# Symmetric Circle Configurations from Regular Skeletal Polyhedra
## 4 Skeletal Polyhedra
For a regular polyhedron \(Q\) in \(\mathbb{E}^{3}\), the symmetry group \(G(Q)\) is generated by three geometric reflections \(r_{0},r_{1},r_{2}\) in points, lines, or planes. Suppose \(\{v,e,f\}\) is a fixed flag of \(Q\), called the _base_ flag of \(Q\). Then the generators of \(G(Q)\) can be chosen in such a way that \(r_{0}\), \(r_{1}\), and \(r_{2}\) map the base flag to the flag differing in \(v\), \(e\), or \(f\), respectively; thus, the generator \(r_{j}\) of \(G(Q)\) maps the base flag to its \(j\)-adjacent flag. The nine classical regular polyhedra (Platonic solids and Kepler-Poinsot polyhedra) are the only regular polyhedra for which all three generators \(r_{0},r_{1},r_{2}\) are plane reflections. If \(Q\) has Schlafli symbol \(\{p,q\}\) (recall that this means that the faces of \(Q\) are \(p\)-gons and the vertex figures are \(q\)-gons), then the symmetry group \(G(P)\) satisfies the Coxeter type relations  
\[r_{0}^{2}=r_{1}^{2}=r_{2}^{2}=(r_{0}r_{1})^{p}=(r_{1}r_{2})^{q}=(r_{0}r_{2})^{ 2}=1, \tag{3}\]  
but usually other independent relations too. It is convenient to describe regular skeletal polyhedra by padded Schlafli symbols obtained from the basic Schlafli symbol \(\{p,q\}\) by adding symbols that signify specific types of information (such as extra defining relations for the symmetry group). The _mirror vector_ (or _dimension vector_) of a regular polyhedron \(Q\) records the dimensions of the three reflection mirrors (fixed point sets) and is denoted as \((\dim r_{0},\dim r_{1},\dim r_{2})\). The chiral skeletal polyhedra in \(\mathbb{E}^{3}\) have also been completely classified (see [18]). Chiral polyhedra do not occur in the classical theory, and no finite, planar, or \"blended\" examples exist. There are six infinite families of chiral polyhedra (apeirohedra) in \(\mathbb{E}^{3}\), each with two or one (integral or real) parameters depending on whether the classification scheme is by congruence or similarity of polyhedra, respectively. In three families, the polyhedra have finite skew faces; in the other three, the polyhedra have helical faces. For certain exceptional parameter values, the respective polyhedra become regular and, in some cases, finite. Each family has two exceptional parameter values, and it is convenient to include the two polyhedra in the family. Then, the six resulting larger families will contain exactly two regular polyhedra, while all other polyhedra are chiral. For a chiral polyhedron \(Q\) in \(\mathbb{E}^{3}\), the symmetry group \(G(Q)\) is generated by two isometries \(s_{1},s_{2}\), which, as in the above, are determined by their action on a base flag \(\{v,e,f\}\) of \(Q\). Suppose \(v^{\prime}\) denotes the vertex of \(e\) distinct from \(v\), and \(f^{\prime}\) the face at \(e\) distinct from \(f\). Then the generator \(s_{1}\) fixes the base face \(f\) while permuting the vertices of \(f\) cyclically so that \(s_{1}(v)=v^{\prime}\); and \(s_{2}\) stabilizes \(v\) while permuting the vertices in the vertex figure at \(v\) cyclically so that \(s_{2}(f^{\prime})=f\). These generators \(s_{1},s_{2}\) are non-involutory and satisfy the relations\[s_{1}^{p}=s_{2}^{q}=(s_{1}s_{2})^{2}=1,\]  
among others, where, as in the above, \(\{p,q\}\) is the basic Schlafli type of \(Q\). Just as a geometric polyhedron \(Q\) can be viewed as a realization of its underlying abstract polyhedron, \(Q^{\prime}\) (say), there is a geometric configuration that realizes the combinatorial E-configuration \((V(Q^{\prime}),B(Q^{\prime}))\) of \(Q^{\prime}\) on \(Q\) itself. Its point set is the vertex set \(V(Q)\) of \(Q\), and its block set \(B(Q)\) consists of the subsets of \(V(Q)\) corresponding to blocks in \(B(Q)\). Thus \((V(Q),B(Q))\) realizes \((V(Q^{\prime}),B(Q^{\prime}))\) on \(Q\) itself. We say that \(Q\) satisfies the _coplanarity condition_ if the points of each block in \(B(Q)\) are coplanar.