# Symmetric Circle Configurations from Regular Skeletal Polyhedra
## 6 V-Constructed Point-Circle Configurations  
In this section, we briefly discuss the V-construction for regular skeletal polyhedra in \(\mathbb{E}^{3}\). By our discreteness assumption, each regular skeletal polyhedron has finite vertex figures, which either are planar (convex or star-) regular polygons or non-planar (skew) regular polygons. The V-construction only applies in cases where the vertex figures are planar. Given a regular skeletal polyhedron with planar vertex figures, the V-construction yields a point-circle configuration whose points are the vertices of \(Q\) and whose circles are the circumcircles of the vertex figures at the vertices of \(Q\). As a regular polyhedron \(Q\) and its Petrie dual \(Q^{\pi}\) share the same underlying edge graph, and the same vertex figures, the corresponding point-circle configurations for \(Q\) and \(Q^{\pi}\) are identical. Theorem 3.2: _Let \(Q\) be a regular polyhedron in \(\mathbb{E}^{3}\) with symmetry group \(G(Q)=\langle r_{0},r_{1},r_{2}\rangle\), and suppose \(Q\) has planar vertex figures._  
_(a) Then the V-construction applied to_ \(Q\) _yields a point-circle configuration_ \(\mathcal{C}\) _on which_ \(G(Q)\) _acts point-circle transitively, in all but two cases with a point-circle stabilizer of order_ \(2\)_._  
_(b) If_ \(\mathcal{C}\) _is the circumcircle of the vertex figure at the base vertex_ \(v\) _of_ \(Q\)_, and_ \(v^{\prime}\) _is the vertex in the base edge of_ \(Q\) _distinct from_ \(v\)_, then the stabilizer in_ \(G(Q)\) _of the point-circle pair_ \((v^{\prime},\mathcal{C})\) _of_ \(\mathcal{C}\) _is given by_ \(\langle r_{2}\rangle\) _except when_ \(Q\) _is the octahedron or its Petrie dual. In the exceptional cases, the stabilizer is given by_ \(\langle r_{2},r_{\mathcal{C}}\rangle\cong C_{2}\times C_{2}\)_, where_ \(r_{\mathcal{C}}\) _denotes the reflection in the plane through_ \(\mathcal{C}\)_._  
Proof: Let \(Q\) be of type \(\{p,q\}\). By the classification of regular skeletal polyhedra, \(q\leq 6\), and \(Q\) is finite if \(q=5\). Thus, the planar vertex figures of \(Q\) are convex \(q\)-gons, or possibly pentagrams if \(q=5\). Let \(\{v,e,f\}\) denote the base flag of \(Q\). Each point-circle pair of the circle configuration is determined by an incident vertex-edge pair of \(Q\), and this vertex-edge pair is unique except in the cases of the octahedron and its Petrie dual. As \(G(Q)\) acts transitively on the incident vertex-edge pairs of \(Q\), it follows that \(G(Q)\) also acts point-circle transitively on the V-constructed point-circle configuration. Suppose \(v^{\prime}\) is the vertex of \(e\) distinct from \(v\), and \((v^{\prime},C)\) is the point-circle pair of \(\mathcal{C}\) associated with the vertex-edge pair \((v,e)\) of \(Q\). Then the stabilizer \(\langle\rho_{2}\rangle\) of the vertex-edge pair \((v,e)\) in \(G(Q)\) also stabilizes the point-circle pair \((v^{\prime},C)\). It remains to show that \(\langle\rho_{2}\rangle\) is the full stabilizer of \((v^{\prime},C)\) in \(G(Q)\) except when \(Q\) is the octahedron or its Petrie dual. Suppose \(s\in G(Q)\) stabilizes \((v^{\prime},C)\), that is, \(s\) fixes the circle \(C\) and the point \(v^{\prime}\) on \(\mathcal{C}\). If we can show that \(s\) also fixes \(v\), then \(s\) fixes the base vertex and base edge of \(Q\) and thus \(s\in\langle r_{2}\rangle\), as claimed. To this end observe that \(s\) permutes the vertices of the vertex figure at \(v\) (that is, the \(q\) points on \(C\)) and thus fixes the center \(z\) of \(C\). As the edges of \(Q\) connecting \(v\) to the points on \(C\) all have the same length, \(v\) must lie on the line \(l\) perpendicular, at \(z\), to the plane containing \(C\); as the polyhedron \(\{6,6\}_{4}\) shows, it is possible that \(v=z\). Now, as \(s\) leaves the plane \(H\) through \(C\) invariant, \(s\) also keeps \(l\) invariant and thus maps \(v\) to itself or to the point \(\vartheta:=2z-v\) on \(l\) on the other side of \(H\) at the same distance from \(z\) as \(v\).