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Concept
Rectification (geometry)
Résumé
In Euclidean geometry, rectification, also known as critical truncation or complete-truncation, 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 vertex figure facets and the rectified facets of the original polytope.
A rectification operator is sometimes denoted by the letter r with a Schläfli symbol. For example, r{4,3} is the rectified cube, also called a cuboctahedron, and also represented as . And a rectified cuboctahedron rr{4,3} is a rhombicuboctahedron, and also represented as .
Conway polyhedron notation uses a for ambo as this operator. In graph theory this operation creates a medial graph.
The rectification of any regular self-dual polyhedron or tiling will result in another regular polyhedron or tiling with a tiling order of 4, for example the tetrahedron {3,3} becoming an octahedron {3,4}. As a special case, a square tiling {4,4} will turn into another square tiling {4,4} under a rectification operation.
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 degree rectification can be performed on higher-dimensional regular polytopes. The highest degree of rectification creates the dual polytope. A rectification truncates edges to points. A birectification truncates faces to points. A trirectification truncates cells to points, and so on.
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:
The dual of a polygon is the same as its rectified form. New vertices are placed at the center of the edges of the original polygon.
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 appropriately scaled concentric version of its dual.
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