Ingeometry, avertex configuration is a shorthand notation for representing apolyhedron ortiling as the sequence offaces around avertex. It has variously been called avertex 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 aCundy and Rollett symbol for its usage for theArchimedean solids in their 1952 bookMathematical Models.[12][13][14] Foruniform 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, alternatingtriangles andpentagons. This vertex configuration defines thevertex-transitiveicosidodecahedron. The notation is cyclic and therefore is equivalent with different starting points, so3.5.3.5 is the same as5.3.5.3. The order is important, so3.3.5.5 is different from3.5.3.5 (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.
 {3,3} = 33 Defect 180° |  {3,4} = 34 Defect 120° |  {3,5} = 35 Defect 60° |  {3,6} = 36 |
 {4,3} Defect 90° |  {4,4} = 44 |  {5,3} = 53 Defect 36° |  {6,3} = 63 |
| A vertex needs at least 3 faces, and anangle defect. A 0° angle defect will fill the Euclidean plane with regular tiling. ByDescartes' theorem, the number of vertices is 720°/defect (4π radians/defect). | |||
A vertex configuration is written as one or more numbers separated by either dots or commas. Each number represents the number of sides in each face that meets at each vertex.[15] Anicosidodecahedron is denoted as because there are four faces at each vertex, alternating betweentriangles (with 3 sides) andpentagons (with 5 sides). This can also be written as.
The vertex configuration can also be considered an expansive form of the simpleSchläfli symbol forregular polyhedra. The Schläfli notation has the form, where is the number of sides in each face and is the number of faces that meet at each vertex. Hence, the Schläfli notation can be written as (where appears times), or simply.[15]
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 forchiral forms. For example, thesnub cube has clockwise and counterclockwise forms which are identical across mirror images. Both have a 3.3.3.3.4 vertex configuration.
The notation also applies for nonconvex regular faces, thestar polygons. For example, apentagram 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. Thesmall stellated dodecahedron has theSchläfli 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. Thegreat stellated dodecahedron, {5/2,3} has a triangular vertex figure and configuration (5/2.5/2.5/2) or (5/2)3. Thegreat dodecahedron, {5,5/2} has a pentagrammic vertex figure, withvertex configuration is (5.5.5.5.5)/2 or (55)/2. Agreat icosahedron, {3,5/2} also has a pentagrammic vertex figure, with vertex configuration (3.3.3.3.3)/2 or (35)/2.
 |  |  |  |  |
| {5/2,5} = (5/2)5 | {5/2,3} = (5/2)3 | 34.5/2 | 34.5/3 | (34.5/2)/2 |
|---|---|---|---|---|
 |  |  |  |  |
| {5,5/2} = (55)/2 | {3,5/2} = (35)/2 | V.34.5/2 | V34.5/3 | V(34.5/2)/2 |
Faces on a vertex figure are considered to progress in one direction. Some uniform polyhedra have vertex figures with inversions where the faces progress retrograde. A vertex figure represents this in thestar polygon notation of sidesp/q such thatp<2q, wherep is the number of sides andq the number of turns around a circle. For example, "3/2" means a triangle that has vertices that go around twice, which is the same as backwards once. Similarly "5/3" is a backwards pentagram 5/2.
Semiregular polyhedra have vertex configurations with positiveangle defect.
NOTE: The vertex figure can represent a regular or semiregular tiling on the plane if its defect is zero. It can represent a tiling of the hyperbolic plane if its defect is negative.
For uniform polyhedra, the angle defect can be used to compute the number of vertices. Descartes' theorem states that all the angle defects in a topological sphere must sum to 4π radians or720 degrees.
Since uniform polyhedra have all identical vertices, this relation allows us to compute the number of vertices, which is 4π/defect or720/defect.
Example: Atruncated cube 3.8.8 has an angle defect of 30 degrees. Therefore, it has720/30 = 24 vertices.
In particular it follows that {a,b} has 4 / (2 −b(1 − 2/a)) vertices.
Every enumerated vertex configuration potentially uniquely defines a semiregular polyhedron. However, not all configurations are possible.
Topological requirements limit existence. Specificallyp.q.r implies that ap-gon is surrounded by alternatingq-gons andr-gons, so eitherp is even orq equalsr. Similarlyq is even orp equalsr, andr is even orp equalsq. Therefore, potentially possible triples are 3.3.3, 3.4.4, 3.6.6, 3.8.8, 3.10.10, 3.12.12, 4.4.n (for anyn>2), 4.6.6, 4.6.8, 4.6.10, 4.6.12, 4.8.8, 5.5.5, 5.6.6, 6.6.6. In fact, all these configurations with three faces meeting at each vertex turn out to exist.
The number in parentheses is the number of vertices, determined by the angle defect.
The uniform dual orCatalan solids, including thebipyramids andtrapezohedra, arevertically-regular (face-transitive) and so they can be identified by a similar notation which is sometimes calledface configuration.[16] Cundy and Rollett prefixed these dual symbols by aV. In contrast,Tilings and patterns uses square brackets around the symbol for isohedral tilings.
This notation represents a sequential count of the number of faces that exist at eachvertex around aface.[12] For example, V3.4.3.4 or V(3.4)2 represents therhombic dodecahedron which is face-transitive: every face is arhombus, and alternating vertices of the rhombus contain 3 or 4 faces each.