Rectification (geometry)

In Euclidean geometry, rectification aka 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.

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:

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:

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 appropriated scaled concentric version of its dual. For this reason, its name is a combination of the names of the original and the dual:

  1. The rectified tetrahedron, whose dual is the tetrahedron, is the tetratetrahedron, better known as the octahedron.
  2. The rectified octahedron, whose dual is the cube, is the cuboctahedron.
  3. The rectified icosahedron, whose dual is the dodecahedron, is the icosidodecahedron.
  4. A rectified square tiling is a square tiling.
  5. 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

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

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

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 polygons

Facets are edges, represented as {2}.

name
{p}
Coxeter diagram t-notation
Schläfli symbol
Vertical Schläfli symbol
Name Facet-1 Facet-2
Parent t0{p} {p} {2}
Rectified t1{p} {p} {2}

Regular polyhedra and tilings

Facets are regular polygons.

name
{p,q}
Coxeter diagram t-notation
Schläfli symbol
Vertical Schläfli 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}

Regular Uniform 4-polytopes and honeycombs

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 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}

Regular 5-polytopes and 4-space honeycombs

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 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}

See also

References

  • 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
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)
Polyhedron operators
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}
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