Rectification (geometry)
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Image:Dual Cube-Octahedron.svg
Image:Rectified cubic honeycomb.jpg
In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by the vertex figures and the rectified facets of the original polytope.
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[edit] Orders of rectification
A first order rectification truncates edges down to points. If a polytope is regular, this form is represented by an extended Schläfli symbol notation t1{p,q,...}.
A second order rectification, or birectification, truncates faces down to points. If regular it has notation t2{p,q,...}. For polyhedra, a birectification creates a dual polyhedron.
Higher order rectifications can be constructed for higher order polytopes. In general an n-rectification truncates n-faces to points.
If an n-polytope is (n-1)-rectified, its facets are reduced to points and the polytope becomes its dual.
[edit] In polygons
The dual of a polygon is the same as its rectified form.
[edit] In polyhedrons and plane tilings
Each platonic solid and its dual have the same rectified polyhedron. (This is not true of polytopes in higher dimensions.)
Examples
The rectified polyhedron turns out to be expressible as the intersection of the original platonic solid with an appropriated scaled concentric version of its dual. For this reason, its name is a combination of the names of the original and the dual:
- The rectified tetrahedron, whose dual is the tetrahedron, is the tetratetrahedron, better known as the octahedron.
- The rectified octahedron, whose dual is the cube, is the cuboctahedron.
- The rectified icosahedron, whose dual is the dodecahedron, is the icosidodecahedron.
[edit] Plane tilings
- A rectified square tiling is a square tiling.
- A rectified triangular tiling or hexagonal tiling is a trihexagonal tiling.
[edit] In polychora and 3d honeycomb tessellations
Each convex regular polychoron has a rectified form as a uniform polychoron.
A regular polychoron {p,q,r} has cells {p,q}. Its rectification will have two cell types, a rectified {p,q} polyhedron left from the original cells and {q,r} polyhedron as new cells formed by each truncated vertex.
A rectified {p,q,r} is not the same as a rectified {r,q,p}, however. A further truncation, called bitruncation, is symmetric between a polychoron and its dual. See Uniform_polychoron#Geometric_derivations.
Examples
