# Symmetric Circle Configurations from Regular Skeletal Polyhedra
## 5 E-Constructed Point-Circle Configurations
### Configurations from Finite Regular Polyhedra  
The eighteen finite regular polyhedra are the five Platonic solids, the four Kepler-Poinsot polyhedra, and their Petrie duals. In each case, the generator \(r_{2}\) is a plane reflection,  
Figure 4: Circle configuration for the cube. The colors of the circles and of their corresponding edges of the cube match. A view from the “top” is shown on the right.  
so, in particular, Lemma 2 (b) applies. In fact, for the classical finite regular polyhedra, all three generators \(r_{0},r_{1},r_{2}\) are plane reflections, and for their Petrie duals, the generators transition to \(r_{0}r_{2},r_{1},r_{2}\), leaving \(r_{2}\) unchanged.  
The Petrie dual pairs of finite polyhedra can be found in [7] (p. 218). Discounting the pairs that contain a polyhedron with triangle faces, only the following five pairs can contribute a circle configuration, with \(\pi\) indicating Petrie duality:  
\[\begin{array}{ccc}\{4,3\}&\leftarrow^{\pi}_{\pi}&\{6,3\}_{4}\\ \{5,3\}&\leftarrow^{\pi}_{\pi}&\{10,3\}\\ \{\frac{5}{2},3\}&\leftarrow^{\pi}_{\pi}&\{10,3\}\\ \{5,\frac{5}{2}\}&\leftarrow^{\pi}_{\pi}&\{6,\frac{5}{2}\}\\ \{\frac{5}{2},5\}&\leftarrow^{\pi}_{\pi}&\{6,5\}.\end{array} \tag{4}\]  
The case of the cube \(\{4,3\}\) (and its Petrie dual \(\{6,3\}_{4}\)) is special in that the circle configuration has fewer circles than the polyhedron has edges. In this case, opposite edges of the cube determine the same circle, with the same four points on it, so in a sense, the circle occurs twice. The configuration is shown in Figure 4.  
The polyhedra \(\{5,3\}\) and \(\{\frac{5}{2},3\}\) in the second and third rows of (4) are geometrically distinct but combinatorially isomorphic. They share the same vertex sets. Still, these two geometrically distinct polyhedra share the same circle configuration, with 20 points and 30 circles. The edge sets of either polyhedron determine the sets of blocks \(N(e)\), and thus, the circle configurations are the same. The configuration for \(\{\frac{5}{2},3\}\) is depicted in Figure 5 and is the same as the configuration shown in Figure 1 relative to \(\{5,3\}\). Again, colors have been added for visual clarity.  
The circle configurations for the pair of dual polyhedra \(\{5,\frac{5}{2}\}\) and \(\{\frac{5}{2},5\}\) in the fourth and fifth rows of (4) also are the same, for the same reason reason as for the previous pair of polyhedra. Each has 12 points and 15 circles, each with multiplicity 2. They are shown side by side in Figure 6. The circles in each case are the equatorial circles in the planes perpendicular to the two-fold symmetry axes (through the midpoints of pairs of antipodal edges), and since the two polyhedra share the same symmetry groups and same vertex sets, the collections of circles must coincide. This provides an alternative explanation of why the two circle configurations are identical for this pair of polyhedra.  
Figure 5: Circle configuration for the great stellated dodecahedron \(\{\frac{5}{2},3\}\). Each red circle is associated with a dotted red edge of the polyhedron.