Vertex configuration

[15] It is also called a Cundy and Rollett symbol for its usage for the Archimedean solids in their 1952 book Mathematical Models.

The notation can also be considered an expansive form of the simple Schläfli symbol for regular polyhedra.

For example, the snub cube has clockwise and counterclockwise forms which are identical across mirror images.

The notation also applies for nonconvex regular faces, the star polygons.

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.

Some uniform polyhedra have vertex figures with inversions where the faces progress retrograde.

Semiregular polyhedra have vertex configurations with positive angle defect.

NOTE: The vertex figure can represent a regular or semiregular tiling on the plane if its defect is zero.

Descartes' theorem states that all the angle defects in a topological sphere must sum to 4π radians or 720 degrees.

Every enumerated vertex configuration potentially uniquely defines a semiregular polyhedron.

The uniform dual or Catalan solids, including the bipyramids and trapezohedra, are vertically-regular (face-transitive) and so they can be identified by a similar notation which is sometimes called face configuration.