Rectification (geometry)
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- For other senses of this word, see rectification.
If the starting polytope is regular, the facets of the resulting polytope will also be regular. However, the resulting polytope itself will generally not be regular.
In polygons
The dual of a polygon is the same as its rectified form.
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
| Parent | Rectification | Dual |
|---|---|---|
 Tetrahedron |  Tetratetrahedron | Tetrahedron |
 Octahedron |  Cuboctahedron |  Cube |
 Icosahedron |  Icosidodecahedron |  Dodecahedron |
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.
Plane tilings
- A rectified square tiling is a square tiling.
- A rectified triangular tiling or hexagonal tiling is a trihexagonal tiling.
In polychora and 3d honeycomb tessellations
Each convex regular polychoron has a rectified form as a uniform polychoron.
A regular polychoron has cells . Its rectification will have two cell types, a rectified polyhedron left from the original cells and polyhedron as new cells formed by each truncated vertex.
A rectified is not the same as a rectified , however. A further truncation, called bitruncation, is symmetric between a polychoron and its dual. See Uniform_polychoron#Geometric_derivations.
Examples
| Parent | Rectification |
|---|---|
 5-cell |  Rectified 5-cell |
 hypercube | (No image) Rectified tesseract |
|
16-cell |  24-cell |
24-cell | (No image) rectified 24-cell |
 120-cell | (No image) rectified 120-cell |
 600-cell |  Rectified 600-cell |
See also
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