Expansion (geometry)

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An example of expanding pentagon into a decagon by moving edges away from the center and inserting new edges in the gaps. The expansion is uniform if all the edges are the same length.

Expansion of a dodecahedron creates a rhombicosidodecahedron and a reverse expansion creates an icosahedron.

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.)

According to Coxeter, this multidimensional term was defined by Alicia Boole Stott^[1] for creating new polytopes, specifically starting from regular polytopes constructs new uniform polytopes.

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 a Wythoff construction.

By dimension:

A regular {p} polygon expands into a regular 2n-gon.
- The operation is identical to truncation for polygons, t_0,1{p} and has Coxeter-Dynkin diagram .
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, t_0,2{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, t_0,3{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, t_0,4{p,q,r,s} and has Coxeter-Dynkin diagram .

The general operator for expansion for an n-polytope is t_0,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.