# Rectification

Rectification
Coxeter-Dynkin diagram...
Prefix${\displaystyle t_{1}}$
Minimum rank2
Extended Schläfli symbol${\displaystyle r}$
Conway polyhedron notation${\displaystyle a}$

Rectification (also called complete truncation) is an operation on polytopes which creates a new polytope by cutting away area around each vertex to create new facets with new vertices at the midpoints of the original edges. It can be thought of a form of truncation with deeper cuts.

## Definition

### Quasiregular polytopes

A quasiregular polytope, that is a polytope with a Coxeter-Dynkin diagram with exactly one ringed node, can be rectified by placing a ring on every node adjacent to its ringed node[note 1] and removing the ring from that node.

Since planar regular polytopes can be represented with a diagram with the first node ringed, this means the rectification of a regular polytope is the same symmetry with exactly the second node ringed. For example the tesseract is rectified to the rectified tesseract .

Since the result of rectification of a regular is itself quasiregular this definition allows regular polytopes to be rectified twice, the result is the cantellation.

## Rectification by dimension

### Polygons

Polygons are the lowest dimension for which rectification is possible. Rectifying an n-gon produces another n-gon.

### Polyhedra

A cube (left) and a rectified cube (right). The newly exposed faces are colored yellow.

Rectifying a polyhedron results in a new polyhedron. For every face F  in the original polyhedron, there is a corresponding face in the rectified polyhedron F'  which is the rectification of the original F . Furthermore, if two faces were adjacent by an edge in the original polyhedron, their rectifications are adjacent by a vertex in the rectified polyhedron. Recfication additionally adds more faces corresponding to vertices in the original polyhedron. These faces are similarly adjacent by a vertex iff the vertexes the correspond to in the initial polyhedron were adjacent by an edge.

The rectification of a regular polyhedron and its dual are isomorphic. This is not true in general for polytopes of higher rank.

#### Element counts

For an initial polyhedron 𝓟 with vertex, edge and face counts ${\displaystyle V({\mathcal {P}})}$, ${\displaystyle E({\mathcal {P}})}$ and ${\displaystyle F({\mathcal {P}})}$, its rectification ${\displaystyle a({\mathcal {P}})}$ has counts ${\displaystyle V(a({\mathcal {P}}))=E({\mathcal {P}})}$, ${\displaystyle E(a({\mathcal {P}}))=2E({\mathcal {P}})}$ and ${\displaystyle F(a({\mathcal {P}}))=F({\mathcal {P}})+V({\mathcal {P}})}$.

The rectification of a polyhedron has twice the number of flags as the initial polyhedron. This can be seen by comparing two flag counting methods for polyhedra:

1. The number of flags in a polyhedron is 4 times the number of edges in the polyhedron.
2. The number of flags in a polyhedron is the sum of the flags in its vertex figures.

Since rectifying a polyhedron replaces a polyhedron's edges with vertices with a irregular tetragonal vertex figure, this replaces groupings of 4 flags with groupings of 8. Thus doubling the flags.

## Maps

The idea of rectification can be generalized to an operation on maps. Maps correspond to graph embeddings and their rectification corresponds is the medial graph.

## Notes

1. Here "adjacent" refers to any node connected by an edge greater than 2. Since edges of value 2 are not drawn this means any node visually connected to the ringed node.