Scheitelstücke

Scheitelstücke, in geometry often translated as vertex figures, describe the local structure of a polyhedron around a vertex. For a chosen vertex V, a small plane (or equivalently a small sphere centered at V) cuts off the vertex and intersects the edges that emanate from V. The polygon formed by these intersections is the Scheitelstück. Its number of sides equals the number of edges (or faces) meeting at V, and its shape reflects how these faces are arranged around the vertex.

Properties: In regular polyhedra all Scheitelstücke are congruent and are regular polygons; the number of sides of the Scheitelstück equals the vertex degree. Examples: the tetrahedron and the cube have three edges meeting at each vertex, so the Scheitelstücke are equilateral triangles. The octahedron has four incident edges per vertex, yielding square Scheitelstücke. The dodecahedron has three incident edges per vertex (triangle vertex figure), and the icosahedron has five (pentagonal vertex figure).

Applications: Vertex figures are used to classify polyhedra and uniform tilings, to infer the Schläfli symbol, and to study angular defects and curvature in polyhedral geometry. They also provide a convenient local picture of the arrangement of faces around each vertex and help in understanding how complex solids are assembled from simpler facial units.

Terminology: In German geometric literature, Scheitelstück is commonly used for the vertex figure; the plural Scheitelstücke refers to the vertex figures at the various vertices of a polyhedron. The concept can be generalized to higher-dimensional polytopes by considering the link of a vertex.