View source for Rectification (geometry)

{{about|an operation on polyhedra|rectification of curves|arc length}}
{{short description|Operation in Euclidean geometry}}
[[Image:Cuboctahedron.png|thumb|A rectified cube is a [[cuboctahedron]] – edges reduced to vertices, and vertices expanded into new faces]]
[[Image:Dual Cube-Octahedron.svg|thumb|A ''birectified'' cube is an octahedron – faces are reduced to points and new faces are centered on the original vertices.]]
[[Image:Rectified cubic honeycomb.jpg|thumb|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 [[Edge (geometry)|edges]], and cutting off its [[Vertex (geometry)|vertices]] at those points.<ref>{{Mathworld | urlname=Rectification | title=Rectification}}</ref> 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 {{mvar|r}} with a [[Schläfli symbol]]. For example, {{nowrap|{{math|''r''{4,3} }}}} is the rectified [[cube]], also called a [[cuboctahedron]], and also represented as <math>\begin{Bmatrix} 4 \\ 3 \end{Bmatrix}</math>. And a rectified cuboctahedron {{math|rr{4,3} }} is a [[rhombicuboctahedron]], and also represented as <math>r\begin{Bmatrix} 4 \\ 3 \end{Bmatrix}</math>.

[[Conway polyhedron notation]] uses {{math|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]] {{math|{3,3} }} becoming an [[octahedron]] {{math|{3,4}.}} As a special case, a [[square tiling]] {{math|{4,4} }} will turn into another square tiling {{math|{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:

[[Image:Cube truncation sequence.svg|520px|class=skin-invert]]

== 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:
:[[Image:Birectified cube sequence.png|480px]]

== 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 ==
{{Further|quasiregular polyhedron}}
Each [[platonic solid]] and its [[dual polyhedron|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 appropriately scaled concentric version of its dual.  For this reason, its name is a combination of the names of the original and the dual:

* The [[tetrahedron]] is its own dual, and its rectification is the ''tetratetrahedron'', better known as the [[octahedron]]. 
* The [[octahedron]] and the [[cube]] are each other's dual, and their rectification is the [[cuboctahedron]].
* The [[icosahedron]] and the [[dodecahedron]] are duals, and their rectification is the [[icosidodecahedron]].

Examples
{| class="wikitable"
!Family
!Parent
!Rectification
!Dual
|-
!{{CDD|node|p|node|q|node}}<BR>[p,q]
!{{CDD|node_1|p|node|q|node}}
!{{CDD|node|p|node_1|q|node}}
!{{CDD|node|p|node|q|node_1}}
|- align=center
![3,3]
|[[Image:Uniform polyhedron-33-t0.png|75px]]<BR>[[Tetrahedron]]
|[[Image:Uniform polyhedron-33-t1.svg|75px]]<BR>[[Tetratetrahedron|Octahedron]]
|[[Image:Uniform polyhedron-33-t2.png|75px]]<BR>[[Tetrahedron]]
|- align=center
![4,3]
|[[Image:Uniform polyhedron-43-t0.svg|75px]]<BR>[[Cube]]
|[[Image:Uniform polyhedron-43-t1.svg|75px]]<BR>[[Cuboctahedron]]
|[[Image:Uniform polyhedron-43-t2.svg|75px]]<BR>[[Octahedron]]
|- align=center
![5,3]
|[[Image:Uniform polyhedron-53-t0.svg|75px]]<BR>[[Dodecahedron]]
|[[Image:Uniform polyhedron-53-t1.svg|75px]]<BR>[[Icosidodecahedron]]
|[[Image:Uniform polyhedron-53-t2.svg|75px]]<BR>[[Icosahedron]]
|- align=center
![6,3]
|[[Image:Uniform tiling 63-t0.svg|75px]]<BR>[[Hexagonal tiling]]
|[[Image:Uniform tiling 63-t1.svg|75px]]<BR>[[Trihexagonal tiling]]
|[[Image:Uniform tiling 63-t2.svg|75px]]<BR>[[Triangular tiling]]
|- align=center
![7,3]
|[[Image:Heptagonal tiling.svg|75px]]<BR>[[Order-3 heptagonal tiling]]
|[[Image:Triheptagonal tiling.svg|75px]]<BR>[[Triheptagonal tiling]]
|[[Image:Order-7 triangular tiling.svg|75px]]<BR>[[Order-7 triangular tiling]]
|- align=center
![4,4]
|[[Image:Uniform tiling 44-t0.svg|75px]]<BR>[[Square tiling]]
|[[Image:Uniform tiling 44-t1.svg|75px]]<BR>[[Square tiling]]
|[[Image:Uniform tiling 44-t2.svg|75px]]<BR>[[Square tiling]]
|- align=center
![5,4]
|[[Image:H2-5-4-dual.svg|75px]]<BR>[[Order-4 pentagonal tiling]]
|[[Image:H2-5-4-rectified.svg|75px]]<BR>[[Tetrapentagonal tiling]]
|[[Image:H2-5-4-primal.svg|75px]]<BR>[[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 [[n-skeleton|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 [[expansion (geometry)|'''expand''']] operation, '''e''', which is the same as Johnson's [[Cantellation (geometry)|cantellation]] operation, t<sub>0,2</sub> 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 (geometry)|bitruncation]], is symmetric between a 4-polytope and its dual. See [[Uniform 4-polytope#Geometric derivations]].

Examples
{| class="wikitable"
!Family
!Parent
!Rectification
!Birectification<BR>(Dual rectification)
!Trirectification<BR>(Dual)
|-
!{{CDD|node|p|node|q|node|r|node}}<BR>[''p'',''q'',''r'']
!{{CDD|node_1|p|node|q|node|r|node}}<BR>{''p'',''q'',''r''}
!{{CDD|node|p|node_1|q|node|r|node}}<BR>r{''p'',''q'',''r''}
!{{CDD|node|p|node|q|node_1|r|node}}<BR>2r{''p'',''q'',''r''}
!{{CDD|node|p|node|q|node|r|node_1}}<BR>3r{''p'',''q'',''r''}
|- align=center
![3,3,3]
|[[Image:Schlegel wireframe 5-cell.png|120px]]<BR>[[5-cell]] 
|[[Image:Schlegel half-solid rectified 5-cell.png|120px]]<BR>[[rectified 5-cell]]
|[[Image:Schlegel half-solid rectified 5-cell.png|120px]]<BR>[[rectified 5-cell]]
|[[Image:Schlegel wireframe 5-cell.png|120px]]<BR>[[5-cell]] 
|- align=center
![4,3,3]
|[[Image:Schlegel wireframe 8-cell.png|150px]]<BR>[[tesseract]]
|[[Image:Schlegel half-solid rectified 8-cell.png|150px]]<BR>[[rectified tesseract]]
|[[Image:Schlegel half-solid rectified 16-cell.png|150px]]<BR>Rectified 16-cell<BR>([[24-cell]])
|[[Image:Schlegel wireframe 16-cell.png|150px]]<BR>[[16-cell]]
|- align=center
![3,4,3]
|[[Image:Schlegel wireframe 24-cell.png|150px]]<BR>[[24-cell]]
|[[File:Schlegel half-solid cantellated 16-cell.png|150px]]<BR>[[rectified 24-cell]]
|[[File:Schlegel half-solid cantellated 16-cell.png|150px]]<BR>[[rectified 24-cell]]
|[[Image:Schlegel wireframe 24-cell.png|150px]]<BR>[[24-cell]]
|- align=center
![5,3,3]
|[[Image:Schlegel wireframe 120-cell.png|150px]]<BR>[[120-cell]]
|[[File:Rectified 120-cell schlegel halfsolid.png|150px]]<BR>[[rectified 120-cell]]
|[[File:Rectified_600-cell_schlegel_halfsolid.png|150px]]<BR>[[rectified 600-cell]]
|[[Image:Schlegel wireframe 600-cell vertex-centered.png|150px]]<BR>[[600-cell]]
|- align=center
![4,3,4]
|[[Image:Partial cubic honeycomb.png|150px]]<BR>[[Cubic honeycomb]]
|[[Image:Rectified cubic honeycomb.jpg|150px]]<BR>[[Rectified cubic honeycomb]]
|[[Image:Rectified cubic honeycomb.jpg|150px]]<BR>[[Rectified cubic honeycomb]]
|[[Image:Partial cubic honeycomb.png|150px]]<BR>[[Cubic honeycomb]]
|- align=center
![5,3,4]
|[[Image:Hyperbolic_orthogonal_dodecahedral_honeycomb.png|150px]]<BR>[[Order-4 dodecahedral honeycomb|Order-4 dodecahedral]]
|[[File:Rectified_order_4_dodecahedral_honeycomb.png|150px]]<BR>[[Rectified order-4 dodecahedral honeycomb|Rectified order-4 dodecahedral]]
|[[File:H3 435 CC center 0100.png|150px]]<BR>[[Rectified order-5 cubic honeycomb|Rectified order-5 cubic]]
|[[Image:Hyperb gcubic hc.png|150px]]<BR>[[Order-5 cubic honeycomb|Order-5 cubic]]
|}

== Degrees of rectification ==

A first rectification truncates edges down to points. If a polytope is [[Regular polytope|regular]], this form is represented by an extended [[Schläfli symbol]] notation '''t'''<sub>1</sub>{p,q,...} or '''r'''{p,q,...}.

A second rectification, or '''birectification''', truncates [[Face (geometry)|faces]] down to points. If regular it has notation '''t'''<sub>2</sub>{p,q,...} or 2'''r'''{p,q,...}. For [[polyhedron|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 [[Facet (mathematics)|facets]] are reduced to points and the polytope becomes its [[Dual polytope|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 [[Facet (mathematics)|facet]]s for each.

==== Regular [[polygon]]s ====

[[Facet (mathematics)|Facet]]s are edges, represented as {}.

{| class="wikitable"
!rowspan=2|name<BR>{p}
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter diagram]]
!rowspan=2|t-notation<BR>[[Schläfli symbol]]
!colspan=3|Vertical [[Schläfli symbol]]
|-
!Name
!Facet-1
!Facet-2
|- align=center
|Parent
|{{CDD|node_1|p|node}}
|t<sub>0</sub>{p}
| {p}
| {}
|
|- align=center
|Rectified
|{{CDD|node|p|node_1}}
|t<sub>1</sub>{p}
| {p}
|
| {}
|}

==== Regular [[Uniform polyhedron|polyhedra]] and [[List of uniform tilings|tiling]]s ====

[[Facet (mathematics)|Facet]]s are regular polygons.
{| class="wikitable"
!rowspan=2|name<BR>{p,q}
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter diagram]]
!rowspan=2|t-notation<BR>[[Schläfli symbol]]
!colspan=3|Vertical [[Schläfli symbol]]
|-
!Name
!Facet-1
!Facet-2
|- align=center
|Parent
|{{CDD|node_1|p|node|q|node}} = {{CDD|node|split1-pq||nodes_10lu}}
|t<sub>0</sub>{p,q}
| {p,q}
| {p}
|
|- align=center
|Rectified
|{{CDD|node|p|node_1|q|node}} = {{CDD|node_1|split1-pq||nodes}}
|t<sub>1</sub>{p,q} 
| r{p,q} = <math>\begin{Bmatrix} p \\ q  \end{Bmatrix}</math>
| {p}
| {q}
|- align=center
|Birectified
|{{CDD|node|p|node|q|node_1}} = {{CDD|node|split1-pq||nodes_01ld}}
|t<sub>2</sub>{p,q}
| {q,p}
|
| {q}
|}

==== Regular [[Uniform 4-polytope]]s and [[honeycomb]]s ====
[[Facet (mathematics)|Facet]]s are regular or rectified polyhedra.
{| class="wikitable"
!rowspan=2|name<BR>{p,q,r}
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter diagram]]
!rowspan=2|t-notation<BR>[[Schläfli symbol]]
!colspan=3|Extended [[Schläfli symbol]]
|-
!Name
!Facet-1
!Facet-2
|- align=center
|Parent
|{{CDD|node_1|p|node|q|node|r|node}}
|t<sub>0</sub>{p,q,r}
| {p,q,r}
| {p,q}
|
|- align=center
|Rectified
|{{CDD|node|p|node_1|q|node|r|node}}
|t<sub>1</sub>{p,q,r}
| <math>\begin{Bmatrix} p \ \ \\ q , r \end{Bmatrix}</math> = r{p,q,r}
| <math>\begin{Bmatrix} p \\ q \end{Bmatrix}</math> = r{p,q}
| {q,r}
|- align=center
|Birectified<BR>(Dual rectified)
|{{CDD|node|p|node|q|node_1|r|node}}
|t<sub>2</sub>{p,q,r}
| <math>\begin{Bmatrix} q , p \\ r \ \ \end{Bmatrix}</math> = r{r,q,p}
| {q,r}
|<math>\begin{Bmatrix} q \\ r \end{Bmatrix}</math> = r{q,r}
|- align=center
|Trirectified<BR>(Dual)
|{{CDD|node|p|node|q|node|r|node_1}}
|t<sub>3</sub>{p,q,r}
| {r,q,p}
|
| {r,q}
|}

==== Regular [[5-polytope]]s and 4-space [[Honeycomb (geometry)|honeycombs]] ====
[[Facet (mathematics)|Facet]]s are regular or rectified 4-polytopes.
{| class="wikitable"
!rowspan=2|name<BR>{p,q,r,s}
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter diagram]]
!rowspan=2|t-notation<BR>[[Schläfli symbol]]
!colspan=3|Extended [[Schläfli symbol]]
|-
!Name
!Facet-1
!Facet-2
|- align=center 
|Parent
|{{CDD|node_1|p|node|q|node|r|node|s|node}}
|t<sub>0</sub>{p,q,r,s}
| {p,q,r,s}
| {p,q,r}
|
|- align=center 
|Rectified
|{{CDD|node|p|node_1|q|node|r|node|s|node}}
|t<sub>1</sub>{p,q,r,s}
| <math>\begin{Bmatrix} p \ \ \ \ \ \\ q , r , s \end{Bmatrix}</math> = r{p,q,r,s}
|<math>\begin{Bmatrix} p \ \ \\ q , r \end{Bmatrix}</math> = r{p,q,r}
|{q,r,s}
|- align=center 
|Birectified<BR>(Birectified dual)
|{{CDD|node|p|node|q|node_1|r|node|s|node}}
|t<sub>2</sub>{p,q,r,s}
| <math>\begin{Bmatrix} q , p \\ r , s \end{Bmatrix}</math> = 2r{p,q,r,s}
|<math>\begin{Bmatrix} q , p \\  r \ \ \end{Bmatrix}</math> = r{r,q,p}
|<math>\begin{Bmatrix} q \ \ \\ r , s \end{Bmatrix}</math>  = r{q,r,s}
|- align=center 
|Trirectified<BR>(Rectified dual)
|{{CDD|node|p|node|q|node|r|node_1|s|node}}
|t<sub>3</sub>{p,q,r,s}
| <math>\begin{Bmatrix} r , q , p \\ s \ \ \ \ \ \end{Bmatrix}</math> = r{s,r,q,p}
|{r,q,p}
|<math>\begin{Bmatrix} r , q \\ s \ \ \end{Bmatrix}</math> = r{s,r,q}
|- align=center 
|Quadrirectified<BR>(Dual)
|{{CDD|node|p|node|q|node|r|node|s|node_1}}
|t<sub>4</sub>{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 ==
{{reflist}}
* [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, {{isbn|0-486-61480-8}} (pp.&nbsp;145–154 Chapter 8: Truncation)
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** [[Norman Johnson (mathematician)|N.W. Johnson]]: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966
* [[John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{isbn|978-1-56881-220-5}} (Chapter 26)

== External links ==
* {{GlossaryForHyperspace | anchor=Rectification | title=Rectification }}

{{Polyhedron_operators}}

[[Category:Polytopes]]
[[Category:Truncated tilings| ]]