Ingeometry, avertex figure, broadly speaking, is the figure exposed when a corner of a generaln-polytope is sliced off.
Take some corner orvertex of apolyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points around the face. When done, these lines form a complete circuit, i.e. a polygon, around the vertex. This polygon is the vertex figure.
More precise formal definitions can vary quite widely, according to circumstance. For exampleCoxeter (e.g. 1948, 1954) varies his definition as convenient for the current area of discussion. Most of the following definitions of a vertex figure apply equally well to infinitetilings or, by extension, tospace-filling tessellation withpolytopecells and other higher-dimensionalpolytopes.
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 (aplane 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 theDorman 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 toconvex 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.
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.
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.
In the theory ofabstract polytopes, the vertex figure at a given vertexV comprises all the elements which are incident on the vertex; edges, faces, etc. More formally it is the (n−1)-sectionFn/V, whereFn is the greatest face.
This set of elements is elsewhere known as avertex star. The geometrical vertex figure and the vertex star may be understood as distinctrealizations of the same abstract section.
A vertex figure of ann-polytope is an (n−1)-polytope. For example, a vertex figure of apolyhedron is apolygon, and the vertex figure for a4-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 havestar polygons for faces and/or for vertex figures.
Vertex figures are especially significant foruniforms and otherisogonal (vertex-transitive) polytopes because one vertex figure can define the entire polytope.
For polyhedra with regular faces, a vertex figure can be represented invertex 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 uniformrhombicuboctahedron.
If the polytope is isogonal, the vertex figure will exist in ahyperplane surface of then-space.
By considering the connectivity of these neighboring vertices, a vertex figure can be constructed for each vertex of a polytope:
For a uniform polyhedron, the face of thedual polyhedron may be found from the original polyhedron's vertex figure using the "Dorman Luke" construction.
If a polytope is regular, it can be represented by aSchläfli symbol and both thecell 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}, andvertex figures as {b,c,...,y,z}.
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 theDorman Luke construction.
The vertex figure of atruncated cubic honeycomb is a nonuniformsquare pyramid. One octahedron and four truncated cubes meet at each vertex form a space-fillingtessellation.
| Vertex figure: A nonuniformsquare pyramid |  Schlegel diagram |  Perspective |
| Created as asquare base from anoctahedron |  (3.3.3.3) | |
| And fourisosceles triangle sides fromtruncated cubes |  (3.8.8) |
Related to thevertex figure, anedge figure is thevertex figure of avertex figure.[3] Edge figures are useful for expressing relations between the elements within regular and uniform polytopes.
Anedge figure will be a (n−2)-polytope, representing the arrangement offacets around a given edge. Regular and single-ringedcoxeter diagram uniform polytopes will have a single edge type. In general, a uniform polytope can have as many edge types as active mirrors in the construction, since each active mirror produces one edge in the fundamental domain.
Regular polytopes (and honeycombs) have a singleedge figure which is also regular. For a regular polytope {p,q,r,s,...,z}, theedge figure is {r,s,...,z}.
In four dimensions, the edge figure of a4-polytope or3-honeycomb is a polygon representing the arrangement of a set of facets around an edge. For example, theedge figure for a regularcubic honeycomb {4,3,4} is asquare, and for a regular 4-polytope {p,q,r} is the polygon {r}.
Less trivially, thetruncated cubic honeycomb t0,1{4,3,4}, has asquare pyramid vertex figure, withtruncated cube andoctahedron cells. Here there are two types ofedge figures. One is a square edge figure at the apex of the pyramid. This represents the fourtruncated cubes around an edge. The other four edge figures are isosceles triangles on the base vertices of the pyramid. These represent the arrangement of two truncated cubes and one octahedron around the other edges.
