Vertex Symbol | polyhedra.mathmos.net

Each regular or semi-regular polyhedron has identical sequences of regular polygons surrounding each vertex. This sequence identifies the polyhedron, and can be used to assign a vertex symbol to it. The symbol is a sequence of numbers that shows the sequence of polygons about each vertex. Each number represents a face that is a regular polygon with that many sides.

So 3.4.5.4 means each vertex of the polyhedron is surrounded by an equilateral triangle, a square, a regular pentagon, and a second square, in that order. For polyhedra where a particular polygon occurs consecutively twice or more, indices can be used to simplify the symbol. Hence 3.4.4.4 is equivalent to 3.4³, and 5.5.5 is equivalent to 5³.

Star Faces

Fractional numbers, p/q (with p and q coprime and p > 2q), may also be seen in vertex symbols. These represent so-called star polygons. Such a polygon has p vertices and is non-convex. The vertices are joined by sides in order, but jumping q ahead each time. This has the effect that the internal angle formula for an n-gon; φ = π (1 - 2/n) still holds with n = p/q. Common star polygons are 5/2 (the pentagram), 8/3, and 10/3.

Star Vertices

In a similar way to how we can have star faces, star vertices are also possible. Image a line drawn across the corner of each face meeting at the vertex. This draws out what is known as the vertex figure. It will generally be a deformed polygon. Star vertices are classified by the symbol of their corresponding vertex figure.

Following the edge of the vertex figure will result in sequence of polygons, whose numbers are written down in the usual way. The whole symbol is then raised to the power of 1/q where q is the number of times around the vertex the line goes. As for normal (non-star) vertices, repeated polygons can be written using indices, so a 5/2 star vertex where 5 triangles meet would be (3.3.3.3.3)^1/2 or more simply (3)^5/2.

Extension to Vertically Regular Solids

The dual of a regular or semi-regular polyhedron is vertically regular or semi-regular, meaning that it has fixed sequences of vertices around each face. They cannot be classified by the faces around each vertex; instead we use the vertices around each face. The symbol of the dual solid is used, but with a 'V' added to distinguish it. The numbers now represent the sequence of vertex types (i.e. how many faces meet there) around each face.