# Symmetric Circle Configurations from Regular Skeletal Polyhedra
## 3 The E-Construction
Note that if the point \(v\) of the configuration \((V(Q),B(Q))\) is a vertex of \(Q\) surrounded by \(q\) 2-faces of \(Q\), where none of which is a triangle, then \(v\) lies in \(2q\) blocks \(N(e)\). In fact, the \(2q\) edges \(e\) occurring here are those edges of \(Q\) two steps away from \(v\), in either direction, along the boundary of a 2-face of \(Q\) at \(v\). If triangle faces occur at \(v\), then each such face contributes only one edge to the count. For example, if \(q\) is even and triangle and non-triangle faces of \(Q\) alternate around \(v\), then the point \(v\) lies in \(3q/2\) blocks \(N(e)\). Bear in mind that we are excluding the possibility that two triangle faces meet at an edge. We are particularly interested in highly symmetric polyhedra and their respective E-configurations. In this case, the configurations usually inherit high symmetry. An abstract polyhedron \(Q\) is _regular_ if its (_combinatorial_) _automorphism group_\(\Gamma(Q)\) is transitive on the flags. In this case, the automorphism group \(\Gamma(Q)\) is generated by three involutions \(\rho_{0},\rho_{1},\rho_{2}\) uniquely determined by the way in which they act on a chosen _base_ flag. More specifically, \(\rho_{0},\rho_{1},\) and \(\rho_{2}\) map this base flag to the flag differing from the base flag in its vertex, edge, or face, respectively; thus, \(\rho_{i}\) map the base flag to its \"i-adjacent\" flag (differing from the original flag in its \(i\)-face). If \(Q\) is of Schlafli type \(\{p,q\}\) (that is, the faces are \(p\)-gons and the vertex figures are \(q\)-gons), then these generators satisfy the standard relations  
\[\rho_{0}^{2}=\rho_{1}^{2}=\rho_{2}^{2}=(\rho_{0}\rho_{1})^{p}=(\rho_{1}\rho_{2 })^{q}=(\rho_{0}\rho_{2})^{2}=1,\]  
but in general, other independent relations too. Observe that, in a natural way, the group of the base face of \(Q\) is \(\langle\rho_{0},\rho_{1}\rangle\), while that of the (base) vertex figure is \(\langle\rho_{1},\rho_{2}\rangle\). For an abstract regular polyhedron \(Q\), the stabilizer of the base edge, \(e\) (say), leaves \(N(e)\) invariant and is given by the subgroup \(\langle\rho_{0},\rho_{2}\rangle\) isomorphic to \(C_{2}\times C_{2}\). In the notation of Figure 3, the generators \(\rho_{0}\) and \(\rho_{2}\) act on \(N(e)\) like the permutations  
\[(v_{1}\,v_{2})(v_{3}\,v_{4}),\ (v_{1}\,v_{3})(v_{2}\,v_{4}),\]  
respectively. In particular, \(\langle\rho_{0},\rho_{2}\rangle\) acts transitively on the four points \(v_{1},\ldots,v_{4}\). Moreover, as \(Q\) is regular, the configuration \((V(Q),B(Q))\) is point-block transitive under the automorphism group of \(Q\). In fact, in this case \(\Gamma(Q)\) acts transitively on the edges of \(Q\) and thus on the blocks in \(B(Q)\), and the stabilizer of an edge in \(Q\) also stabilizes the corresponding block in \(B(Q)\) and acts transitively on the four points of this block. If \(Q\) is of type \(\{p,q\}\) (with \(p\
eq 3\)), then each point of \((V(Q),B(Q))\) lies in \(2q\) blocks. An abstract polyhedron \(Q\) is called (combinatorially) _chiral_ if \(\Gamma(Q)\) has two orbits on the flags, such that adjacent flags are in distinct orbits. If \(Q\) is chiral, then each point also lies in \(2q\) blocks if \(Q\) has type \(\{p,q\}\) (with \(p\
eq 3\)), but now an edge stabilizer in \(\Gamma(Q)\) has two orbits on the points of the corresponding block. A _Petrie polygon_ of an abstract regular polyhedron \(Q\) is a path along edges such that any two, but no three, successive edges belong to a 2-face of \(Q\). Any two Petrie polygons of an abstract regular polyhedron \(Q\) are equivalent under \(\Gamma(Q)\). The _Petrie dual_ of an abstract regular polyhedron \(Q\) has the same vertices and edges as \(Q\), but has as its 2-faces the Petrie polygons of \(Q\). This is again a polyhedron, denoted \(Q^{\pi}\), if and only if a Petrie polygon visits any vertex of \(Q\) at most once (see [7] (Lemma 7B3)). In this case \(Q^{\pi}\) is an abstract regular polyhedron. The following lemma follows directly from the definition of the Petrie dual.