A vertex figure or vertex star of an n-polytope at a specified vertex roughly refers to the section exposed by shaving off that vertex. The definition varies depending on context, but a common definition produces an (n - 1)-polytope, so a vertex figure of a polyhedron is a polygon, a vertex figure of a polychoron is a polyhedron, etc. Users of Bowers-style acronyms sometimes abbreviate the name to verf.
In the context of isogonal polytopes, there is only one vertex figure, so it is sensible to speak of "the" vertex figure of such polytopes. However, there are polytopes with a single vertex figure that are not isogonal, most notably the pseudo-uniform polytopes.
Definitions[edit | edit source]
As the set of incident elements[edit | edit source]
The simplest definition of a vertex figure is the set of all proper elements incident on the given vertex. In abstract polytopes, this is a purely combinatorial definition. In geometrical polytopes, vertex figures comprise both the combinatorial structure and the shapes, sizes, and relative spatial positions of the elements, such as angles between edges and faces.
The set of all elements incident on a given vertex, minus that vertex v, can be partitioned into equivalence classes under the relationship of incidence. If there is more than one such equivalence class, it is a compound vertex figure. The presence of one or more compound vertex figures is a sufficient but not necessary condition for the polytope to be fissary (a term used in the enthusiast community).
In the case of polyhedra, the faces in a vertex figure connect to each other by adjacency into a cyclic sequence, or more than one cyclic sequence for compound vertex figures. This is a consequence of dyadicity. In polychora and above, there is no inherent cyclic ordering of facets about a vertex.
As the polytope connecting adjacent vertices[edit | edit source]
A simple definition is to find all adjacent vertices to a selected vertex of an n-polytope and connect them together into an (n - 1)-polytope. This is related to truncation and rectification of a single vertex, but cuts even deeper until the adjacent edges completely vanish.
Less commonly, the midpoints of the adjacent edges can also be connected.
Using a hyperplane cross section[edit | edit source]
Given a finite planar n-polytope in Euclidean n-space, for a given vertex v there may be a hyperplane that separates v from all of v's adjacent vertices. This hyperplane is not guaranteed to exist, but when it does, it intersects the elements incident on v to form an (n - 1)-polytope (possibly a compound). Such a hyperplane is not unique, and will result in geometrically different polytopes, but they are combinatorially equivalent to each other.
If the polytope has a well-defined center (such as in an isogonal polytope), we may require that this hyperplane is orthogonal to the line connecting the center of the polytope to v. In such cases, the polytopes formed by the cross section are all similar in the geometrical sense, and it is meaningful to pick any one of them to visualize the vertex figure.
If no hyperplane exists that separates v from adjacent vertices, this is another possible definition of a skew vertex figure.
Using a spherical section[edit | edit source]
Given a finite planar polytope in Euclidean n-space, for every vertex v there is an n-ball that is centered on v and contains none of its adjacent vertices. The intersection of the boundary of that n-ball and the elements incident on v form an (n - 1)-polytope realized on the (n - 1)-sphere that can be called the vertex figure for v.
The resulting "k-polytope on a k-sphere" is not a traditional spherical polytope, which would be rank k + 1.
Higher-order figures[edit | edit source]
For a polytope and an element e of rank r, the rank-r figure of e is the set of all proper elements incident on e which have greater rank than e. This gives rise to edge figures, face figures, etc.
Such a figure is compound if, omitting e itself, the relation of incidence partitions the set of elements into multiple equivalence classes. A polytope is fissary iff it is itself compound or the figure of any of its proper elements is compound.
If a polytope is transitive on its rank-r elements for a given r, it will have a single rank-r figure. For example, if a polytope is isotoxal it has a single edge figure, if a polytope is face-transitive it has a single face figure, and so on. As is the case with vertex figures, the converses of each of these statements are not true.
This definition of higher-order figures is purely combinatorial, but it is theoretically possible to extend some of the above geometric definitions of vertex figures as well.
For specific polytope classes[edit | edit source]
Uniform polyhedra[edit | edit source]
All uniform polyhedra have non-compound vertex figures, and all faces are regular polygons. Thus, a common notation for the vertex figures of uniform polyhedra is to list off the cyclic sequence of faces in order, using n/d as a stand-in for the regular n/d-gon (Schlafli symbol). For example, the cuboctahedron has two squares and two triangles meeting alternately at each vertex, so its vertex figure can be notated 188.8.131.52 or 184.108.40.206. Such cyclic lists do not uniquely identify the uniform polyhedron; for example, the tetrahemihexahedron also has the same cyclic list. This notation is also used for general polyhedra with all regular faces, see acrohedra. Uniform polychora and above do not have an inherent cyclic order in their facets about a vertex, so this notation does not readily generalize to them.
If the vertex figure of a uniform polyhedron is viewed as a polygon (via connecting adjacent vertices), the edge lengths of that polygon may be read off. There is a simple relationship between the edge lengths of the polygon and the entries of the above cyclic list: the edge corresponding to the entry n/d has edge length 2 cos(πd/n), assuming the polyhedron has edge length 1.
Scaliform polytopes[edit | edit source]
Vertex figures do not uniquely identify a scaliform polytope. For instance, the small rhombihexahedral prism and the chasmic cuboctachoron have exactly the same vertex figure, but are different polychora.