Rectification (geometry)

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A rectified cube is a cuboctahedron - edges reduced to vertices, and vertices expanded into new faces Cuboctahedron.png
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. Dual Cube-Octahedron.svg
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. Rectified cubic honeycomb.jpg
A rectified cubic honeycomb – edges reduced to vertices, and vertices expanded into new cells.

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.

Contents

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:

Cube truncation sequence.svg

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:

Birectified cube sequence.png

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

FamilyParentRectificationDual
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png
[p,q]
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel 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.svg
Cube
Uniform polyhedron-43-t1.svg
Cuboctahedron
Uniform polyhedron-43-t2.svg
Octahedron
[5,3] Uniform polyhedron-53-t0.svg
Dodecahedron
Uniform polyhedron-53-t1.svg
Icosidodecahedron
Uniform polyhedron-53-t2.svg
Icosahedron
[6,3] Uniform tiling 63-t0.svg
Hexagonal tiling
Uniform tiling 63-t1.svg
Trihexagonal tiling
Uniform tiling 63-t2.svg
Triangular tiling
[7,3] Heptagonal tiling.svg
Order-3 heptagonal tiling
Triheptagonal tiling.svg
Triheptagonal tiling
Order-7 triangular tiling.svg
Order-7 triangular tiling
[4,4] Uniform tiling 44-t0.svg
Square tiling
Uniform tiling 44-t1.svg
Square tiling
Uniform tiling 44-t2.svg
Square tiling
[5,4] H2-5-4-dual.svg
Order-4 pentagonal tiling
H2-5-4-rectified.svg
Tetrapentagonal tiling
H2-5-4-primal.svg
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

FamilyParentRectificationBirectification
(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
{p,q,r}
CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png
r{p,q,r}
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.png
2r{p,q,r}
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.png
3r{p,q,r}
[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

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
NameFacet-1Facet-2
ParentCDel node 1.pngCDel p.pngCDel node.pngt0{p}{p}{2}
RectifiedCDel node.pngCDel p.pngCDel node 1.pngt1{p}{p}{2}

Regular polyhedra and tilings

Facets are regular polygons.

name
{p,q}
Coxeter diagram t-notation
Schläfli symbol
Vertical Schläfli symbol
NameFacet-1Facet-2
ParentCDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png = CDel node.pngCDel split1-pq.pngCDel nodes 10lu.pngt0{p,q}{p,q}{p}
RectifiedCDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png = CDel node 1.pngCDel split1-pq.pngCDel nodes.pngt1{p,q}r{p,q} = {p}{q}
BirectifiedCDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png = CDel node.pngCDel split1-pq.pngCDel nodes 01ld.pngt2{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
NameFacet-1Facet-2
ParentCDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngt0{p,q,r}{p,q,r}{p,q}
RectifiedCDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngt1{p,q,r} = r{p,q,r} = r{p,q}{q,r}
Birectified
(Dual rectified)
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.pngt2{p,q,r} = r{r,q,p}{q,r} = r{q,r}
Trirectified
(Dual)
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.pngt3{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
NameFacet-1Facet-2
ParentCDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngCDel s.pngCDel node.pngt0{p,q,r,s}{p,q,r,s}{p,q,r}
RectifiedCDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngCDel s.pngCDel node.pngt1{p,q,r,s} = r{p,q,r,s} = 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.pngt2{p,q,r,s} = 2r{p,q,r,s} = r{r,q,p} = 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.pngt3{p,q,r,s} = r{s,r,q,p}{r,q,p} = 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.pngt4{p,q,r,s}{s,r,q,p}{s,r,q}

See also

Related Research Articles

Cuboctahedron Polyhedron with 8 triangular faces and 6 square faces

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.

Cube Solid object with six equal square faces

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

Octahedron Polyhedron with 8 triangular faces

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.

Truncated cube

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

Vertex figure Shape made by slicing off a corner of a polytope

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

Tetrahedral-octahedral honeycomb Quasiregular space-filling tesselation

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.

Truncation (geometry) Operation that cuts polytope vertices, creating a new facet in place of each vertex

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.

Alternation (geometry) Operation on a polyhedron or tiling that removes alternate vertices

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

Conway polyhedron notation Method of describing higher-order polyhedra

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.

Cantellation (geometry) Geometric operation on a regular polytope

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.

Expansion (geometry) Geometric operation on convex polytopes

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.

Uniform polytope Isogonal polytope with uniform facets

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.

Simple polytope N-dimensional polytope with vertices adjacent to N facets

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.

Snub (geometry) Geometric operation applied to a polyhedron

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.

Order-4 hexagonal tiling honeycomb

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.

Square tiling honeycomb

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.

Truncated rhombicuboctahedron

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.

References

  1. Weisstein, Eric W. "Rectification". MathWorld .
Polyhedron operators
Seed Truncation Rectification Bitruncation Dual Expansion Omnitruncation Alternations
CDel node 1.pngCDel p.pngCDel node n1.pngCDel q.pngCDel node n2.pngCDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel node h.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel node.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.pngCDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png
Uniform polyhedron-43-t0.svg Uniform polyhedron-43-t01.svg Uniform polyhedron-43-t1.svg Uniform polyhedron-43-t12.svg Uniform polyhedron-43-t2.svg Uniform polyhedron-43-t02.png Uniform polyhedron-43-t012.png Uniform polyhedron-33-t0.png Uniform polyhedron-43-h01.svg 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}
ht0{p,q}
h{q,p}
ht12{p,q}
s{q,p}
ht012{p,q}
sr{p,q}