In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way.
One kind of vertex figure is the polygon resulting when, taking a single vertex of a polyhedron, lines are drawn between the midpoints of all the edges through the vertex. If we slice through the corner following these lines, the vertex figure is exposed.
Taking the cube as an example, there are three edges through any one vertex. Three lines can be drawn between their midpoints, and the vertex figure is therefore a triangle. As the angles of the three faces meeting at a vertex are equal, the triangle is equilateral.
Vertex figures can be generalised to higher dimensional polytopes, simply by noting that the vertex figure of an N-dimensional polytope is of dimension N-1. Thus, the vertex figure of both the hypercube and the 4-simplex are tetrahedra.
The vertex figure of a regular polytope is itself regular, and this is one of the two rules needed to categorise a regular polytope, the other being that it must have regular cells. This is easiest to see when writing Schläfli symbols describing the polytope, as for a polytope described by
- {j,k,...,x,y},
both
- {j,k,...,x}
and
- {k,...,x,y}
must be regular polytopes.
Various kinds of vertex figure have been described, for various purposes. Rather than take the midpoint of each edge, one may wish to take points a unit distance along each edge, or even points the whole way along each edge, i.e. the adjacent vertices, and draw lines between those.
For regular and uniform (semi-regular) polytopes, these types of vertex figure are all flat, but for irregular polytopes such as toroids they are often skew.
