# Symmetric Circle Configurations from Regular Skeletal Polyhedra
## 1 Introduction  
Configurations of points and lines or other geometric objects have been studied extensively in the literature (see [PERSON] [1] and [PERSON] [2]). Because of their conceptual simplicity, the configurations occur in many different mathematical contexts and have been investigated from many different points of view--geometric, combinatorial, algebraic, topological, and others. Most work deals with the classical geometric case of point-line configurations in the plane or in space, or with combinatorial or topological configurations being viewed as incidence structures; for example, see [PERSON] [3], [PERSON], [PERSON], [PERSON] [4], and [PERSON] [5].  
In this article, we explore point-circle configurations in ordinary Euclidean 3-space associated with regular skeletal polyhedra or related discrete structures. The study of skeletal polyhedra was initiated in [PERSON] [6], although the term \"skeletal polyhedron\" was not used at that time. These skeletal polyhedra (or more generally, skeletal complexes) are finite or infinite periodic geometric graphs, which are equipped with additional polyhedral super-structure imposed by its polygon faces and are allowed to be finite (planar or skew) or infinite (linear, zigzag, or helical) polygons. A polyhedron is regular if its geometric symmetry group is transitive on the flags (incident vertex-edge-face triples). There are 48 regular skeletal polyhedra comprising 18 finite polyhedra (the five Platonic solids and four Kepler-Poinsot polyhedra, and their Petrie duals), and 30 infinite aperiodic; for the full classification, see [PERSON] and [PERSON] [7] (Ch. 7E) or [PERSON] [6], [PERSON] [8], [PERSON] [9]. These relatively new regular figures exhibit fascinating geometric, combinatorial, and algebraic properties and have applications in crystal chemistry [10].  
The regular skeletal polyhedra and complexes give rise to highly symmetric point-circle configurations in \(\mathbb{E}^{3}\). In this paper, we investigate the geometric properties of point-circle configurations, which are derived from a polyhedron by what we call the E-construction. Our approach and construction methods were inspired by the work inGevay and [PERSON] [11] and [PERSON], [PERSON], [PERSON], and [PERSON] [12], specifically by the V-construction of [11]. As we shall see, when applied to regular skeletal polyhedra with planar vertex figures, the V-construction also leads to highly symmetric point-circle configurations. The point-circle configurations obtained by our methods inherit the symmetry of the underlying polyhedron or complex and, in most cases, are point-circle transitive under their symmetry group.  
The paper is organized as follows. In Section 2, we begin by reviewing the basic notions for configurations. In Section 3, E-construction is introduced in both combinatorial and geometric contexts. The skeletal regular polyhedra and their basic properties and classification are described in Section 4. In Section 5, we discuss the point-circle configurations associated with these regular polyhedra and investigate in detail the configurations derived via the E-construction from the polyhedra in the four possible classes: finite, planar, blended, and pure polyhedra. We provide attractive illustrations (generated using Mathematica [13]). In Section 5.4, we explain why in spite of their extremely high symmetry, chiral polyhedra will never lead to point-circle configurations via the E-construction or the \(V\)-construction. Section 6 concerns V-constructed circle configurations from regular polyhedra. The final Section 7 summarizes concluding remarks and proposes two open research problems.