Neutron cross section

A rectified cube is a cuboctahedron – edges reduced to vertices, and vertices expanded into new faces

A birectified cube is an octahedron – faces are reduced to points and new faces are centered on the original vertices.

A rectified cubic honeycomb – edges reduced to vertices, and vertices expanded into new cells.

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.

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

Template:See 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
Template:CDD [p,q]	Template:CDD	Template:CDD	Template:CDD
[3,3]	File:Uniform polyhedron-33-t0.png Tetrahedron	File:Uniform polyhedron-33-t1.png Octahedron	File:Uniform polyhedron-33-t2.png Tetrahedron
[4,3]	File:Uniform polyhedron-43-t0.png Cube	File:Uniform polyhedron-43-t1.png Cuboctahedron	File:Uniform polyhedron-43-t2.png Octahedron
[5,3]	File:Uniform polyhedron-53-t0.png Dodecahedron	File:Uniform polyhedron-53-t1.png Icosidodecahedron	File:Uniform polyhedron-53-t2.png Icosahedron
[6,3]	File:Uniform tiling 63-t0.png Hexagonal tiling	File:Uniform tiling 63-t1.png Trihexagonal tiling	File:Uniform tiling 63-t2.png Triangular tiling
[7,3]	File:Uniform tiling 73-t0.png Order-3 heptagonal tiling	File:Uniform tiling 73-t1.png Triheptagonal tiling	File:Uniform tiling 73-t2.png Order-7 triangular tiling
[4,4]	File:Uniform tiling 44-t0.png Square tiling	File:Uniform tiling 44-t1.png Square tiling	File:Uniform tiling 44-t2.png Square tiling
[5,4]	File:Uniform tiling 54-t0.png Order-4 pentagonal tiling	File:Uniform tiling 54-t1.png tetrapentagonal tiling	File:Uniform tiling 54-t2.png 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, t_0,2 generated from regular polyhedral and tilings.

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)
Template:CDD [p,q,r]	Template:CDD	Template:CDD	Template:CDD	Template:CDD
[3,3,3]	File:Schlegel wireframe 5-cell.png 5-cell	File:Schlegel half-solid rectified 5-cell.png rectified 5-cell	File:Schlegel half-solid rectified 5-cell.png rectified 5-cell	File:Schlegel wireframe 5-cell.png 5-cell
[4,3,3]	File:Schlegel wireframe 8-cell.png tesseract	File:Schlegel half-solid rectified 8-cell.png rectified tesseract	File:Schlegel half-solid rectified 16-cell.png Rectified 16-cell (24-cell)	File:Schlegel wireframe 16-cell.png 16-cell
[3,4,3]	File:Schlegel wireframe 24-cell.png 24-cell	File:Schlegel half-solid cantellated 16-cell.png rectified 24-cell	File:Schlegel half-solid cantellated 16-cell.png rectified 24-cell	File:Schlegel wireframe 24-cell.png 24-cell
[5,3,3]	File:Schlegel wireframe 120-cell.png 120-cell	File:Rectified 120-cell schlegel halfsolid.png rectified 120-cell	File:Rectified 600-cell schlegel halfsolid.png rectified 600-cell	File:Schlegel wireframe 600-cell vertex-centered.png 600-cell
[4,3,4]	File:Partial cubic honeycomb.png Cubic honeycomb	Error creating thumbnail: Rectified cubic honeycomb	Error creating thumbnail: Rectified cubic honeycomb	File:Partial cubic honeycomb.png Cubic honeycomb
[5,3,4]	File:Hyperbolic orthogonal dodecahedral honeycomb.png Order-4 dodecahedral	File:Rectified order 4 dodecahedral honeycomb.png Rectified order-4 dodecahedral	(No image) Rectified order-5 cubic	File:Hyperb gcubic hc.png 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,...}.

A second 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 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-Dynkin	t-notation Schläfli symbol	Vertical Schläfli symbol
			Name	Facet-1	Facet-2
Parent	Template:CDD	t0{p}	{p}	{2}
Rectified	Template:CDD	t1{p}	{p}		{2}

Regular polyhedra and tilings

Facets are regular polygons.

name {p,q}	Coxeter-Dynkin	t-notation Schläfli symbol	Vertical Schläfli symbol
			Name	Facet-1	Facet-2
Parent	Template:CDD	t0{p,q}	{p,q}	{p}
Rectified	Template:CDD	t1{p,q}	$\left\{\begin{array}{c}p\\ q\end{array}\right\}$ = r{p,q}	{p}	{q}
Birectified	Template:CDD	t2{p,q}	{q,p}		{q}

Regular polychora and honeycombs

Facets are regular or rectified polyhedra.

name {p,q,r}	Coxeter-Dynkin	t-notation Schläfli symbol	Extended Schläfli symbol
			Name	Facet-1	Facet-2
Parent	Template:CDD	t0{p,q,r}	{p,q,r}	{p,q}
Rectified	Template:CDD	t1{p,q,r}	$\left\{\begin{array}{c}p\\ q,r\end{array}\right\}$ = r{p,q,r}	$\left\{\begin{array}{c}p\\ q\end{array}\right\}$ = r{p,q}	{q,r}
Birectified (Dual rectified)	Template:CDD	t2{p,q,r}	$\left\{\begin{array}{c}q,p\\ r\end{array}\right\}$ = r{r,q,p}	{q,r}	$\left\{\begin{array}{c}q\\ r\end{array}\right\}$ = r{q,r}
Trirectified (Dual)	Template:CDD	t3{p,q,r}	{r,q,p}		{r,q}

Regular polyterons and 4-space honeycombs

Facets are regular or rectified polychora.

name {p,q,r,s}	Coxeter-Dynkin	t-notation Schläfli symbol	Extended Schläfli symbol
			Name	Facet-1	Facet-2
Parent	Template:CDD	t0{p,q,r,s}	{p,q,r,s}	{p,q,r}
Rectified	Template:CDD	t1{p,q,r,s}	$\left\{\begin{array}{c}p\\ q,r,s\end{array}\right\}$ = r{p,q,r,s}	$\left\{\begin{array}{c}p\\ q,r\end{array}\right\}$ = r{p,q,r}	{q,r,s}
Birectified (Birectified dual)	Template:CDD	t2{p,q,r,s}	$\left\{\begin{array}{c}q,p\\ r,s\end{array}\right\}$ = 2r{p,q,r,s}	$\left\{\begin{array}{c}q,p\\ r\end{array}\right\}$ = r{r,q,p}	$\left\{\begin{array}{c}q\\ r,s\end{array}\right\}$ = r{q,r,s}
Trirectified (Rectified dual)	Template:CDD	t3{p,q,r,s}	$\left\{\begin{array}{c}r,q,p\\ s\end{array}\right\}$ = r{s,r,q,p}	{r,q,p}	$\left\{\begin{array}{c}r,q\\ s\end{array}\right\}$ = r{s,r,q}
Quadrirectified (Dual)	Template:CDD	t4{p,q,r,s}	{s,r,q,p}		{s,r,q}

See also

• Dual polytope
• Quasiregular polyhedron
• List of regular polytopes
• Truncation (geometry)
• Conway polyhedron notation

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

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