The vertex figure, or verf of an n-D polytope is intuitively the (n-1)-D shape exposed when you slice off a corner of said polytope.
The vertex figure can be constructed by considering points on all adjacent edges some arbitrary distance away from a vertex (usually taken to be the midpoint) of the polytope and connecting them such that all edges of the figure are subsets of a face, the faces of the figure are subsets of a cell, and so on. Generally the n-D subfacets of the vertex figure of a polytope are embedded in an (n+1)-D subfacets of the polytope.
The vertex figure can alternatively be constructed by taking a small hypersphere centered at a vertex. The vertex figure can be defined the spherical polygon inside the polytope.
For any regular polytope or tiling with Schläfli symbol , the vertex figure of the polytope has symbol .
The operation that replaces all vertices of a polytope with the vertex figure of said polytope is called truncation. The facets of the truncated polytope also have their vertices replaced with their vertex figures.
The vertex figure of a polytope's vertex figure is called the edge figure, the vertex figure of the edge figure is called the face figure, and so on.