Rectification (geometry)

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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[edit]

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:

Cube truncation sequence.svg


Higher degree rectifications[edit]

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[edit]

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:

Birectified cube sequence.png

In polygons[edit]

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[edit]

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
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png
[p,q]
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png
[3,3] Uniform polyhedron-33-t0.png
Tetrahedron
Uniform polyhedron-33-t1.png
Octahedron
Uniform polyhedron-33-t2.png
Tetrahedron
[4,3] Uniform polyhedron-43-t0.png
Cube
Uniform polyhedron-43-t1.png
Cuboctahedron
Uniform polyhedron-43-t2.png
Octahedron
[5,3] Uniform polyhedron-53-t0.png
Dodecahedron
Uniform polyhedron-53-t1.png
Icosidodecahedron
Uniform polyhedron-53-t2.png
Icosahedron
[6,3] Uniform tiling 63-t0.png
Hexagonal tiling
Uniform tiling 63-t1.png
Trihexagonal tiling
Uniform tiling 63-t2.png
Triangular tiling
[7,3] Uniform tiling 73-t0.png
Order-3 heptagonal tiling
Uniform tiling 73-t1.png
Triheptagonal tiling
Uniform tiling 73-t2.png
Order-7 triangular tiling
[4,4] Uniform tiling 44-t0.png
Square tiling
Uniform tiling 44-t1.png
Square tiling
Uniform tiling 44-t2.png
Square tiling
[5,4] Uniform tiling 54-t0.png
Order-4 pentagonal tiling
Uniform tiling 54-t1.png
tetrapentagonal tiling
Uniform tiling 54-t2.png
Order-5 square tiling

In nonregular polyhedra[edit]

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 polychora and 3d honeycomb tessellations[edit]

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)
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png
[p,q,r]
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.png CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.png
[3,3,3] Schlegel wireframe 5-cell.png
5-cell
Schlegel half-solid rectified 5-cell.png
rectified 5-cell
Schlegel half-solid rectified 5-cell.png
rectified 5-cell
Schlegel wireframe 5-cell.png
5-cell
[4,3,3] Schlegel wireframe 8-cell.png
tesseract
Schlegel half-solid rectified 8-cell.png
rectified tesseract
Schlegel half-solid rectified 16-cell.png
Rectified 16-cell
(24-cell)
Schlegel wireframe 16-cell.png
16-cell
[3,4,3] Schlegel wireframe 24-cell.png
24-cell
Schlegel half-solid cantellated 16-cell.png
rectified 24-cell
Schlegel half-solid cantellated 16-cell.png
rectified 24-cell
Schlegel wireframe 24-cell.png
24-cell
[5,3,3] Schlegel wireframe 120-cell.png
120-cell
Rectified 120-cell schlegel halfsolid.png
rectified 120-cell
Rectified 600-cell schlegel halfsolid.png
rectified 600-cell
Schlegel wireframe 600-cell vertex-centered.png
600-cell
[4,3,4] Partial cubic honeycomb.png
Cubic honeycomb
Rectified cubic honeycomb.jpg
Rectified cubic honeycomb
Rectified cubic honeycomb.jpg
Rectified cubic honeycomb
Partial cubic honeycomb.png
Cubic honeycomb
[5,3,4] Hyperbolic orthogonal dodecahedral honeycomb.png
Order-4 dodecahedral
Rectified order 4 dodecahedral honeycomb.png
Rectified order-4 dodecahedral
H3 435 CC center 0100.png
Rectified order-5 cubic
Hyperb gcubic hc.png
Order-5 cubic

Degrees of rectification[edit]

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[edit]

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[edit]

Facets are edges, represented as {2}.

name
{p}
Coxeter-Dynkin t-notation
Schläfli symbol
Vertical Schläfli symbol
Name Facet-1 Facet-2
Parent CDel node 1.pngCDel p.pngCDel node.png t0{p} {p} {2}
Rectified CDel node.pngCDel p.pngCDel node 1.png t1{p} {p} {2}

Regular polyhedra and tilings[edit]

Facets are regular polygons.

name
{p,q}
Coxeter-Dynkin t-notation
Schläfli symbol
Vertical Schläfli symbol
Name Facet-1 Facet-2
Parent CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png t0{p,q} {p,q} {p}
Rectified CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png t1{p,q} \begin{Bmatrix} p \\ q  \end{Bmatrix} = r{p,q} {p} {q}
Birectified CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png t2{p,q} {q,p} {q}

Regular polychora and honeycombs[edit]

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 CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png t0{p,q,r} {p,q,r} {p,q}
Rectified CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png 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)
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.png 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)
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.png t3{p,q,r} {r,q,p} {r,q}

Regular polyterons and 4-space honeycombs[edit]

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 CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngCDel s.pngCDel node.png t0{p,q,r,s} {p,q,r,s} {p,q,r}
Rectified CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngCDel s.pngCDel node.png 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)
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.pngCDel s.pngCDel node.png 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)
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.pngCDel s.pngCDel node.png 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)
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngCDel s.pngCDel node 1.png t4{p,q,r,s} {s,r,q,p} {s,r,q}

See also[edit]

References[edit]

External links[edit]

Seed
CDel node 1.pngCDel p.pngCDel node n1.pngCDel q.pngCDel node n2.png
Truncation
CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png
Rectification
CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png
Bitruncation
CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png
Dual
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png
Expansion
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png
Omnitruncation
CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png
Snub
CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png
Uniform polyhedron-43-t0.png Uniform polyhedron-43-t01.png Uniform polyhedron-43-t1.png Uniform polyhedron-43-t12.png Uniform polyhedron-43-t2.png Uniform polyhedron-43-t02.png Uniform polyhedron-43-t012.png Uniform polyhedron-43-s012.png
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}
ht012{p,q}
sr{p,q}