Rectification (geometry)

In Euclidean geometry, rectification 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 the vertex figures and the rectified facets of the original polytope.

1 Example of rectification as a final truncation to an edge
2 Higher-order rectifications
3 Example of birectification as a final truncation to a face
4 In polygons
5 In polyhedra and plane tilings
- 5.1 In nonregular polyhedra
6 In polychora and 3d honeycomb tessellations
7 Orders of rectification
- 7.1 Notations and facets
8 See also
9 References
10 External links

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-order rectifications

Higher-order rectification can be performed on higher-dimensional regular polytopes. The highest order of rectification creates the dual polytope. A rectification truncates edges to points. A birectification truncates faces to points. A trirectification truncates cells to points. ... The final rectification is the dual polytope.

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.

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:

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
[3,3]	Tetrahedron	Tetratetrahedron	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.

In polychora and 3d honeycomb tessellations

Each convex regular polychoron has a rectified form as a uniform polychoron.

A regular polychoron {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 polychoron and its dual. See Uniform_polychoron#Geometric_derivations.

Examples

Family	Parent	Rectification	Birectification (Dual rectification)	Trirectification (Dual)
[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	(No image) Rectified order-4 dodecahedral	(No image) Rectified order-5 cubic	Order-5 cubic

Orders of rectification

A first order rectification truncates edges down to points. If a polytope is regular, this form is represented by an extended Schläfli symbol notation t₁{p,q,...}.

A second order rectification, or birectification, truncates faces down to points. If regular it has notation t₂{p,q,...}. For polyhedra, a birectification creates a dual polyhedron.

Higher order rectifications can be constructed for higher order 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 order 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-Dynkin	t-notation Schläfli symbol	Vertical Schläfli symbol
name {p}	Coxeter-Dynkin	t-notation Schläfli symbol	Name	Facet-1	Facet-2
Parent		t₀{p}	$\begin{Bmatrix} p \end{Bmatrix}$	$\begin{Bmatrix} 2 \end{Bmatrix}$
Rectified		t₁{p}	$\begin{Bmatrix} p \end{Bmatrix}$		$\begin{Bmatrix} 2 \end{Bmatrix}$

Regular polyhedra and tilings

Facets are regular polygons.

name {p,q}	Coxeter-Dynkin	t-notation Schläfli symbol	Vertical Schläfli symbol
name {p,q}	Coxeter-Dynkin	t-notation Schläfli symbol	Name	Facet-1	Facet-2
Parent		t₀{p,q}	$\begin{Bmatrix} p , q \end{Bmatrix}$	$\begin{Bmatrix} p \end{Bmatrix}$
Rectified		t₁{p,q}	$\begin{Bmatrix} p \\ q \end{Bmatrix}$	$\begin{Bmatrix} p \end{Bmatrix}$	$\begin{Bmatrix} q \end{Bmatrix}$
Birectified		t₂{p,q}	$\begin{Bmatrix} q , p \end{Bmatrix}$		$\begin{Bmatrix} q \end{Bmatrix}$

Regular polychora and honeycombs

Facets are regular or rectified polyhedra.

name {p,q,r}	Coxeter-Dynkin	t-notation Schläfli symbol	Vertical Schläfli symbol
name {p,q,r}	Coxeter-Dynkin	t-notation Schläfli symbol	Name	Facet-1	Facet-2
Parent		t₀{p,q,r}	$\begin{Bmatrix} p , q , r \end{Bmatrix}$	$\begin{Bmatrix} p , q \end{Bmatrix}$
Rectified		t₁{p,q,r}	$\begin{Bmatrix} p \ \ \\ q , r \end{Bmatrix}$	$\begin{Bmatrix} p \\ q \end{Bmatrix}$	$\begin{Bmatrix} q , r \end{Bmatrix}$
Birectified		t₂{p,q,r}	$\begin{Bmatrix} q , p \\ r \ \ \end{Bmatrix}$	$\begin{Bmatrix} q , p \end{Bmatrix}$	$\begin{Bmatrix} q \\ r \end{Bmatrix}$
Trirectified		t₃{p,q,r}	$\begin{Bmatrix} r, q , p \end{Bmatrix}$		$\begin{Bmatrix} r , q \end{Bmatrix}$

Regular polyterons and 4-space honeycombs

Facets are regular or rectified polychora.

name {p,q,r,s}	Coxeter-Dynkin	t-notation Schläfli symbol	Vertical Schläfli symbol
name {p,q,r,s}	Coxeter-Dynkin	t-notation Schläfli symbol	Name	Facet-1	Facet-2
Parent		t₀{p,q,r,s}	$\begin{Bmatrix} p , q , r , s \end{Bmatrix}$	$\begin{Bmatrix} p , q , r \end{Bmatrix}$
Rectified		t₁{p,q,r,s}	$\begin{Bmatrix} p \ \ \ \ \ \\ q , r , s \end{Bmatrix}$	$\begin{Bmatrix} p \ \ \\ q , r \end{Bmatrix}$	$\begin{Bmatrix} q , r , s \end{Bmatrix}$
Birectified		t₂{p,q,r,s}	$\begin{Bmatrix} q , p \\ r , s \end{Bmatrix}$	$\begin{Bmatrix} q , p \\ r \ \ \end{Bmatrix}$	$\begin{Bmatrix} q \ \ \\ r , s \end{Bmatrix}$
Trirectified		t₃{p,q,r,s}	$\begin{Bmatrix} r , q , p \\ s \ \ \ \ \ \end{Bmatrix}$	$\begin{Bmatrix} r , q , p \end{Bmatrix}$	$\begin{Bmatrix} r , q \\ s \ \ \end{Bmatrix}$
Quadrirectified		t₄{p,q,r,s}	$\begin{Bmatrix} s, r, q , p \end{Bmatrix}$		$\begin{Bmatrix} s , r , q \end{Bmatrix}$

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

External links

Weisstein, Eric W., "Rectification" from MathWorld.
Olshevsky, George, Rectification at Glossary for Hyperspace.