- Expansion (geometry)
[
dodecahedron creates arhombicosidodecahedron and a reverse expansion creates anicosahedron .] Ingeometry , expansion is apolytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements (vertices, edges, etc). (Equivalently this operation can be imagined by keeping facets in the same location, but reducing their size.)According to
Coxeter , this multidimensional term was defined byAlicia Boole Stott [Coxeter, H. S. M., "Regular Polytopes". 3rd edition, Dover, 1973, p. 123. ISBN 0-486-61480-8. p.210] for creating new polytopes, specifically starting fromregular polytope s constructs newuniform polytope s.The "expansion" operation is symmetric with respects to a regular polytope and its dual. The resulting figure contains the facets of both the regular and regular dual, along with various prismatic facets filling the gaps created between intermediate dimensional elements.
It has somewhat different meanings by
dimension , and correspond to reflections from the first and last mirrors in aWythoff construction .By dimension:
* A regular {p}polygon "expands" into a regular 2n-gon.
** The operation is identical to truncation for polygons, t0,1{p} and hasCoxeter-Dynkin diagram .
* A regular {p,q}polyhedron (3-polytope) "expands" into a polyhedron withvertex figure "p.4.q.4".
**This operation for polyhedra is also called cantellation, t0,2{p,q} and hasCoxeter-Dynkin 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}polychoron (4-polytope) "expands" into a new polychoron with the original {p,q} cells, new cells {q,r} in place of he old vertices and q-gonal prisms in place of the old edges.
** This operation for polychora is also called runcination, t0,3{p,q,r} and hasCoxeter-Dynkin diagram .
* Similar a regular {p,q,r,s}polyteron (5-polytope) "expands" into a new polyteron with facets {p,q,r}, {q,r,s}, {q,r} hyperprisms, {p}duoprism s, {q} duoprisms.
** This operation is calledsterication , t0,4{p,q,r,s} and hasCoxeter-Dynkin diagram .The general operator for "expansion" for an n-polytope is t0,n-1{p,q,r,...}. New
simplex facets are added at each vertex, and new prismatic polytopes are added at each divided edge, face, ... ridge, etc.References
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