# Symmetric Circle Configurations from Regular Skeletal Polyhedra
## 2 Basic Notions  
In this section, we briefly review the basics of combinatorial and geometrical configurations and introduce notation used later in the text. Although most combinatorial configurations we encounter can be described in terms of a set of distinct \"points\" and a set of \"blocks\" in which the blocks can be identified uniquely with subsets of the set of points, it is useful to introduce terminology that allows us to deal with more general situations. Following [2], an _incidence structure_ is a triple \((\mathcal{P},\mathcal{B},\mathcal{I})\) where \(\mathcal{P}\) and \(\mathcal{B}\) are disjoint sets whose elements are called _points_ or _blocks_, respectively, and \(\mathcal{I}\subseteq\mathcal{P}\times\mathcal{B}\) is an incidence relation. The _degree_ of a point \(p\in\mathcal{P}\) is defined to be the number of blocks incident to \(p\), and the _degree_ of a block \(b\in\mathcal{B}\) is defined to be the number of points incident to \(b\). We allow these numbers to be infinite if \(\mathcal{P}\) or \(\mathcal{B}\) are infinite sets. An incidence structure naturally determines a bipartite graph with vertex set \(\mathcal{P}\cup\mathcal{B}\) and parts \(\mathcal{P}\) and \(\mathcal{B}\), such that \(p\in\mathcal{P}\) and \(b\in\mathcal{B}\) are joined by an edge if and only if \(p\) is incident to \(b\). This graph is called the _Levi graph_ of the incidence structure [2]. A _combinatorial configuration_\((\mathcal{P},\mathcal{B})\) is an incidence structure, \((\mathcal{P},\mathcal{B},\mathcal{I})\), for which \(\mathcal{B}\) is a collection of non-empty subsets of \(\mathcal{P}\) and the incidence relation \(\mathcal{I}\) is given by standard inclusion. Thus, for point \(p\in\mathcal{P}\) and block \(b\in\mathcal{B}\), point \(p\) and block \(b\) are \"incident\" if and only if \(p\in b\). If \(|\mathcal{P}|=m\), \(|\mathcal{B}|=n\), then \((\mathcal{P},\mathcal{B})\) is called an \((m_{r},n_{s})\)-_configuration_ if every point of \(P\) lies in exactly \(r\) blocks and every block in \(B\) contains exactly \(s\) points of \(P\). We allow the numbers \(m,n,r,s\) to be infinite if \(\mathcal{P}\) is infinite. An _automorphism_ of a combinatorial configuration \((\mathcal{P},\mathcal{B})\) is a bijection \(\gamma:\mathcal{P}\cup\mathcal{B}\to\mathcal{P}\cup\mathcal{B}\) such \(\gamma(\mathcal{P})=\mathcal{P}\), \(\gamma(\mathcal{B})=\mathcal{B}\), and for all \(p\in P\), \(b\in B\) we have \(p\in b\) if and only if \(\gamma(p)\in\gamma(b)\). The _automorphism group_ of \((\mathcal{P},\mathcal{B})\) consists of all of its automorphisms. Geometric configurations are essentially geometrically realized combinatorial configurations. Here, \"realization\" will mean that the elements of \(\mathcal{P}\) are represented by actual points in a Euclidean space, in our case of dimension 3, while the blocks are represented by lines, planes, circles, ellipses, spheres, ellipsoids, etc., along with standard geometric incidence (inclusion) as underlying incidence structure. We are primarily interested in representations of the blocks by circles. While our terminology largely aligns with that in [2], there will be instances where we use different terms. A _point-circle configuration_\(\mathcal{C}=(P,C)\) in \(\mathbb{E}^{3}\) consists of a set of distinct points \(P\) in \(\mathbb{E}^{3}\) and a set of distinct circles \(C\) in \(\mathbb{E}^{3}\) such that every point of \(P\) lies on a circle of \(C\), and every circle of \(C\) contains a point of \(P\). As for combinatorial configurations, if \(|P|=m\), \(|C|=n\), then \(\mathcal{C}\) is said to be an \((m_{r},n_{s})\)-_configuration_ if every point of \(P\) lies on \(r\) circles and every circle of \(C\) contains \(s\) points of \(P\). Again \(m,n,r,s\) are allowed to be infinite. Most research on circle configurations concerns planar configurations.