Rectification (geometry)
Operation in Euclidean geometry
title: "Rectification (geometry)" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["polytopes", "truncated-tilings"] description: "Operation in Euclidean geometry" topic_path: "general/polytopes" source: "https://en.wikipedia.org/wiki/Rectification_(geometry)" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Operation in Euclidean geometry ::
::figure[src="https://upload.wikimedia.org/wikipedia/commons/a/a6/Cuboctahedron.png" caption="A rectified cube is a [[cuboctahedron]] – edges reduced to vertices, and vertices expanded into new faces"] ::
::figure[src="https://upload.wikimedia.org/wikipedia/commons/e/e7/Dual_Cube-Octahedron.svg" caption="A ''birectified'' cube is an octahedron – faces are reduced to points and new faces are centered on the original vertices."] ::
::figure[src="https://upload.wikimedia.org/wikipedia/commons/b/b9/Rectified_cubic_honeycomb.jpg" caption="A [[rectified cubic honeycomb]] – edges reduced to vertices, and vertices expanded into new cells."] ::
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 \begin{Bmatrix} 4 \ 3 \end{Bmatrix}. And a rectified cuboctahedron rr{4,3} is a rhombicuboctahedron, and also represented as r\begin{Bmatrix} 4 \ 3 \end{Bmatrix}.
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.
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:
::figure[src="https://upload.wikimedia.org/wikipedia/commons/e/ec/Cube_truncation_sequence.svg"] ::
Higher degree rectifications
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.
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: :[[Image:Birectified cube sequence.png|480px]]
In polygons
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.
In polyhedra 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 appropriately scaled concentric version of its dual. For this reason, its name is a combination of the names of the original and the dual:
- The tetrahedron is its own dual, and its rectification is the tetratetrahedron, better known as the octahedron.
- The octahedron and the cube are each other's dual, and their rectification is the cuboctahedron.
- The icosahedron and the dodecahedron are duals, and their rectification is the icosidodecahedron.
Examples ::data[format=table]
| Family | Parent | Rectification | Dual | [p,q] | [3,3] | [4,3] | [5,3] | [6,3] | [7,3] | [4,4] | [5,4] |
|---|---|---|---|---|---|---|---|---|---|---|---|
| [[Image:Uniform polyhedron-33-t0.svg | 75px]]Tetrahedron | [[Image:Uniform polyhedron-33-t1.svg | 75px]]Octahedron | [[Image:Uniform polyhedron-33-t2.svg | 75px]]Tetrahedron | ||||||
| [[Image:Uniform polyhedron-43-t0.svg | 75px]]Cube | [[Image:Uniform polyhedron-43-t1.svg | 75px]]Cuboctahedron | [[Image:Uniform polyhedron-43-t2.svg | 75px]]Octahedron | ||||||
| [[Image:Uniform polyhedron-53-t0.svg | 75px]]Dodecahedron | [[Image:Uniform polyhedron-53-t1.svg | 75px]]Icosidodecahedron | [[Image:Uniform polyhedron-53-t2.svg | 75px]]Icosahedron | ||||||
| [[Image:Uniform tiling 63-t0.svg | 75px]]Hexagonal tiling | [[Image:Uniform tiling 63-t1.svg | 75px]]Trihexagonal tiling | [[Image:Uniform tiling 63-t2.svg | 75px]]Triangular tiling | ||||||
| [[Image:Heptagonal tiling.svg | 75px]]Order-3 heptagonal tiling | [[Image:Triheptagonal tiling.svg | 75px]]Triheptagonal tiling | [[Image:Order-7 triangular tiling.svg | 75px]]Order-7 triangular tiling | ||||||
| [[Image:Uniform tiling 44-t0.svg | 75px]]Square tiling | [[Image:Uniform tiling 44-t1.svg | 75px]]Square tiling | [[Image:Uniform tiling 44-t2.svg | 75px]]Square tiling | ||||||
| [[Image:H2-5-4-dual.svg | 75px]]Order-4 pentagonal tiling | [[Image:H2-5-4-rectified.svg | 75px]]Tetrapentagonal tiling | [[Image:H2-5-4-primal.svg | 75px]]Order-5 square tiling | ||||||
| :: |
In nonregular polyhedra
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, t0,2 generated from regular polyhedral and tilings.
In 4-polytopes and 3D honeycomb tessellations
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 ::data[format=table]
| 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] | [4,3,3] | [3,4,3] | [5,3,3] | [4,3,4] | [5,3,4] |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| [[Image:Schlegel wireframe 5-cell.png | 120px]]5-cell | [[Image:Schlegel half-solid rectified 5-cell.png | 120px]]rectified 5-cell | [[Image:Schlegel half-solid rectified 5-cell.png | 120px]]rectified 5-cell | [[Image:Schlegel wireframe 5-cell.png | 120px]]5-cell | ||||||||
| [[Image:Schlegel wireframe 8-cell.png | 150px]]tesseract | [[Image:Schlegel half-solid rectified 8-cell.png | 150px]]rectified tesseract | [[Image:Schlegel half-solid rectified 16-cell.png | 150px]]Rectified 16-cell(24-cell) | [[Image:Schlegel wireframe 16-cell.png | 150px]]16-cell | ||||||||
| [[Image:Schlegel wireframe 24-cell.png | 150px]]24-cell | [[File:Schlegel half-solid cantellated 16-cell.png | 150px]]rectified 24-cell | [[File:Schlegel half-solid cantellated 16-cell.png | 150px]]rectified 24-cell | [[Image:Schlegel wireframe 24-cell.png | 150px]]24-cell | ||||||||
| [[Image:Schlegel wireframe 120-cell.png | 150px]]120-cell | [[File:Rectified 120-cell schlegel halfsolid.png | 150px]]rectified 120-cell | [[File:Rectified_600-cell_schlegel_halfsolid.png | 150px]]rectified 600-cell | [[Image:Schlegel wireframe 600-cell vertex-centered.png | 150px]]600-cell | ||||||||
| [[Image:Partial cubic honeycomb.png | 150px]]Cubic honeycomb | [[Image:Rectified cubic honeycomb.jpg | 150px]]Rectified cubic honeycomb | [[Image:Rectified cubic honeycomb.jpg | 150px]]Rectified cubic honeycomb | [[Image:Partial cubic honeycomb.png | 150px]]Cubic honeycomb | ||||||||
| [[Image:Hyperbolic_orthogonal_dodecahedral_honeycomb.png | 150px]]Order-4 dodecahedral | [[File:Rectified_order_4_dodecahedral_honeycomb.png | 150px]]Rectified order-4 dodecahedral | [[File:H3 435 CC center 0100.png | 150px]]Rectified order-5 cubic | [[Image:Hyperb gcubic hc.png | 150px]]Order-5 cubic | ||||||||
| :: |
Degrees of rectification
A first rectification truncates edges down to points. If a polytope is regular, this form is represented by an extended Schläfli symbol notation t1{p,q,...} or r{p,q,...}.
A second rectification, or birectification, truncates faces down to points. If regular it has notation t2{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.
Notations and facets
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.
Regular [[polygon]]s
Facets are edges, represented as {}.
::data[format=table]
| name{p} | Coxeter diagram | t-notationSchläfli symbol | Vertical Schläfli symbol | Name | Facet-1 | Facet-2 |
|---|---|---|---|---|---|---|
| Parent | t0{p} | {p} | {} | |||
| Rectified | t1{p} | {p} | {} | |||
| :: |
Regular [[Uniform polyhedron|polyhedra]] and [[List of uniform tilings|tiling]]s
Facets are regular polygons. ::data[format=table]
| name{p,q} | Coxeter diagram | t-notationSchläfli symbol | Vertical Schläfli symbol | Name | Facet-1 | Facet-2 |
|---|---|---|---|---|---|---|
| Parent | = {{CDD | node | split1-pq | nodes_10lu}} | t0{p,q} | {p,q} |
| Rectified | = {{CDD | node_1 | split1-pq | nodes}} | t1{p,q} | r{p,q} = \begin{Bmatrix} p \ q \end{Bmatrix} |
| Birectified | = {{CDD | node | split1-pq | nodes_01ld}} | t2{p,q} | {q,p} |
| :: |
Regular [[Uniform 4-polytope]]s and [[honeycomb]]s
Facets are regular or rectified polyhedra. ::data[format=table]
| name{p,q,r} | Coxeter diagram | t-notationSchläfli symbol | Extended Schläfli symbol | Name | Facet-1 | Facet-2 |
|---|---|---|---|---|---|---|
| Parent | t0{p,q,r} | {p,q,r} | {p,q} | |||
| Rectified | t1{p,q,r} | \begin{Bmatrix} p \ \ \ q , r \end{Bmatrix} = r{p,q,r} | \begin{Bmatrix} p \ q \end{Bmatrix} = r{p,q} | {q,r} | ||
| Birectified(Dual rectified) | t2{p,q,r} | \begin{Bmatrix} q , p \ r \ \ \end{Bmatrix} = r{r,q,p} | {q,r} | \begin{Bmatrix} q \ r \end{Bmatrix} = r{q,r} | ||
| Trirectified(Dual) | t3{p,q,r} | {r,q,p} | {r,q} | |||
| :: |
Regular [[5-polytope]]s and 4-space [[Honeycomb (geometry)|honeycombs]]
Facets are regular or rectified 4-polytopes. ::data[format=table]
| name{p,q,r,s} | Coxeter diagram | t-notationSchläfli symbol | Extended Schläfli 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} | \begin{Bmatrix} p \ \ \ \ \ \ q , r , s \end{Bmatrix} = r{p,q,r,s} | \begin{Bmatrix} p \ \ \ q , r \end{Bmatrix} = r{p,q,r} | {q,r,s} | ||
| Birectified(Birectified dual) | t2{p,q,r,s} | \begin{Bmatrix} q , p \ r , s \end{Bmatrix} = 2r{p,q,r,s} | \begin{Bmatrix} q , p \ r \ \ \end{Bmatrix} = r{r,q,p} | \begin{Bmatrix} q \ \ \ r , s \end{Bmatrix} = r{q,r,s} | ||
| Trirectified(Rectified dual) | t3{p,q,r,s} | \begin{Bmatrix} r , q , p \ s \ \ \ \ \ \end{Bmatrix} = r{s,r,q,p} | {r,q,p} | \begin{Bmatrix} r , q \ s \ \ \end{Bmatrix} = r{s,r,q} | ||
| Quadrirectified(Dual) | t4{p,q,r,s} | {s,r,q,p} | {s,r,q} | |||
| :: |
References
- Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, (pp. 145–154 Chapter 8: Truncation)
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, (Chapter 26)
References
- "Rectification".
::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::