# Truncation

Truncation
Coxeter-Dynkin diagram...
Prefix${\displaystyle t_{0,1}}$
Minimum rank2
Conway polyhedron notationt

Truncation is an operation on polytopes which creates a new facet at every vertex of a polytope by "cutting away" the vertex and some of the surrounding material.

## Truncation of regular polytopes

### Parameterized truncation

Some truncations of the cube along with the cube itself and a rectified cube.

Although it makes no difference for abstract polytopes, the cuts used to truncate a polytope can be of different depths resulting in polytopes with different measures. If truncating on regular polytopes with cuts of the same depth, the resulting polytope will have two distinct lengths of edges. This type of truncation can be parameterized by a single real value. Although there is no agreed upon standard for what this value represents. Uniform truncation is truncation in which both types of edges in the result have the same length. It is called so because the uniform truncation of of a regular polytope is a uniform polytope.

### Coxeter-Dynkin diagrams

Every regular polytope has a linear Coxeter-Dynkin diagram with one ring on the first node (...). Its truncation is has the same diagram but with a ring on the first two nodes as well (...).

## Truncation by dimension

### Polygons

Polygons are the lowest dimension of polytope for which truncation is possible. Truncating a polygon places an edge at every vertex of the original polygon. Since polygons have an equal number of edges and vertices this has the effect of doubling the number of edges and vertices. For example a truncated square is an octagon.

Written in terms of Coxeter-Dynkin diagrams this is:

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### Polyhedra

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

Truncating a polyhedron places a face at every vertex of the original polyhedron. 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 truncation ${\displaystyle t(\mathcal{P})}$ has counts ${\displaystyle V(t(\mathcal{P})) = 2E(\mathcal{P})}$, ${\displaystyle E(t(\mathcal{P}))=2E(\mathcal{P})}$ and ${\displaystyle F(t(\mathcal{P}))=F(\mathcal{P})+V(\mathcal{P})}$.