# Symmetric Circle Configurations from Regular Skeletal Polyhedra
## 3 The E-Construction
However, in almost all cases we are interested in, the edge neighborhoods \(N(e)\) contain 3 or 4 vertices and uniquely determine the respective edges, \(e\). In these cases \((V(Q),B(Q))\) gives a combinatorial configuration referred to as the _combinatorial E-configuration_ of \(Q\). If we adopted a broader definition of combinatorial configuration and allowed blocks to be counted with multiplicity, then \((V(Q),B(Q))\) would also be a combinatorial configuration in the cases where edge neighborhoods do not uniquely determine the respective edge. We are particularly interested in the case when no two triangular faces of \(Q\) share an edge (so that each edge neighborhood has at least three points), and the points in \(N(e)\) are coplanar and lie on a circle \(c(e)\) for each edge \(e\) of \(Q\). Then, up to allowing a multiplicity of circles, this gives a point-circle configuration \(\mathcal{C}(Q):=(V(Q),C(Q))\), with  
\[C(Q):=\{c(e)|e\in E(Q)\}. \tag{2}\]  
It is straightforward to generalize the E-construction from convex polyhedra to abstract polyhedra (see [7]). An _abstract polyhedron_ is a (finite or infinite) partially ordered set \(Q\) with elements of _rank_\(j\) for \(j=-1\), \(0\), \(1\), \(2\), or \(3\), called _\(j\)-faces_ of \(Q\); for \(j=0\), \(1\), or \(2\), the \(j\)-faces are also called vertices, edges, and faces (\(2\)-faces), respectively. The faces of ranks \(-1\) and \(3\) are unique and are minimal and maximal, respectively; that is, if these faces are \(F_{-1}\) and \(F_{3}\), respectively, then \(F_{-1}\leq F\leq F_{3}\) for all \(F\) in \(Q\). The faces \(F_{-1}\) and \(F_{3}\) are called the _improper_ faces of \(Q\). (For convex polyhedra, the \((-1)\)-face \(F_{-1}\) is the empty set, \(\emptyset\), and the \(3\)-face \(F_{3}\) is the polyhedron itself. In discussing polyhedra, we will usually suppress the improper faces of ranks \(-1\) and \(3\)). A _flag_ of \(Q\) is an incident triple consisting of a vertex, an edge, and a face. Further, \(Q\) is _strongly flag-connected_, meaning that any two flags \(\Phi\) and \(\Psi\) of \(Q\) can be joined by a sequence of flags \(\Phi=\Phi_{0},\Phi_{1},\ldots,\Phi_{k}=\Psi\), where \(\Phi_{i-1}\) and \(\Phi_{i}\) are _adjacent_ (differ by one face), and \(\Phi\cap\Psi\subseteq\Phi_{i}\) for each \(i\). Finally, if \(F\) and \(G\) are a \((j-1)\)-face and a \((j+1)\)-face with \(F<G\) and \(0\leq j\leq 2\), then there are exactly _two_\(j\)-faces \(H\) such that \(F<H<G\). Thus, edges have two vertices, and edges lie in two faces. We usually identify a face \(F\) of \(Q\) with \(\{G\mid G\leq F\}\) and note that this partially ordered set is isomorphic to (the face lattice of) a convex polygon or the apeirogon, again ignoring improper faces. Similarly, if \(F\) is a vertex, then \(\{G\mid F<G\}\), called the _vertex figure_ at \(F\), is also isomorphic to a convex polygon or the apeirogon. Note that an abstract polyhedron with finite faces and vertex figures gives rise to a map on a surface, and it is often convenient to identify the polyhedron with this map. The E-construction for abstract polyhedra works in much the same way as for convex polytopes. Suppose \(Q\) is an abstract polyhedron with vertex set \(V(Q)\) and edge set \(E(Q)\), and no two triangular faces of \(Q\) share an edge. For an edge \(e\) of \(Q\), we let \(N(e)\) denote the _edge neighborhood_ of \(e\), that is, the set of vertices of \(Q\) which are adjacent to a vertex of \(e\) and lie in a face of \(Q\) at edge \(e\), but are not vertices of \(e\); see again Figure 2. As before, we let \(B(Q)\) denote the family of the edge neighborhoods,  
\[B(Q):=\{N(e)\mid e\in E(Q)\}.\]  
Then, with the same reservations with regards to multiplicity of blocks as in the case of convex polyhedra, \((V(Q),B(Q))\) is a combinatorial configuration referred to as the _combinatorial E-configuration_ of \(Q\). Figure 2: The edge neighborhood \(N(e)\) of edge \(e\), which consists of \(v_{1},\ldots,v_{4}\).