# Symmetric Circle Configurations from Regular Skeletal Polyhedra
## 3 The E-Construction  
Many geometric structures give rise to interesting circle configurations. For example, if \(Q\) is a 3-dimensional convex polyhedron (with all its vertices) inscribed in a sphere in \(\mathbb{E}^{3}\) (see [14]), then each face of \(Q\) is inscribed in a circle lying in this sphere, and the family of all such circles for faces of \(Q\) forms a circle configuration with point set given by the vertex set of \(Q\). In this configuration, a point lies on a circle if and only if the point is the vertex of the corresponding face of \(Q\). Thus, the combinatorics of the configuration is entirely  
Figure 1: A circle configuration in \(\mathbb{E}^{3}\) associated with the regular dodecahedron. determined by \(Q\). Every vertex-transitive convex polyhedron is naturally inscribed in a sphere and gives a circle configuration of this kind. Examples are the Platonic solids and Archimedean solids. Similarly, if all faces of a convex polyhedron in \(\mathbb{E}^{3}\) are regular polygons, then the family of in-circles of the faces of \(Q\) form a circle configuration whose point set is given by the edge midpoints of \(Q\). In this configuration, only two circles meet at a point. A more interesting construction involves the vertex figures of convex polytopes or other geometric figures. This is a particular instance of the V-construction discussed in [PERSON] and Pisanski [11] and [PERSON] [15]. Let \(Q\) be a convex polyhedron such that the vertices of \(Q\) adjacent to any given vertex \(v\) of \(Q\) lie in a plane and thus span a convex polygon called the vertex figure of \(Q\) at \(v\). Suppose the vertex figure at each vertex of \(Q\) is inscribed in a circle. Then the family of all such circles for vertices of \(Q\) forms a circle configuration whose underlying point set is the vertex set of \(Q\). For example, every simple convex polyhedron (with only 3-valent vertices) gives rise to such a configuration. In Section 6, we revisit the V-construction in the context of regular skeletal polyhedra and derive highly symmetric finite or infinite circle configurations. This article primarily investigates point-circle configurations obtained by what we call the _E-construction_. Here, \"E\" stands for \"edge\". At the combinatorial level, the E-construction proceeds from a combinatorial object such as a convex or abstract polyhedron, a map on a surface, or just a graph (with certain properties) and derives from it a combinatorial configuration whose point set is the vertex set of the original object and whose blocks are determined by the edges of the original object and consist of the vertices in \"edge-neighborhoods\". This combinatorial configuration may or may not be realizable by a circle configuration. The E-construction works in fairly general contexts, and variations of the construction are possible as well. We begin by explaining the E-construction for convex polyhedra, and we explore other kinds of polyhedra after they have been introduced below. Suppose \(Q\) is a convex polyhedron in \(\mathbb{E}^{3}\) with vertex set \(V(Q)\) and edge set \(E(Q)\). For an edge \(e\) of \(Q\), denote by \(N(e)\) the set of vertices of \(Q\) which are adjacent to endpoints of \(e\) and lie in a face of \(Q\) containing \(e\), but are not vertices of \(e\). We call \(N(e)\) the _edge neighborhood_ of \(e\), and let \(B(Q)\) denote the family of these edge neighborhoods,  
\[B(Q):=\{N(e)|e\in E(Q)\}. \tag{1}\]  
In Figure 2, the two faces \(F_{1}\) and \(F_{2}\) of \(Q\) meet at edge \(e\), and \(N(e)=\{v_{1},\cdots,v_{4}\}\). As every edge of \(Q\) lies in two faces of \(Q\), each set \(N(e)\) contains 2, 3, or 4 vertices, according to whether both faces at \(e\) are triangles, one face at \(e\) is a triangle, or no face at \(e\) is a triangle. There are instances of polytopes in which two edges \(e\) and \(e^{\prime}\) can share the same edge neighborhood, that is, \(N(e^{\prime})=N(e)\); for example, each pair of antipodal edges of the 3-cube has this property.