# Symmetric Circle Configurations from Regular Skeletal Polyhedra
## 5 E-Constructed Point-Circle Configurations
### Configurations from Planar or Blended Regular Polyhedra  
The circle configurations for the planar regular polyhedra are straightforward. These polyhedra are the three regular tessellations \(\{4,4\}\), \(\{3,6\}\), and \(\{6,3\}\), and their Petrie duals Lemma 2. The triangular tessellation \(\{3,6\}\) and its Petrie dual will not give a circle configuration, but the other pairs of Petrie duals will. The two corresponding circle configurations are shown in Figure 7.  
Figure 6: Circle configuration for the pair of polyhedra \(\{5,\frac{5}{2}\}\), \(\{\frac{5}{2},5\}\), shown on the left relative to the great dodecahedron \(\{5,\frac{5}{2}\}\) and on the right relative to the small stellated dodecahedron \(\{\frac{5}{2},5\}\). The circles are the equatorial circles in planes perpendicular to the axis of two-fold symmetry for the polyhedra and have been colored with different colors. In the figure on the left, the edges labelled \(e_{1}\) and \(e_{2}\) determine the same circle, colored blue, lying in the equatorial plane halfway between \(e_{1}\) and \(e_{2}\). Figure 7: Circle configurations for the tessellations \(\{4,4\}\) and \(\{6,3\}\), each shown in two ways: relative to the tessellation itself and relative to its Petrie dual. The figures on the left highlight a pair of adjacent faces of the tessellation, along with their common edge and the circle corresponding to this edge. The twelve blended regular polyhedra \(P\#L\) are orthogonal blends of two components: a planar regular polyhedron \(P\) and a linear regular polygon \(L\) (a line segment \(\{\ \}\) or an apeirogon \(\{\infty\}\)). In each case, the generator \(r_{2}\) is a plane reflection. In fact, for a planar regular polyhedron, the generator \(r_{2}\) is a reflection in a line so that, under orthogonal blending with either a line segment or an apeirogon, the generator \(r_{2}\) becomes a plane reflection symmetry for the blended polyhedron. Thus, Lemma 2 (b) applies and shows that the blended polyhedra contributes six circle configurations. More explicitly, suppose \(P\) lies in \(\mathbb{E}^{2}\) and has symmetry group \(G(P)=\langle r_{0}^{\prime},r_{1}^{\prime},r_{2}^{\prime}\rangle\), and \(L\) lies on the line \(\mathbb{E}^{1}\) and has symmetry group \(G(L)=\langle r_{0}^{\prime\prime}\rangle\) or \(G(L)=\langle r_{0}^{\prime\prime},r_{1}^{\prime\prime}\rangle\) if \(L=\{\ \}\) or \(L=\{\infty\}\), respectively. If 3-space is taken in the form \(\mathbb{E}^{3}=\mathbb{E}^{2}\perp\mathbb{E}^{1}\) with coordinates \(x,y,z\) (and with \(((x,y),z)\) identified with \((x,y,z)\)), then the generators \(r_{0},r_{1},r_{2}\) of the symmetry group of \(G(P\#L)\) are given by  
\[r_{0}(x,y,z) = (r_{0}^{\prime}(x,y),r_{0}^{\prime\prime}(z)),\] \[r_{1}(x,y,z) = \left\{ \begin{array}{l}(r_{1}^{\prime}(x,y),y,z)\text{ if }L=\{\ \}\\ (r_{1}^{\prime}(x,y),r_{1}^{\prime\prime}(x,y),z)\text{ if }L=\{\infty\},\\ r_{2}(x,y,z)\text{ }= (r_{2}^{\prime}(x,y),z).\end{array} \right.\]  
The polyhedron \(P\#L\) itself is obtained from these generators of the group by applying [PERSON]'s construction with the initial (base) vertex \(v:=(x_{0},y_{0},z_{0})\), where \((x_{0},y_{0})\) and \(z_{0}\) are the initial vertices of \(P\) and \(L\), respectively. Note that taking Petrie duals commutes with blending. In other words, \((P\#L)^{\pi}\ =\ P^{\pi}\#L\); here, the linear component remains unchanged. It follows that the six blends with line segments and the six blends with apeirogons each give three circle configurations, up to relative scaling of the components of the blend. The three blended polyhedra \(\{4,4\}\#\{\ \}\), \(\{6,3\}\#\{\ \}\), and \(\{3,6\}\#\{\ \}\) all lie in a slab of \(\mathbb{E}^{3}\) bounded by a pair of parallel planes \(H_{1}\) and \(H_{2}\) (say), and are obtained from a pair of tessellations \(\{4,4\}\), \(\{6,3\}\), or \(\{3,6\}\) in \(H_{1}\) and \(H_{2}\), respectively, positioned so that one tessellation is a translate of the other under a translation perpendicular to the planes.