# Symmetric Circle Configurations from Regular Skeletal Polyhedra
## 4 Skeletal Polyhedra  
In this section, we briefly review the regular and chiral skeletal polyhedra in \(\mathbb{E}^{3}\) (see [7] (Ch. 7E) and [18; 19]). Informally, a skeletal polyhedron in \(\mathbb{E}^{3}\) is a collection of vertices, edges, and finite or infinite polygons, all assembled in a manner typical for geometric polyhedra. By a _finite polygon_, or an \(n\)-_gon_, we mean a figure formed by distinct points \(v_{1},\ldots,v_{n}\), called the _vertices_ of the polygon, along with the line segments \((v_{i},v_{i+1})\), for each \(i\), and the line segment \((v_{n},v_{1})\), called the _edges_ of the polygon. Similarly, an _infinite polygon_ is a figure consisting of infinitely distinct points, \(\ldots,v_{-2},v_{-1},v_{0},v_{1},v_{2},\ldots\), called the _vertices_ of the polygon, and of line segments \((v_{i},v_{i+1})\) for each \(i\), called the _edges_ of the polygon, such that each compact subset intersects only finitely many line segments. A _skeletal polyhedron_, or a _geometric polyhedron_, \(Q\) in \(\mathbb{E}^{3}\) similarly has three kinds of elements: vertices, edges, and faces. More precisely, \(Q\) consists of a set \(V\) of points, called the _vertices_, a set \(E\) of line segments, called the _edges_, and a set \(F\) of polygons, called the _faces_, with the following properties. The graph \((V,E)\) determined by \(V\) and \(E\) is connected; this graph is called the _edge graph_ of \(Q\). The vertex figure at the vertices of \(Q\) are cycles. Recall that the _vertex figure_ at a vertex \(v\) is a graph whose vertex set consists of the neighbors of \(v\) in \(Q\) and whose edge set is given by the line segments \((u,w)\) where \(u,w\) are neighbors of \(v\) such that \((u,v)\) and \((v,w)\) are edges of a common face of \(Q\). Further, each edge of \(Q\) lies in exactly two faces of \(Q\). Finally, \(Q\) is _discrete_, in the sense that each compact set meets only finitely many faces of \(Q\). Given a polyhedron \(Q\), the _symmetry group_\(G=G(Q)\) of \(Q\) consists of all isometries of the ambient space, \(\mathbb{E}^{3}\), or the affine hull of \(Q\) if \(Q\) is planar, which leave \(Q\) invariant. By a _flag_ of \(Q\) we mean an incident triple consisting of a vertex, an edge, and a face of \(Q\). Two flags of \(Q\) are called _0-adjacent_, _1-adjacent_, or _2-adjacent_, respectively, if they differ in their their vertices, edges, or faces. Two flags of \(Q\) are said to be _adjacent_ if they are \(j\)-adjacent for some \(j=0,1,2\). A skeletal polyhedron \(Q\) is called (geometrically) _regular_ if \(G(Q)\) is transitive on the flags of \(Q\). A skeletal polyhedron \(Q\) is said to be (geometrically) _chiral_ if \(G(Q)\) acts with two orbits on the flags of \(Q\) such that any two adjacent flags are in different orbits. The regular skeletal polyhedra have been completely classified and are also known as the _Grinbaum-Dress polyhedra_ (see [PERSON] [6], [PERSON] [8], McMullen & [PERSON] [7; 20] (Ch. 7E), and [PERSON] [9]. Using the classification scheme of [7] (Ch. 7E), the altogether 48 regular polyhedra naturally fall into the following classes. There are eighteen finite polyhedra, namely the five Platonic solids and the four Kepler-Poinsot polyhedra (see [21]), which are also referred to as the _classical regular polyhedra_, and their Petrie duals, as well as six planar polyhedra, namely the regular tessellations by triangles, squares, and hexagons, and their Petrie-duals. There also are twelve \"blended\" polyhedra, denoted \(P\#L\), obtained by blending the six planar polyhedra \(P\) with either a line segment or a linear apeirogon \(L\) positioned on a line perpendicular to the plane of the planar polyhedron. Strictly speaking, the geometry of the blended polyhedra depends on the ratio of the edge lengths of the planar and the linear components, but if these ratios are discounted, there are twelve polyhedra. Finally, there are twelve \"pure\" regular polyhedra. The geometry of some regular skeletal polyhedra is somewhat more involved.