vertexfigures

A vertex figure is a geometric construction related to polyhedra and polytopes. For a given vertex of a polyhedron, its vertex figure is the polygon formed by connecting the midpoints of the edges that meet at that vertex. Alternatively, it can be viewed as the intersection of a plane slicing through the vertex with all the incident edges. The shape of the vertex figure provides information about the local structure around a vertex.

For a convex polyhedron, the vertex figure is always a convex polygon. The number of sides of the vertex figure is equal to the number of edges meeting at the vertex, which is also equal to the number of faces meeting at the vertex. For example, a cube has a vertex figure that is an equilateral triangle, as three edges and three square faces meet at each vertex. The vertex figure of a regular tetrahedron is also an equilateral triangle.

In the context of uniform polyhedra and tessellations, vertex figures are particularly important. A vertex figure can be used to describe the symmetry around a vertex. If all vertex figures of a polyhedron are congruent and regular polygons, the polyhedron is vertex-transitive. For regular polyhedra, the vertex figures are regular polygons. The Schlafli symbol, a notation used to describe regular polyhedra and tessellations, often incorporates information about the vertex figure. The concept can be extended to higher-dimensional polytopes, where the vertex figure becomes a (d-1)-dimensional polytope for a d-dimensional polytope.