As a flat slice

Make a slice through the corner of the polyhedron, cutting through all the edges connected to the vertex. The cut surface is the vertex figure. This is perhaps the most common approach, and the most easily understood. Different authors make the slice in different places. Wenninger (2003) cuts each edge a unit distance from the vertex, as does Coxeter (1948). For uniform polyhedra the Dorman Luke construction cuts each connected edge at its midpoint. Other authors make the cut through the vertex at the other end of each edge.^[1][2]

For an irregular polyhedron, cutting all edges incident to a given vertex at equal distances from the vertex may produce a figure that does not lie in a plane. A more general approach, valid for arbitrary convex polyhedra, is to make the cut along any plane which separates the given vertex from all the other vertices, but is otherwise arbitrary. This construction determines the combinatorial structure of the vertex figure, similar to a set of connected vertices (see below), but not its precise geometry; it may be generalized to convex polytopes in any dimension. However, for non-convex polyhedra, there may not exist a plane near the vertex that cuts all of the faces incident to the vertex.

As a spherical polygon

Cromwell (1999) forms the vertex figure by intersecting the polyhedron with a sphere centered at the vertex, small enough that it intersects only edges and faces incident to the vertex. This can be visualized as making a spherical cut or scoop, centered on the vertex. The cut surface or vertex figure is thus a spherical polygon marked on this sphere. One advantage of this method is that the shape of the vertex figure is fixed (up to the scale of the sphere), whereas the method of intersecting with a plane can produce different shapes depending on the angle of the plane. Additionally, this method works for non-convex polyhedra.

As the set of connected vertices

Many combinatorial and computational approaches (e.g. Skilling, 1975) treat a vertex figure as the ordered (or partially ordered) set of points of all the neighboring (connected via an edge) vertices to the given vertex.

Abstract definition

In the theory of abstract polytopes, the vertex figure at a given vertex V comprises all the elements which are incident on the vertex; edges, faces, etc. More formally it is the (n−1)-section F_n/V, where F_n is the greatest face.

This set of elements is elsewhere known as a vertex star. The geometrical vertex figure and the vertex star may be understood as distinct realizations of the same abstract section.

Isogonal figures

Vertex figures are especially significant for uniforms and other isogonal (vertex-transitive) polytopes because one vertex figure can define the entire polytope.

For polyhedra with regular faces, a vertex figure can be represented in vertex configuration notation, by listing the faces in sequence around the vertex. For example 3.4.4.4 is a vertex with one triangle and three squares, and it defines the uniform rhombicuboctahedron.

If the polytope is isogonal, the vertex figure will exist in a hyperplane surface of the n-space.

From the adjacent vertices

By considering the connectivity of these neighboring vertices, a vertex figure can be constructed for each vertex of a polytope:

• Each vertex of the vertex figure coincides with a vertex of the original polytope.
• Each edge of the vertex figure exists on or inside of a face of the original polytope connecting two alternate vertices from an original face.
• Each face of the vertex figure exists on or inside a cell of the original n-polytope (for n > 3).
• ... and so on to higher order elements in higher order polytopes.

Dorman Luke construction

For a uniform polyhedron, the face of the dual polyhedron may be found from the original polyhedron's vertex figure using the "Dorman Luke" construction.

Regular polytopes

The vertex figure of the great icosahedron is a regular pentagram or star polygon {5/2}.

If a polytope is regular, it can be represented by a Schläfli symbol and both the cell and the vertex figure can be trivially extracted from this notation.

In general a regular polytope with Schläfli symbol {a,b,c,...,y,z} has cells as {a,b,c,...,y}, and vertex figures as {b,c,...,y,z}.

1. For a regular polyhedron {p,q}, the vertex figure is {q}, a q-gon.
   • Example, the vertex figure for a cube {4,3}, is the triangle {3}.
2. For a regular 4-polytope or space-filling tessellation {p,q,r}, the vertex figure is {q,r}.
   • Example, the vertex figure for a hypercube {4,3,3}, the vertex figure is a regular tetrahedron {3,3}.
   • Also the vertex figure for a cubic honeycomb {4,3,4}, the vertex figure is a regular octahedron {3,4}.

Since the dual polytope of a regular polytope is also regular and represented by the Schläfli symbol indices reversed, it is easy to see the dual of the vertex figure is the cell of the dual polytope. For regular polyhedra, this is a special case of the Dorman Luke construction.

Notes

Bibliography

• H. S. M. Coxeter, Regular Polytopes, Hbk (1948), ppbk (1973).
• H.S.M. Coxeter (et al.), Uniform Polyhedra, Phil. Trans. 246 A (1954) pp. 401–450.
• P. Cromwell, Polyhedra, CUP pbk. (1999).
• H.M. Cundy and A.P. Rollett, Mathematical Models, Oxford Univ. Press (1961).
• J. Skilling, The Complete Set of Uniform Polyhedra, Phil. Trans. 278 A (1975) pp. 111–135.
• M. Wenninger, Dual Models, CUP hbk (1983) ppbk (2003).
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ISBN 978-1-56881-220-5 (p289 Vertex figures)