Vertex figure
In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions – theme and variations
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.
As a flat slice
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 most common 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, as does Coxeter (1948). For uniform polyhedra the Dorman Luke construction cuts each connected edge at its midpoint. Other authors make the cut through the vertex at the other end of each edge.