Vertex figure

The vertex figure of a polytope, often shortened as verf, is roughly speaking, the section exposed when shaving off a vertex. As such it is the sectioning facet underneath a 0-dimensional element of the polytope. A vertex figure of an n-dimensional polytope is always (n–1)-dimensional. For example, a polyhedron's vertex figure is a polygon, a polychoron's vertex figure is a polyhedron, and so on.

An isogonal polytope has only one vertex figure type, however, not every polytope with one vertex figure type is isogonal. To be isogonal, vertices need to be symmetrically equivalent. The elongated square gyrobicupola is an example of a non-isogonal polytope with one vertex figure type.

When denoting vertex figures, numbers can be used, each representing an edge length of 2cos(π/n). This comes from the fact that a regular n-gon of edge length 1 has a vertex figure of a dyad with that edge length. For example, the vertex figure of a small rhombicuboctahedron is denoted 3.4.4.4, representing an isosceles trapezoid of side lengths 2cos(π/3) (1), 2cos(π/4) ($\sqrt{2}$), 2cos(π/4) ($\sqrt{2}$), and  2cos(π/4) ($\sqrt{2}$).

For regular-faced polyhedra when numbers are used to represent the vertex figures the number correspond to the n-gon that is a face around that vertex and the order the numbers are listed in correspond to which faces share an edge. For example, the vertex figure of a small rhombicuboctahedron is denoted 3.4.4.4 which shows that each vertex has one triangle face and three square faces. Furthermore the augmented hexagonal prism has 4 vertices with vertex figures 3.3.4.6 whereas the augmented truncated tetrahedron has 6 vertices with vertex figure 3.4.3.6 these are different because in the augmented hexagonal prism the triangular faces share an edge but in the augmented truncated tetrahedron they do not.

If a polytope contains only triangles, then the vertex figure is directly represented by the edges of the polytope, although there may be extra edges that are not part of the vertex figure, such as the case for the 7-2 step prism.

Contrary to popular belief, vertex figures do not uniquely identify a scaliform polytope. For instance, the small rhombihexahedral prism and the chasmic cuboctachoron have exactly the same verf, but are different polychora. That said, these cases are rare, and most of the time a verf suffices to build a polytope.