Symmetric Circle Configurations from Regular Skeletal Polyhedra

Point-circle configurations provide a natural generalization of classical point-line configurations and connect incidence geometry with the theory of symmetric polyhedra. In this work, we investigate highly symmetric point-circle configurations arising from regular skeletal polyhedra in Euclidean three-space. Our goal is to understand how geometric and combinatorial symmetry can be transferred from polyhedral structures to circle configurations. This work can lead to the investigation of restrictions on geometric realizations of combinatorial configurations.

We introduce the E-construction, a procedure that associates a circle configuration with a polyhedron through the neighborhoods of its edges. For each edge, the adjacent vertices in the surrounding faces determine a candidate block, and coplanarity conditions are analyzed to determine when these blocks can be realized as circles. The construction is applied systematically to the complete classification of regular skeletal polyhedra, including finite, planar, blended, and pure infinite examples. Symmetry properties are studied through the action of the corresponding polyhedral symmetry groups.

The E-construction produces a large family of finite and infinite point-circle configurations that inherit the full symmetry of the underlying regular polyhedra. In most cases, the resulting configurations are point-circle transitive and consist of congruent circles. A complete classification is obtained for the regular skeletal polyhedra that satisfy the required coplanarity condition. In contrast, chiral polyhedra fail this condition and therefore do not yield point-circle configurations through the E-construction. The study also reveals relationships between Petrie-dual polyhedra and their associated configurations.

Regular skeletal polyhedra constitute a rich source of highly symmetric point-circle configurations in three-dimensional space. The results establish a unified framework linking polyhedral symmetry, incidence structures, and geometric configurations, and suggest several open directions concerning the classification of point-circle transitive configurations in Euclidean 3-space.