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.

- Example of rectification as a final truncation to an edge
- Higher degree rectifications
- Example of birectification as a final truncation to a face
- In polygons
- In polyhedra and plane tilings
- In nonregular polyhedra
- In 4-polytopes and 3D honeycomb tessellations
- Degrees of rectification
- Notations and facets
- See also
- References
- External links

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:

- 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

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 2**r**{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} |

A **cuboctahedron** is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral.

In geometry, a **cube** is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

In geometry, an **octahedron** is a polyhedron with eight faces. The term is most commonly used to refer to the **regular** octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

In geometry, a **polyhedral compound** is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.

In geometry, the **truncated cube**, or **truncated hexahedron**, is an Archimedean solid. It has 14 regular faces, 36 edges, and 24 vertices.

In geometry, a **vertex figure**, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.

The **tetrahedral-octahedral honeycomb**, **alternated cubic honeycomb** is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.

In geometry, a **truncation** is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids.

In geometry, an **alternation** or *partial truncation*, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.

In geometry, **Conway polyhedron notation**, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations.

In geometry, a **cantellation** is a 2nd-order truncation in any dimension that bevels a regular polytope at its edges and at its vertices, creating a new facet in place of each edge and of each vertex. Cantellation also applies to regular tilings and honeycombs. Cantellating is also rectifying its rectification.

In geometry, **expansion** is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements. Equivalently this operation can be imagined by keeping facets in the same position but reducing their size.

In geometry, a **uniform polytope** of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.

In geometry, a **quasiregular polyhedron** is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.

In geometry, a d-dimensional **simple polytope** is a d-dimensional polytope each of whose vertices are adjacent to exactly d edges. The vertex figure of a simple d-polytope is a (*d* – 1)-simplex.

In geometry, a **snub** is an operation applied to a polyhedron. The term originates from Kepler's names of two Archimedean solids, for the snub cube and snub dodecahedron. In general, snubs have chiral symmetry with two forms: with clockwise or counterclockwise orientation. By Kepler's names, a snub can be seen as an expansion of a regular polyhedron: moving the faces apart, twisting them about their centers, adding new polygons centered on the original vertices, and adding pairs of triangles fitting between the original edges.

In the field of hyperbolic geometry, the **order-4 hexagonal tiling honeycomb** arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is *paracompact* because it has cells composed of an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.

In the geometry of hyperbolic 3-space, the **square tiling honeycomb** is one of 11 paracompact regular honeycombs. It is called *paracompact* because it has infinite cells, whose vertices exist on horospheres and converge to a single ideal point at infinity. Given by Schläfli symbol {4,4,3}, it has three square tilings, {4,4}, around each edge, and six square tilings around each vertex, in a cubic {4,3} vertex figure.

The **truncated rhombicuboctahedron** is a polyhedron, constructed as a truncation of the rhombicuboctahedron. It has 50 faces consisting of 18 octagons, 8 hexagons, and 24 squares. It can fill space with the truncated cube, truncated tetrahedron and triangular prism as a truncated runcic cubic honeycomb.

- Coxeter, H.S.M.
*Regular Polytopes*, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 (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

- N.W. Johnson:
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss,
*The Symmetries of Things*2008, ISBN 978-1-56881-220-5 (Chapter 26)

- Olshevsky, George. "Rectification".
*Glossary for Hyperspace*. Archived from the original on 4 February 2007.

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} |

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