# Symmetric Circle Configurations from Regular Skeletal Polyhedra
## 7 Concluding Remarks and Open Problems  
In this paper, we investigated the highly symmetric circle configurations obtained from the regular skeletal polyhedra in \(\mathbb{E}^{3}\). These circle configurations inherit their symmetry from the polyhedron and are point-circle transitive. However, as the family of chiral polyhedra shows, the high symmetry of the underlying polyhedron alone is not a guarantee for the applicability of the E-construction as the coplanarity condition may fail. If a convex or skeletal polyhedron satisfies the coplanarity condition and its symmetry group has \(k\) orbits on the edges, then the E-construction yields a circle configuration with at most \(k\) orbits on the circles. In particular, the edge transitivity of a polyhedron will imply the circle transitivity of the configuration. Among the Archimedean convex polyhedra, exactly four have an edge transitive symmetry group: the cuboctahedron and icosidodecahedron, and their duals (the rhombic dodecahedron and rhombic triacontahedron, respectively). See Figure 15 for the configuration derived from the cuboctahedron. These four polyhedra yield circle-transitive circle configurations via the E-construction. However, these configurations are not point-circle transitive. In fact, in each case, the symmetry group of the polyhedron is the full octahedral or icosahedral group, with edge stabilizers (for the 24 or 60 edges, respectively) of order 2, and must necessarily coincide with the full symmetry group of the configuration since the full octahedral and icosahedral groups are not proper subgroups of any finite group of isometries in \(\mathbb{E}^{3}\). Thus, as the edge stabilizers contribute only two symmetries, the stabilizer of a circle will not act transitively on the points of the circle. For more general polyhedra-like structures such as the regular polygonal complexes \(K\) of [22], there are generalizations of the E-construction method. For example, if the faces at every edge \(e\) of \(K\) occur in cyclic order such that every pair of successive faces in this order satisfies the coplanarity condition relative to \(e\), then a circle configuration can often be obtained in much the same way as before, namely by choosing the circles that pass through the four (or three) vertices determined by each pair of successive faces. The simplest such complex is the 2-skeleton of the cubical tessellation in \(\mathbb{E}^{3}\), that is, the complex formed by the vertices, edges, and square faces of the tessellation. The resulting circular configuration is point-circle transitive and is shown in Figure 16.  
Tilings of space can also be an interesting source of circle configurations. Let \(\mathcal{T}\) be a face-to-face tiling of \(\mathbb{E}^{3}\) whose tiles are convex polyhedra, and suppose each tile of \(T\) permits the construction of a circle configuration \(C(T)\) via the E-construction. Then, the union of all circle configurations \(C(T)\), taken over all tiles \(T\) of \(\mathcal{T}\), is again a circle configuration in \(\mathbb{E}^{3}\). For example, the above circle configuration associated with the 2-skeleton of the cubical tessellation occurs as a special case when \(\mathcal{T}\) is the cubical tessellation. The symmetry group of the tessellations acts point-circle transitively on the configuration, with a point stabilizer of order 48, a circle stabilizer of order 16 (the reflection in the plane of a circle fixes every point on the circle), and a point-circle stabilizer of order 2 (generated by this reflection). The circle configurations presented in this paper suggest the following two attractive open research problems. Figure 16: Circle configuration for the 2-skeleton of the cubical tessellation of \(\mathbb{E}^{3}\). Each cube of the tessellation contributes six circles. Figure 15: Circle configurations for the cuboctahedron, with a top view shown on the right. **Problem 1**. : _Classify all finite or infinite discrete circle configurations in \(\mathbb{E}^{3}\), which are point-circle transitive._  
**Problem 2**.