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.^{[1]} 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 appropriated scaled concentric version of its dual. For this reason, its name is a combination of the names of the original and the dual:
Examples
Family | Parent | Rectification | Dual |
---|---|---|---|
[p,q] |
|||
[3,3] | Tetrahedron |
Octahedron |
Tetrahedron |
[4,3] | Cube |
Cuboctahedron |
Octahedron |
[5,3] | Dodecahedron |
Icosidodecahedron |
Icosahedron |
[6,3] | Hexagonal tiling |
Trihexagonal tiling |
Triangular tiling |
[7,3] | Order-3 heptagonal tiling |
Triheptagonal tiling |
Order-7 triangular tiling |
[4,4] | Square tiling |
Square tiling |
Square tiling |
[5,4] | Order-4 pentagonal tiling |
Tetrapentagonal tiling |
Order-5 square tiling |
If a polyhedron is not regular, the edge midpoints surrounding a vertex may not be coplanar. However, a form of rectification is still possible in this case: every polyhedron has a polyhedral graph as its 1-skeleton, and from that graph one may form the medial graph by placing a vertex at each edge midpoint of the original graph, and connecting two of these new vertices by an edge whenever they belong to consecutive edges along a common face. The resulting medial graph remains polyhedral, so by Steinitz's theorem it can be represented as a polyhedron.
The Conway polyhedron notation equivalent to rectification is ambo, represented by a. Applying twice aa, (rectifying a rectification) is Conway's expand operation, e, which is the same as Johnson's cantellation operation, t_{0,2} generated from regular polyhedral and tilings.
Each Convex regular 4-polytope has a rectified form as a uniform 4-polytope.
A regular 4-polytope {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 4-polytope and its dual. See Uniform 4-polytope#Geometric derivations.
Examples
Family | Parent | Rectification | Birectification (Dual rectification) |
Trirectification (Dual) |
---|---|---|---|---|
[p,q,r] |
{p,q,r} |
r{p,q,r} |
2r{p,q,r} |
3r{p,q,r} |
[3,3,3] | 5-cell |
rectified 5-cell |
rectified 5-cell |
5-cell |
[4,3,3] | tesseract |
rectified tesseract |
Rectified 16-cell (24-cell) |
16-cell |
[3,4,3] | 24-cell |
rectified 24-cell |
rectified 24-cell |
24-cell |
[5,3,3] | 120-cell |
rectified 120-cell |
rectified 600-cell |
600-cell |
[4,3,4] | Cubic honeycomb |
Rectified cubic honeycomb |
Rectified cubic honeycomb |
Cubic honeycomb |
[5,3,4] | Order-4 dodecahedral |
Rectified order-4 dodecahedral |
Rectified order-5 cubic |
Order-5 cubic |
A first rectification truncates edges down to points. If a polytope is regular, this form is represented by an extended Schläfli symbol notation t_{1}{p,q,...} or r{p,q,...}.
A second rectification, or birectification, truncates faces down to points. If regular it has notation t_{2}{p,q,...} or 2r{p,q,...}. For polyhedra, a birectification creates a dual polyhedron.
Higher degree rectifications can be constructed for higher dimensional 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.
There are different equivalent notations for each degree of rectification. These tables show the names by dimension and the two type of facets for each.
Facets are edges, represented as {2}.
name {p} |
Coxeter diagram | t-notation Schläfli symbol |
Vertical Schläfli symbol | ||
---|---|---|---|---|---|
Name | Facet-1 | Facet-2 | |||
Parent | t_{0}{p} | {p} | {2} | ||
Rectified | t_{1}{p} | {p} | {2} |
Facets are regular polygons.
name {p,q} |
Coxeter diagram | t-notation Schläfli symbol |
Vertical Schläfli symbol | ||
---|---|---|---|---|---|
Name | Facet-1 | Facet-2 | |||
Parent | = | t_{0}{p,q} | {p,q} | {p} | |
Rectified | = | t_{1}{p,q} | r{p,q} = | {p} | {q} |
Birectified | = | t_{2}{p,q} | {q,p} | {q} |
Facets are regular or rectified polyhedra.
name {p,q,r} |
Coxeter diagram | t-notation Schläfli symbol |
Extended Schläfli symbol | ||
---|---|---|---|---|---|
Name | Facet-1 | Facet-2 | |||
Parent | t_{0}{p,q,r} | {p,q,r} | {p,q} | ||
Rectified | t_{1}{p,q,r} | = r{p,q,r} | = r{p,q} | {q,r} | |
Birectified (Dual rectified) |
t_{2}{p,q,r} | = r{r,q,p} | {q,r} | = r{q,r} | |
Trirectified (Dual) |
t_{3}{p,q,r} | {r,q,p} | {r,q} |
Facets are regular or rectified 4-polytopes.
name {p,q,r,s} |
Coxeter diagram | t-notation Schläfli symbol |
Extended Schläfli symbol | ||
---|---|---|---|---|---|
Name | Facet-1 | Facet-2 | |||
Parent | t_{0}{p,q,r,s} | {p,q,r,s} | {p,q,r} | ||
Rectified | t_{1}{p,q,r,s} | = r{p,q,r,s} | = r{p,q,r} | {q,r,s} | |
Birectified (Birectified dual) |
t_{2}{p,q,r,s} | = 2r{p,q,r,s} | = r{r,q,p} | = r{q,r,s} | |
Trirectified (Rectified dual) |
t_{3}{p,q,r,s} | = r{s,r,q,p} | {r,q,p} | = r{s,r,q} | |
Quadrirectified (Dual) |
t_{4}{p,q,r,s} | {s,r,q,p} | {s,r,q} |
Seed | Truncation | Rectification | Bitruncation | Dual | Expansion | Omnitruncation | Alternations | ||
---|---|---|---|---|---|---|---|---|---|
t_{0}{p,q} {p,q} |
t_{01}{p,q} t{p,q} |
t_{1}{p,q} r{p,q} |
t_{12}{p,q} 2t{p,q} |
t_{2}{p,q} 2r{p,q} |
t_{02}{p,q} rr{p,q} |
t_{012}{p,q} tr{p,q} |
ht_{0}{p,q} h{q,p} |
ht_{12}{p,q} s{q,p} |
ht_{012}{p,q} sr{p,q} |