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In polyhedral geometry a vertex configuration is a short-hand notation for representing a vertex 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.
A vertex configuration is given as a sequence of numbers representing the number of sides of the faces going around the vertex. A a.b.c means a vertex has 3 faces around it, with a, b, and c sides.
For example 3.5.3.5 means a vertex has 4 faces, alternating triangles and pentagons. This vertex configuration defines the vertex-uniform icosidodecahedron polyhedron.
Vertex Figures
A vertex configuration can also be represented graphically as 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.
See image category: Category:Polyhedra-vf image
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 or (3.5)2.
The order is important and so 3.3.5.5 is different from 3.5.3.5. The first has two triangles followed by two pentagons.
The notation can also be considered an expansive form of the simple Schläfli symbol for regular polyhedra. {p,q} means q p-agons around each vertex. So this can be written as p.p.p... (q times). For example an icosahedron is {3,5} = 3.3.3.3.3 = 3^ 5= 35.
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. To be unique, usually the smallest face (or sequence of smallest faces) are listed first.
This notation applies to polygon tiles as well as polyhedra. A planar vertex configuration can imply a uniform tiling just like a nonplanar vertex configuration can imply a uniform polyhedron.
The notation is ambiguous for chiral forms. For example, the snub cube has a clockwise and counterclockwise form 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 5/2 sides meaning 5 vertex going around the vertex twice. For example, the nonconvex regular polyhedron small stellated dodecahedron has a vertex configuration of Schläfli symbol of {5/2,5} which expands to an explicit vertex configuration as 5/2.5/2.5/2.5/2.5/2.
The last, U75, nonconvex uniform polyhedron great dirhombicosidodecahedron has a vertex figure of (4.5/3.4.3.4.5/2.4.3/2)/2. This complex vertex figure has 8 faces that pass around the vertex twice.
Inverted polygons
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 the Star polygon notation of sides p/q as an improper fraction (greater than one), where p is the number of sides and q 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.
Face Configuration for duals
The dual polyhedron are can also be listed by this notation, but prefixed by a V. See face configuration.
The faces of semiregular polyhedral duals are not regular polygons, but edges vary in length in relation regular polygons in the dual. For example, you can tell a face configuration of V3.4.3.4 represents a rhombus face since every edge is a V3-V4 type, and V3.4.5.4 will be a kite with two types of edges: V3-V4 and V4-V5.
Notation used in articles
- Vertex Configuration
- Face configuration
- Catalan solid - Archimedean duals
- Bipyramid - prism duals
- Trapezohedron - antiprism duals
Reference
"Vertex configuration"