Vertex configuration
In geometry, a vertex configuration is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron. (Chiral polyhedra exist in mirror-image pairs with the same vertex configuration.) A vertex configuration is given as a sequence of numbers representing the number of sides of the faces going around the vertex. The notation "" describes a vertex that has 3 faces around it, faces with , , and sides. For example, "" indicates a vertex belonging to 4 faces, alternating triangles and pentagons. This vertex configuration defines the vertex-transitive icosidodecahedron. The notation is cyclic and therefore is equivalent with different starting points, so is the same as The order is important, so is different from (the first has two triangles followed by two pentagons). Repeated elements can be collected as exponents so this example is also represented as . It has variously been called a vertex description, vertex type, vertex symbol, vertex arrangement, vertex pattern, face-vector. It is also called a Cundy and Rollett symbol for its usage for the Archimedean solids in their 1952 book Mathematical Models. Vertex figures A vertex configuration can also be represented as a polygonal vertex figure showing the faces around the vertex. This vertex figure has a 3-dimensional structure since the faces are not in the same plane for polyhedra, but for vertex-uniform polyhedra all the neighboring vertices are in the same plane and so this plane projection can be used to visually represent the vertex configuration. Variations and uses Different notations are used, sometimes with a comma (,) and sometimes a period (.) separator. The period operator is useful because it looks like a product and an exponent notation can be used. For example, 3.5.3.5 is sometimes written as (3.5)2. The notation can also be considered an expansive form of the simple Schläfli symbol for regular polyhedra. The Schläfli notation {p,q} means q p-gons around each vertex. So {p,q} can be written as p.p.p... (q times) or pq. For example, an icosahedron is {3,5} = 3.3.3.3.3 or 35. This notation applies to polygonal tilings as well as polyhedra. A planar vertex configuration denotes a uniform tiling just like a nonplanar vertex configuration denotes a uniform polyhedron. The notation is ambiguous for chiral forms. For example, the snub cube has clockwise and counterclockwise forms which are identical across mirror images. Both have a 3.3.3.3.4 vertex configuration. Star polygons The notation also applies for nonconvex regular faces, the star polygons. For example, a pentagram has the symbol {5/2}, meaning it has 5 sides going around the centre twice. For example, there are 4 regular star polyhedra with regular polygon or star polygon vertex figures. The small stellated dodecahedron has the Schläfli symbol of {5/2,5} which expands to an explicit vertex configuration 5/2.5/2.5/2.5/2.5/2 or combined as (5/2)5. The great stellated dodecahedron, {5/2,3} has a triangular vertex figure and configuration