Rectification (geometry)

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.

Example of rectification as a final truncation to an edge

Rectification is the final point of a truncation process. For example on a cube this sequence shows four steps of a continuum of truncations between the regular and rectified form:

Higher order rectification can be performed on higher dimensional regular polytopes. The highest order of rectification creates the dual polytope. A rectification truncates edges to points. A birectification truncates faces to points. A trirectification truncates cells to points.

Example of birectification as a final truncation to a face

This sequence shows a "birectified cube" as the final sequence from a cube to the dual where the original faces are truncated down to a single point::

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.)

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.
# A rectified square tiling is a square tiling.
# A rectified triangular tiling or hexagonal tiling is a trihexagonal tiling.

Examples

Regular polyhedra and tilings

Facets are regular polygons.

ee also

* Dual polytope
* Quasiregular polyhedron
* List of regular polytopes
* Truncation (geometry)
* Conway polyhedron notation

References

* Coxeter, H.S.M. "Regular Polytopes", (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 (pp.145-154 Chapter 8: Truncation)

External links

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