Truncation

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.

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 with vertex, edge and face counts $$V(\mathcal{P})$$, $$E(\mathcal{P})$$ and $$F(\mathcal{P})$$, its truncation $$t(\mathcal{P})$$ has counts $$V(t(\mathcal{P})) = 2E(\mathcal{P})$$, $$E(t(\mathcal{P}))=2E(\mathcal{P})$$ $$F(t(\mathcal{P}))=F(\mathcal{P})+V(\mathcal{P})$$.