Vertex figure

Definition and constructions

There are several equivalent constructions used in practice:

• Slicing: truncate a vertex by cutting with a plane (or hyperplane) close to the vertex; the newly exposed face is the vertex figure.
• Spherical intersection: intersect the polytope with a sufficiently small sphere around the vertex; the result is a spherical polygon or polytope representing local adjacency.
• Combinatorial description: list the sequence of faces around the vertex, often written as a vertex configuration like 4.4.4 for a cube (three squares meeting).

Examples and properties

For three-dimensional solids (polyhedra), a vertex figure is a polygon: if k faces meet at a vertex the vertex figure is a k‑gon. Examples: a cube has triangular vertex figures (three squares meet), an octahedron has square vertex figures (four triangles meet). In four or more dimensions the vertex figure is an (n−1)-dimensional polytope: the tesseract (4-cube) has a tetrahedral vertex figure because four cubes meet at each vertex.

Uses and significance

Vertex figures are used to classify polytopes and tilings, to describe local symmetry (uniform polytopes have identical vertex figures at every vertex), and to construct duals: facets of the dual correspond to vertex figures of the original. They also reveal whether a local arrangement produces convex or star-shaped geometry; nonconvex and star polytopes can produce star polygon vertex figures.

History and related concepts

The idea appears in classical studies of polyhedra and in modern polytope theory. It's closely related to the notions of vertex configuration and truncation operations. For abstract polytopes the vertex figure can be defined combinatorially as the section of the poset of faces above a given vertex, a concept useful in algebraic and topological treatments of polytopes (polytope theory).