vertices, p-gonal prisms in place of the old faces, and r-gonal prisms in place of the old edges. • This operation for 4-polytopes is also called
runcination, e{p,q,r} = e3{p,q,r} = t0,3{p,q,r}, and has Coxeter diagram . • Similarly a regular {p,q,r,s}
5-polytope expands into a new 5-polytope with facets {p,q,r}, {s,r,q}, {p,q}×{ }
prisms, {s,r}×{ } prisms, and {p}
×{s}
duoprisms. • This operation is called
sterication, e{p,q,r,s} = e4{p,q,r,s} = t0,4{p,q,r,s} = 2r2r{p,q,r,s} and has Coxeter diagram . The general operator for expansion of a regular n-polytope is t0,n−1{p,q,r,...}. New regular facets are added at each vertex, and new prismatic polytopes are added at each divided edge, face, ...
ridge, etc. == See also==