vertexconfiguration

Vertex configuration is a concise descriptor used in geometry to name the arrangement of faces that meet at a vertex in a polyhedron or a tiling. It records the cyclic sequence of the face sizes around that vertex, with each number representing the number of sides of a face. The order around the vertex matters, and for uniform structures the same configuration occurs at every vertex.

For example, in a regular tetrahedron three triangular faces meet at each vertex, giving the configuration 3.3.3. A cube or a square tiling has three square faces meeting at a vertex, described as 4.4.4. The dodecahedron has three pentagons around each vertex, 5.5.5. The soccer ball, a truncated icosahedron, has a vertex configuration of 5.6.6, meaning a pentagon and two hexagons meet at each vertex in that order. Plane tilings illustrate other configurations as well, such as 3.12.12, where a triangle and two dodecagons meet at each vertex.

Vertex configurations are commonly used to classify regular and semi-regular (Archimedean) solids and uniform tilings, helping to distinguish local structure without specifying the entire global shape. They are related to, but distinct from, the concept of a vertex figure, which describes the local geometry around a vertex more geometrically, often by considering a small cut or slice near the vertex. While the configuration provides a compact shorthand, different global realizations can share the same local vertex configuration.