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 Fn/V, where Fn 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. General properties A vertex figure of an n-polytope is an (n−1)-polytope. For example, a vertex figure of a polyhedron is a polygon, and the vertex figure for a 4-polytope is a polyhedron. In general a vertex figure need not be planar. For nonconvex polyhedra, the vertex figure may also be nonconvex. Uniform polytopes, for instance, can have star polygons for faces and/or for vertex figures. 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. Constructions 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