InEuclidean geometry,rectification, also known ascritical truncation orcomplete-truncation, is the process of truncating apolytope by marking the midpoints of all itsedges, and cutting off itsvertices at those points.[1] The resulting polytope will be bounded byvertex figure facets and the rectified facets of the original polytope.
A rectification operator is sometimes denoted by the letterr with aSchläfli symbol. For example,r{4,3} is the rectifiedcube, also called acuboctahedron, and also represented as. And a rectified cuboctahedronrr{4,3} is arhombicuboctahedron, and also represented as.
Conway polyhedron notation usesa forambo as this operator. Ingraph theory this operation creates amedial graph.
The rectification of any regularself-dual polyhedron or tiling will result in another regular polyhedron or tiling with atiling order of 4, for example thetetrahedron{3,3} becoming anoctahedron{3,4}. As a special case, asquare 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 thedual 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 abirectified 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.
Eachplatonic solid and itsdual 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. For this reason, its name is a combination of the names of the original and the dual:
Examples
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 apolyhedral graph as its1-skeleton, and from that graph one may form themedial 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 bySteinitz's theorem it can be represented as a polyhedron.
TheConway polyhedron notation equivalent to rectification isambo, represented bya. Applying twiceaa, (rectifying a rectification) is Conway'sexpand operation,e, which is the same as Johnson'scantellation operation, t0,2 generated from regular polyhedral and tilings.
EachConvex regular 4-polytope has a rectified form as auniform 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, calledbitruncation, is symmetric between a 4-polytope and its dual. SeeUniform 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 isregular, this form is represented by an extendedSchläfli symbol notationt1{p,q,...} orr{p,q,...}.
A second rectification, orbirectification, truncatesfaces down to points. If regular it has notationt2{p,q,...} or 2r{p,q,...}. Forpolyhedra, a birectification creates adual polyhedron.
Higher degree rectifications can be constructed for higher dimensional polytopes. In general an n-rectification truncatesn-faces to points.
If an n-polytope is (n-1)-rectified, itsfacets are reduced to points and the polytope becomes itsdual.
There are different equivalent notations for each degree of rectification. These tables show the names by dimension and the two type offacets for each.
Facets are edges, represented as {}.
| name {p} | Coxeter diagram | t-notation Schläfli symbol | VerticalSchläfli symbol | ||
|---|---|---|---|---|---|
| Name | Facet-1 | Facet-2 | |||
| Parent | | t0{p} | {p} | {} | |
| Rectified | | t1{p} | {p} | {} | |
Facets are regular polygons.
| name {p,q} | Coxeter diagram | t-notation Schläfli symbol | VerticalSchläfli symbol | ||
|---|---|---|---|---|---|
| Name | Facet-1 | Facet-2 | |||
| Parent | =  | t0{p,q} | {p,q} | {p} | |
| Rectified | = | t1{p,q} | r{p,q} = | {p} | {q} |
| Birectified | = | t2{p,q} | {q,p} | {q} | |
Facets are regular or rectified polyhedra.
| name {p,q,r} | Coxeter diagram | t-notation Schläfli symbol | ExtendedSchläfli symbol | ||
|---|---|---|---|---|---|
| Name | Facet-1 | Facet-2 | |||
| Parent | | t0{p,q,r} | {p,q,r} | {p,q} | |
| Rectified | | t1{p,q,r} | = r{p,q,r} | = r{p,q} | {q,r} |
| Birectified (Dual rectified) | | t2{p,q,r} | = r{r,q,p} | {q,r} | = r{q,r} |
| Trirectified (Dual) | | t3{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 | ExtendedSchläfli symbol | ||
|---|---|---|---|---|---|
| Name | Facet-1 | Facet-2 | |||
| Parent |  | t0{p,q,r,s} | {p,q,r,s} | {p,q,r} | |
| Rectified | | t1{p,q,r,s} | = r{p,q,r,s} | = r{p,q,r} | {q,r,s} |
| Birectified (Birectified dual) | | t2{p,q,r,s} | = 2r{p,q,r,s} | = r{r,q,p} | = r{q,r,s} |
| Trirectified (Rectified dual) | | t3{p,q,r,s} | = r{s,r,q,p} | {r,q,p} | = r{s,r,q} |
| Quadrirectified (Dual) | | t4{p,q,r,s} | {s,r,q,p} | {s,r,q} | |
| Seed | Truncation | Rectification | Bitruncation | Dual | Expansion | Omnitruncation | Alternations | ||
|---|---|---|---|---|---|---|---|---|---|
  | | | | | | |  | | |
 |  |  |  |  |  |  |  |  |  |
| t0{p,q} {p,q} | t01{p,q} t{p,q} | t1{p,q} r{p,q} | t12{p,q} 2t{p,q} | t2{p,q} 2r{p,q} | t02{p,q} rr{p,q} | t012{p,q} tr{p,q} | ht0{p,q} h{q,p} | ht12{p,q} s{q,p} | ht012{p,q} sr{p,q} |
