# Symmetric Circle Configurations from Regular Skeletal Polyhedra
## 5 E-Constructed Point-Circle Configurations  
In this section we study circle configurations derived from regular skeletal polyhedra \(Q\) in \(\mathbb{E}^{3}\) via the E-construction. In each case, we proceed from the configuration \((V(Q),B(Q))\) of \(Q\) and attempt to represent the blocks by circles, each passing through the points of the block. First recall from Lemma 1 that it suffices to consider only one polyhedron from each pair of Petrie duals. Leaving aside the pairs involving a polyhedron with triangle faces, which are excluded here, we find that for all but four pairs of Petrie duals, the point sets of blocks are coplanar and lie on a circle, and thus the pair gives a highly symmetric circle configuration. Further, as \(Q\) is edge-transitive, the coplanarity of the point sets of blocks can be verified for a single edge. The following lemma helps verify the coplanarity condition for most regular polyhedra. Lemma 2: _Let \(Q\) be a regular polyhedron in \(\mathbb{E}^{3}\) of type \(\{p,q\}\), \(p\
eq 3\), with symmetry group \(G(Q)=\langle r_{0},r_{1},r_{2}\rangle\). (a) The points of each block in \(B(Q)\) lie on a sphere centered at the midpoint of the respective edge of \(Q\), of radius independent of the respective edge. (b) If \(r_{0}\) or \(r_{2}\) are plane reflections, then \(Q\) satisfies the coplanarity condition, and the points of each block in \(B(Q)\) lie on a circle of radius independent of the respective edge._  
Proof: Let \(e\) be the base of \(Q\). Then the stabilizer of \(e\) in \(G(Q)\) is given by \(\langle r_{0},r_{2}\rangle\), and in the notation of Figure 3, the generators \(r_{0}\) and \(r_{2}\) act on \(N(e)=\{v_{1},\ldots,v_{4}\}\) like the permutations \((v_{1}\,v_{2})(v_{3}\,v_{4})\) and \((v_{1}\,v_{3})(v_{2}\,v_{4})\), respectively. In particular, \(\langle r_{0},r_{2}\rangle\) fixes the midpoint of \(e\) and acts simply transitively on \(N(e)\). Thus, the points of \(N(e)\) are equidistant from the midpoint of \(e\) and lie on a sphere centered at this point. If \(r_{0}\) is a plane reflection, its reflection plane is the perpendicular bisector of \(e\) and the lines \(\overline{v_{1}v_{3}}\) and \(\overline{v_{2}v_{4}}\) are mapped onto each other under \(r_{0}\). Thus, \(v_{1},\ldots,v_{4}\) are coplanar. Similarly, if \(r_{2}\) is a plane reflection, then \(e\) lies in the reflection plane and \(r_{2}\) maps the lines \(\overline{v_{1}v_{2}}\) and \(\overline{v_{3}v_{4}}\) onto each other, which proves that \(v_{1},\ldots,v_{4}\) is coplanar. In either case, as the points \(v_{1},\ldots,v_{4}\) lie on a sphere about the midpoint of \(e\), they also lie on a circle. The remaining statements follow from the edge transitivity of \(G(Q)\) and the fact that a symmetry of \(Q\) taking \(e\) to an edge \(e^{\prime}\) also takes \(N(e)\) to \(N(e^{\prime})\). Note that Lemma 2(a) shows that, regardless of coplanarity, each regular polyhedron \(Q\) gives rise to an E-constructed point-sphere configuration (or point-circle configuration if \(Q\) is planar), with congruent spheres (circles) centered at the midpoints of the edges of \(Q\). Theorem 5.1: _Let \(Q\) be a regular polyhedron in \(\mathbb{E}^{3}\) of type \(\{p,q\}\). \(p\
eq 3\), with symmetry group \(G(Q)=\langle r_{0},r_{1},r_{2}\rangle\), and suppose \(Q\) satisfies the coplanarity condition. (a) Then the E-construction applied to \(Q\) yields a circle configuration \(\mathcal{C}(Q)=(V(Q),C(Q))\) on which \(G(P)\) acts point-circle transitively.__(b) If \(Q\) has the property that different edges determine different blocks \(N(e)\), then \(G(P)\) acts simply point-circle transitively on \(\mathcal{C}(Q)\) (that is, with trivial point-circle stabilizers)._  
Proof. : Let \(e\) denote the base edge of \(Q\), and let \(c(e)\) denote the circle determined by \(e\). By the edge-transitivity of \(G(Q)\) on \(Q\), the circle configuration is circle transitive under \(G(Q)\). Now consider the stabilizer of \(c(e)\) in \(G(Q)\).