Vertex configuration

The icosidodecahedron's vertex configeration is ${\displaystyle 3.5.3.5}$ or ${\displaystyle (3.5)^{2}}$.

In geometry, a vertex configuration is a shorthand notation for representing a polyhedron or tiling as the sequence of faces around a vertex. It has variously been called a vertex description,^[1][2][3] vertex type,^[4][5] vertex symbol,^[6][7] vertex arrangement,^[8] vertex pattern,^[9] face-vector,^[10] vertex sequence.^[11] It is also called a Cundy and Rollett symbol for its usage for the Archimedean solids in their 1952 book Mathematical Models.^[12][13][14] 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.)

For example, "3.5.3.5" 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 3.5.3.5 is the same as 5.3.5.3. The order is important, so 3.3.5.5 is different from 3.5.3.5 (the first has two triangles followed by two pentagons). Repeated elements can be collected as exponents so this example is also represented as (3.5)₂.