In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.
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Take some vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points. 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 example Coxeter (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 infinite tilings, or space-filling tessellation with polytope cells.
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 commonest 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. For uniform polyhedra the Dorman Luke construction cuts each connected edge at its mid-point. Other authors make the cut through the vertex at the other end of each edge.
Cromwell (1999) makes a spherical cut or scoop, centered on the vertex. The cut surface or vertex figure is thus a spherical polygon marked on this sphere.
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 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.
A vertex figure for an n-polytope is an (n-1)-polytope. For example, a vertex figure for a polyhedron is a polygon figure, and the vertex figure for a polychoron is a polyhedron figure.
By considering the connectivity of these neighboring vertices an (n-1)-polytope, the vertex figure, can be constructed for each vertex of a polytope:
Vertex figures are the most useful for uniform polytopes because one vertex figure can imply the entire polytope.
For polyhedra, the vertex figure can be represented by a vertex configuration notation, by listing the faces in sequence around a vertex. For example 3.4.4.4 is a vertex with one triangle and 3 squares, and it represents the rhombicuboctahedron.
If the polytope is vertex-transitive, the vertex figure will exist in a hyperplane surface of the n-space. In general the vertex figure need not be planar.
Also nonconvex polyhedra, the vertex figure may also be nonconvex. Uniform polytopes can have either star polygon faces and vertex figures for instance.
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.
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}.
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.
The vertex figure of a truncated cubic honeycomb is a nonuniform square pyramid. One octahedron and four truncated cube meet at each vertex for form a space-filling tessellation.
| Vertex figure: A nonuniform square pyramid | Schlegel diagram |
Perspective |
| Created as a square base from an octahedron | (3.3.3.3) |
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| And four isosceles triangle sides from truncated cubes | (3.8.8) |
Related to the vertex figure, an edge figure is the vertex figure of a vertex figure.[1] Edge figures are useful for expressing relations between the elements within regular and uniform polytopes.
An edge figure will be a (n-2)-polytope, representing the arrangement of facets around a given edge. Regular and single-ringed uniform polytopes will have a single edge figure type, while in general, a uniform polytope can have as many edges as active mirrors in the construction, since each active mirror produces one edge in the fundamental domain.
Regular polytopes (and honeycombs) have a single edge figure which is also regular. For a regular polytope {p,q,r,s,...,z}, the edge figure is {r,s,...,z}.
In four dimensions, the edge figure of a 4-polytope or 3-honeycomb is a polygon representing the arrangement of a set of facets around an edge. For example, the edge figure for a regular cubic honeycomb {4,3,4} is a square, and for a regular polychoron {p,q,r} is the polygon {r}.
Less trivially, the truncated cubic honeycomb t0,1{4,3,4}, has a square pyramid vertex figure, with truncated cube and octahedron cells. Here there are 2 types of edge figures. One is a square edge figure at the apex of the pyramid. This represents the four truncated cubes around an edge. The other four edge figures are isosceles triangle on the base vertices of the pyramid. These represent the arrangement of two truncated cubes and one octahedron around the other edges.