# Symmetric Circle Configurations from Regular Skeletal Polyhedra
## 5 E-Constructed Point-Circle Configurations
### Configurations from Pure Regular Polyhedra  
The geometry of the twelve pure polyhedra is more involved. Their mirror vectors can be found in [7] (p. 225).  
In the case of six polyhedra, the generator \(r_{2}\) is a plane reflection, and we can appeal again to Lemma 2. These six polyhedra fall into three pairs of Petrie duals,  
\[\begin{array}{ccc}\{4,6\,|\,4\}&\leftarrow\stackrel{{\pi}}{{ \longrightarrow}}&\{\infty,6\}_{4,4}\\ \{6,4\,|\,4\}&\leftarrow\stackrel{{\pi}}{{\longrightarrow}}&\{ \infty,4\}_{6,4}\\ \{6,6\,|\,3\}&\leftarrow\stackrel{{\pi}}{{\longrightarrow}}&\{ \infty,6\}_{6,3}\,\end{array} \tag{5}\]  
each contributing a circle configuration. The polyhedra on the left in (5) are the three Petrie-Coxeter polyhedra, with convex faces and skew vertex figures. The configuration for \(\{4,6\,|\,4\}\) is shown in Figure 14.  
Figure 12: Circle configurations for \(\{3,6\}\#\{\infty\}\), the blend of the triangular tessellation with an apeirogon. The figure on the left indicates how the circles associated with the edges of a single (triangle based) helical face follow this face as its edges are traversed. The figure in the middle shows the circles generated by six neighboring helical faces; the figure on the right gives a top view of this family of circles.  
Figure 13: Circle configurations for \(\{6,3\}\#\{\infty\}\), the blend of the hexagonal tessellation with an apeirogon. The helical faces of the polyhedron “rise above” the hexagons of the hexagonal plane tessellation \(\{6,3\}\), with two helical faces per hexagon, and with these faces sharing every third vertex. The figure on the left indicates the circles associated with the edges of the two helical faces “over” the same hexagon, with red and blue circles arising from the red or blue helical face, respectively. The figure in the middle shows a top view of this arrangement of circles. The figure on the right shows a top view of the family of circles generated by the six helical faces sharing a vertex of the polyhedron.  
The six remaining pure polyhedra, as well as the three Petrie-Coxeter polyhedra in (5), are precisely the nine infinite regular polyhedra that occur in the extended families of chiral and regular polyhedra described in Section 4. The coplanarity condition for these polyhedra can be analyzed using the methods described in the next section, and we refer to these here; of course, one can also check the coplanarity condition directly.  
Two of these six polyhedra, \(\{6,4\}_{6}\) and \(\{\infty,4\}_{\cdot,\cdot,3}\), are self-Petrie, meaning that their Petrie duals are congruent to the original polyhedron. Interestingly, these two polyhedra will not lead to circle configurations via the E-construction because they fail the coplanarity condition.  
The other four polyhedra fall into two pairs of Petrie duals,  
\[\begin{array}{ccc}\{4,6\}_{6}&\leftarrow^{\pi}&\{6,6\}_{4}\\ \{\infty,3\}^{(a)}&\leftarrow^{\pi}&\{\infty,3\}^{(b)}\,.\end{array} \tag{6}\]  
The second pair also does not meet the coplanarity condition and will not give a circle configuration. On the other hand, the first pair does have the coplanarity property and yields a circle configuration via the E-construction.