Rectification (geometry)

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Rectification (geometry)

In Euclidean geometry

Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Greek mathematics Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry....
, rectification is the process of truncating a polytope

Polytope

In geometry, polytope is a generic term that can refer to a two-dimensional polygon, a three-dimensional polyhedron, or any of the various generalizations thereof, including generalizations to higher dimensions and other abstractions ....
by marking the midpoints of all its edges, and cutting off its vertices at those points.

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In Euclidean geometry

Euclidean geometry

Polytope

Vertex figure

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

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 polyhedrons and plane tilings

Each platonic solid

Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
and its dual

Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
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
Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
, whose dual is the tetrahedron, is the tetratetrahedron, better known as the octahedron
Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
.
The rectified octahedron
Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
, whose dual is the cube
Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
, is the cuboctahedron
Cuboctahedron

In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....
.
The rectified icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
, whose dual is the dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
, is the icosidodecahedron
Icosidodecahedron

An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....
.
A rectified square tiling
Square tiling

In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of .John Horton Conway calls it a quadrille....
is a square tiling
Square tiling

In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of .John Horton Conway calls it a quadrille....
.
A rectified triangular tiling
Triangular tiling

In geometry, the triangular tiling is one of the three regular tessellations of the Euclidean plane. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees....
or hexagonal tiling
Hexagonal tiling

In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of or t .John Horton Conway calls it a hextille....
is a trihexagonal tiling
Trihexagonal tiling

In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. There are two triangles and two hexagons alternating on each vertex ....
.

Examples

Family	Parent	Rectification	Dual
[3,3]	Tetrahedron Tetrahedron A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....	Tetratetrahedron Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....	Tetrahedron Tetrahedron A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
[4,3]	Cube Cube A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....	Cuboctahedron Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....	Octahedron Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
[5,3]	Dodecahedron Dodecahedron A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....	Icosidodecahedron Icosidodecahedron An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....	Icosahedron Icosahedron In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
[6,3]	Hexagonal tiling Hexagonal tiling In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of or t .John Horton Conway calls it a hextille....	Trihexagonal tiling Trihexagonal tiling In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. There are two triangles and two hexagons alternating on each vertex ....	Triangular tiling Triangular tiling In geometry, the triangular tiling is one of the three regular tessellations of the Euclidean plane. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees....
[7,3]	Order-3 heptagonal tiling Order-3 heptagonal tiling In geometry, the order-3 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schl?fli symbol of .The image shows a Poincar? disk model projection of the hyperbolic plane....	Triheptagonal tiling Triheptagonal tiling In geometry, the triheptagonal tiling is a semiregular tiling of the hyperbolic plane. There are two triangles and two heptagons alternating on each vertex ....	Order-7 triangular tiling Order-7 triangular tiling In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schl?fli symbol of .The image shows a Poincar? disk model projection of the hyperbolic plane....
[4,4]	Square tiling Square tiling In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of .John Horton Conway calls it a quadrille....	Square tiling Square tiling In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of .John Horton Conway calls it a quadrille....	Square tiling Square tiling In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of .John Horton Conway calls it a quadrille....
[5,4]	Order-4 pentagonal tiling Order-4 pentagonal tiling In geometry, the order-4 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schl?fli symbol of .The image shows a Poincar? disk model projection of the hyperbolic plane....	tetrapentagonal tiling	Order-5 square tiling Order-5 square tiling In geometry, the order-5 square tiling is a regular tiling of the hyperbolic plane. It has Schl?fli symbol of .The image shows a Poincar? disk model projection of the hyperbolic plane....

In polychora and 3d honeycomb tessellations

Each convex regular polychoron

Convex regular 4-polytope

In mathematics, a convex regular 4-polytope is 4-dimensional polytope which is both regular polytope and convex set. These are the four-dimensional analogs of the Platonic solids and the regular polygons ....
has a rectified form as a uniform polychoron

Uniform polychoron

In geometry, a Uniform polytope polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedron.This article contains the complete list of 64 non-prismatic convex uniform polychora, and describes two infinite sets of convex prismatic forms....
.

A regular polychoron has cells . Its rectification will have two cell types, a rectified polyhedron left from the original cells and polyhedron as new cells formed by each truncated vertex.

A rectified is not the same as a rectified , however. A further truncation, called bitruncation, is symmetric between a polychoron and its dual. See Uniform_polychoron#Geometric_derivations

Uniform polychoron

Family	Parent	Rectification	Birectification (Dual rectification)	Trirectification (Dual)
[3,3,3]	5-cell	rectified 5-cell Rectified 5-cell In Fourth dimension geometry, the Rectification 5-cell is a uniform polychoron composed of 5 regular tetrahedral and 5 regular octahedral cell ....	rectified 5-cell Rectified 5-cell In Fourth dimension geometry, the Rectification 5-cell is a uniform polychoron composed of 5 regular tetrahedral and 5 regular octahedral cell ....	5-cell
[4,3,3]	tesseract Tesseract In geometry, the tesseract, also called an 8-cell or regular octachoron, is the Fourth dimension analog of the cube. The tesseract is to the cube as the cube is to the square ....	rectified tesseract Rectified tesseract In geometry, the rectified tesseract, or rectified 8-cell is a uniform polychoron bounded by 24 cell_: 8 cuboctahedron, and 16 tetrahedron....	Rectified 16-cell (24-cell 24-cell In geometry, the 24-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . It is also called an octaplex and polyoctahedron, being constructed of Octahedron Cell .... )	16-cell 16-cell In Fourth dimension geometry, a 16-cell, is a regular convex polychora, or polytope existing in four dimensions. It is also known as the hexadecachoron....
[3,4,3]	24-cell 24-cell In geometry, the 24-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . It is also called an octaplex and polyoctahedron, being constructed of Octahedron Cell ....	rectified 24-cell Rectified 24-cell In geometry, the rectified 24-cell is a uniform 4-dimensional polytope , which is bounded by 48 cell_: 24 cubes, and 24 cuboctahedron.It can also be considered a cantellated 16-cell with the lower symmetries B4 = [3,3,4], or even D4....	rectified 24-cell Rectified 24-cell In geometry, the rectified 24-cell is a uniform 4-dimensional polytope , which is bounded by 48 cell_: 24 cubes, and 24 cuboctahedron.It can also be considered a cantellated 16-cell with the lower symmetries B4 = [3,3,4], or even D4....	24-cell 24-cell In geometry, the 24-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . It is also called an octaplex and polyoctahedron, being constructed of Octahedron Cell ....
[5,3,3]	120-cell 120-cell In geometry, the 120-cell is the convex regular 4-polytope with Schl?fli symbol .The boundary of the 120-cell is composed of 120 dodecahedral cell with 4 meeting at each vertex....	rectified 120-cell Rectified 120-cell In geometry, the Rectification 120-cell is a convex uniform polychoron composed of 600 regular tetrahedron and 120 icosidodecahedron cell . Its vertex figure is a triangular prism, with 3 icosidodecahedra and 2 tetrahedra meeting at each vertex....	rectified 600-cell Rectified 600-cell In geometry, the Rectification 600-cell is a convex uniform polychoron composed of 600 regular octahedra and 120 icosahedra cell . Each edge has two octahedra and one icosahedron....	600-cell 600-cell In geometry, the 600-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . Its boundary is composed of 600 tetrahedron cell with 20 meeting at each vertex....
[4,3,4]	Cubic honeycomb Cubic honeycomb The cubic honeycomb is the only regular space-filling tessellation in Euclidean 3-space, made up of cubes. It is an analog of the square tiling of the plane, and part of a dimensional family called hypercube honeycombs....	Rectified cubic honeycomb Rectified cubic honeycomb The rectified cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is comprised of octahedron and cuboctahedron in a ratio of 1:1....	Rectified cubic honeycomb Rectified cubic honeycomb The rectified cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is comprised of octahedron and cuboctahedron in a ratio of 1:1....	Cubic honeycomb Cubic honeycomb The cubic honeycomb is the only regular space-filling tessellation in Euclidean 3-space, made up of cubes. It is an analog of the square tiling of the plane, and part of a dimensional family called hypercube honeycombs....
[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

Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flag , thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension = n....
, this form is represented by an extended Schläfli symbol

Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
notation t₁.

A second order rectification, or birectification, truncates faces

Face (geometry)

In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the square s that bound a cube is a face of the cube....
down to points. If regular it has notation t₂. For polyhedra

Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
, a birectification creates a dual polyhedron

Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original 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

Facet (mathematics)

A facet of a simplicial complex is a maximal simplex.In the general theory of polyhedra and polytopes, two conflicting meanings are currently jostling for acceptability:...
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 facet

Facet (mathematics)

A facet of a simplicial complex is a maximal simplex.In the general theory of polyhedra and polytopes, two conflicting meanings are currently jostling for acceptability:...
s for each.

Regular polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
s

Facet

Facet (mathematics)

A facet of a simplicial complex is a maximal simplex.In the general theory of polyhedra and polytopes, two conflicting meanings are currently jostling for acceptability:...
s are edges, represented as .

name	t-notation Schläfli symbol Schläfli symbol In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....	Vertical Schläfli symbol Schläfli symbol In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
name		Name	Facet-1	Facet-2
Parent	t₀
Rectified	t₁

Regular polyhedra
Uniform polyhedron

A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
and tilings

Facet

Facet (mathematics)

A facet of a simplicial complex is a maximal simplex.In the general theory of polyhedra and polytopes, two conflicting meanings are currently jostling for acceptability:...
s are regular polygons.

name	t-notation Schläfli symbol Schläfli symbol In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....	Vertical Schläfli symbol Schläfli symbol In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
name		Name	Facet-1	Facet-2
Parent	t₀
Rectified	t₁
Birectified	t₂

Regular polychora
Uniform polychoron

In geometry, a Uniform polytope polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedron.This article contains the complete list of 64 non-prismatic convex uniform polychora, and describes two infinite sets of convex prismatic forms....
and honeycomb
Honeycomb

A honeycomb is a mass of hexagonal waxcells built by honey bees in their beehive to contain their larva and stores of honey and pollen.beekeeping may remove the entire honeycomb to harvest honey....
s

Facet

Facet (mathematics)

name	t-notation Schläfli symbol Schläfli symbol In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....	Vertical Schläfli symbol Schläfli symbol In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
name		Name	Facet-1	Facet-2
Parent	t₀
Rectified	t₁
Birectified	t₂
Trirectified	t₃

Regular polyterons and 4-space honeycombs
Honeycomb (geometry)

In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions....

Facet

Facet (mathematics)

name	t-notation Schläfli symbol Schläfli symbol In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....	Vertical Schläfli symbol Schläfli symbol In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
name		Name	Facet-1	Facet-2
Parent	t₀
Rectified	t₁
Birectified	t₂
Trirectified	t₃
Tetrarectified	t₄

Rectification (geometry)

Example of rectification as a final truncation to an edge