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.

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) ($\sqrt{2}$).

If a polytope contains only triangles, then the vertex figure is directly represented by the edges of the polytope.