Expansion (geometry)
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In geometry, expansion is a polytope 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.)
This multidimensional term was defined by Alicia Boole Stott for creating new polytopes, specifically starting from regular polytopes.
It has somewhat different meanings by dimension, and correspond to reflections from the first and last mirrors in a Wythoff construction.
By dimension:
- A regular {p} polygon expands into a regular 2n-gon. (The operation is identical to truncation for polygons, t0,1{p}.)
- A regular {p,q} polyhedron (3-polytope) expands into a polyhedron with vertex figure p.4.q.4 (This operation for polyhedra is also called cantellation, t0,2{p,q} and has Coxeter-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 has Coxeter-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} duoprisms, {q} duoprisms. (This operation is called sterication, t0,4{p,q,r,s}.)
The general operator for expansion for an n-polytope is t0,n-1{p,q,r,...}.
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