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\begin{document}


\title{\bf Geometric constructions for 3-configurations with non-trivial geometric symmetry}{}

\author{Leah Wrenn Berman\\
\small Department of Mathematics \& Statistics\\[-0.8ex]
\small University of Alaska Fairbanks\\[-0.8ex] 
\small Fairbanks, Alaska, USA\\
\small {\tt lwberman@alaska.edu}\\
}

\date{\dateline{May 10, 2012}{July 25, 2013}{Aug 2, 2013}\\
\small Mathematics Subject Classifications: 51E30, 05B30}

\maketitle


\begin{abstract}
A geometric $3$-configuration is a collection of points and straight lines, typically in the Euclidean plane, in which every point has $3$ lines passing through it and every line has $3$ points lying on it; that is, it is an $(n_{3})$ configuration for some number $n$ of points and lines. We will say that such configuration is \emph{symmetric} if there are non-trivial isometries of the plane that map the configuration to itself. Many symmetric 3-configurations may be easily constructed with computer algebra systems using algebraic techniques: e.g.,  constructing a number of symmetry classes of points and lines, by various means, and then determining the position of a final class of points or lines by solving some polynomial equation. In contrast, this paper presents a number of ruler-and-compass-type constructions for exactly constructing various types of symmetric $3$-configurations, as long as the vertices of an initial regular $m$-gon are explicitly provided. In addition, it provides methods for constructing chirally symmetric 3-configurations given an underlying unlabelled reduced Levi graph, for extending these constructions to produce dihedrally symmetric 3-configurations, and for constructing 3-configurations corresponding to all 3-orbit and 4-orbit reduced Levi graphs that contain a pair of parallel arcs. Notably, most of the configurations described are movable: that is, they have at least one continuous parameter.
\end{abstract}


\section{Introduction}
A geometric $(q,k)$-configuration is a collection of points and straight lines, typically in the Euclidean or projective plane, in which $k$ points lie on every line and $q$ lines pass through every point. If there are $p$ points and $n$ lines, the configuration is called an $(p_{q}, n_{k})$ configuration; if $q =k$ and consequently $p = n$, then we refer to a $(n_{k})$ configuration or a $k$-configuration; the case where $q = k = 3$ is the first non-trivial case. 3-configurations have been studied extensively since the mid-1800s, and there are many results, both existential and enumerative, classifying various types of 3-configurations. Initial investigations focused primarily on classification of combinatorial configurations. We say that a combinatorial configuration is \emph{geometrically realizable} if there exists at least one  realization of the combinatorial configuration using points and straight lines. Note that it is possible for there to be two realizations of the same combinatorial configuration that are not even affinely congruent.

We will say that a geometric $(n_{k})$ configuration is \emph{symmetric} if it has non-trivial geometric symmetry; that is, under the group of isometries of the plane that map the configuration to itself, the configuration has fewer than $n$ transitivity classes of points and $n$ transitivity classes of lines. This usage of the word ``symmetric'' follows Branko Gr\"{u}nbaum \cite[p. 16]{Gru2009b} in reserving the word ``symmetric'' to refer to geometric properties of configurations. In other places in the literature (e.g. \cite{BetBriPis2000} and in works connecting the study of configurations to design theory), the word symmetric has been used to refer to $(n_{k})$ configurations, which are `symmetric' in the numbers of points and lines; however,  following Gr\"{u}nbaum, we shall call general $(n_{k})$ configurations \emph{balanced}, and restrict `symmetric' to refer to geometric properties of a particular embedding.  

Given a geometric configuration with symmetry group $S$, which consists of all isometries of the plane that map the configuration to itself, we say that the configuration has \emph{chiral} symmetry if $S$ is isomorphic to a cyclic group and \emph{dihedral} symmetry if $S$ is isomorphic to a dihedral group. The action of the symmetry group on the points of the configuration partitions the points into orbits, called the \emph{symmetry classes} of the points, and likewise partitions the lines into symmetry classes of lines; we also can refer to the symmetry classes induced by the rotation subgroup of $S$, if $S$ is isomorphic to a dihedral group.
 Typically, symmetric configurations have only a small number of symmetry classes of points and lines, relative to the total number of points and lines of the configuration.  Note that the definition of a symmetric configuration given above is slightly broader than that of \emph{polycyclic} configurations \cite{BobPis2003}, which requires that every symmetry class of points and of lines determined by the rotation subgroup of the symmetry group must have the same number of elements. Figure \ref{symPoly} shows several embeddings of the $(9_{3})$ Pappus configuration, illustrating chirally and dihedrally symmetric non-polycyclic and polycyclic embeddings.
 All the configurations constructed in this paper are polycyclic.  
 
  Given a $(q,k)$-configuration, and following Gr\"{u}nbaum \cite[Section 1.5]{Gru2009b}, we say that the configuration is $(h_{1}, h_{2})$-astral if it has $h_{1}$ symmetry classes of lines and $h_{2}$ symmetry classes of points; if $h_{1} = h_{2} = h$, we call the configuration $h$-astral. If $h_{1} = \lfloor \frac{q+1}{2} \rfloor$ and $h_{2} = \lfloor \frac{k+1}{2} \rfloor$, then we say the configuration is  \emph{astral} \cite{Gru2006a, Gru2009b}.  For example, a 3-configuration is astral if it has 2 symmetry classes of points and lines. Figure \ref{firstEx} shows two symmetric 3-configurations; the configuration shown in Figure \ref{10-3ex} is astral, while the configuration shown in Figure \ref{papEx} is 3-astral.

\newpage

\begin{figure}[htbp]
\begin{center}
\ffigbox{
\begin{subfloatrow*}[4]
\subfloat[]{\includegraphics[width=.2\linewidth]{pap1}}
%
\subfloat[]{\includegraphics[width=.2\linewidth]{pap2}}
\subfloat[]{\includegraphics[width=.2\linewidth]{pap3}}
\subfloat[]{\includegraphics[width=.2\linewidth]{pap4}}
\end{subfloatrow*}}{
\caption{Four embeddings of the Pappus Configuration, illustrating various kinds of symmetric embeddings. In each figure, symmetry classes of points and of lines are distinguished by color. (a) A chirally symmetric, but not polycyclic, embedding; note that the symmetry class of blue points contains 2 elements, while the symmetry class of black points contains only one. (b) A dihedrally symmetric, but not polycyclic, embedding; the symmetry class of green points  contains four elements, but the symmetry class of red points contains only 2. (c) A polycyclic chiral embedding. (d) A polycyclic dihedral embedding; while the black symmetry class of lines (shown with solid and dashed lines) contains six lines under the dihedral symmetry group, under the rotation subgroup, the symmetry class is partitioned into two sets (solid and dashed) that each contain three lines.}
\label{symPoly}}
\end{center}
\end{figure}

%
%\begin{figure}[htbp]
%\begin{center}
%
%\caption{Four embeddings of the Pappus Configuration. (a) A chirally symmetric, but not polycyclic, embedding; note that the symmetry class of blue points contains 4 elements, while the symmetry class of red points contains only one (b) a dihedrally symmetric, but not polycyclic embedding; (c) a polycyclic chiral embedding; a polycyclic dihedral embedding.}
%\label{symPolyFig}
%\end{center}
%\end{figure}


%It is well-known that there are no geometric $(7_{3})$ and $(8_{3})$ configurations (the Fano plane and the M\"{o}bius-Kantor configuration, respectively, are the only combinatorial $(7_{3})$ and $(8_{3})$ configurations, and neither is realizable geometrically with straight lines), while there are three combinatorial $(9_{3})$ configurations which are all geometrically realizable, including the Pappus configuration, and 9 geometrically realizable combinatorial $(10_{3})$ configurations, including the Desargues configuration, plus an additional non-realizable combinatorial $(10_{3})$ configuration. (For a discussion of these and many other results, the reader is referred to Branko Gr\"{u}nbaum's excellent recent monograph on configurations of points and lines \cite{Gru2009b}, especially chapters 1 and 2.)


In 
\cite[Section 2.7]{Gru2009b}, Gr\"{u}nbaum presented an exact, ruler-and-compass method of constructing chiral astral $3$-configurations (possibly beginning with the vertices of a regular convex $m$-gon, if such an $m$-gon is not constructible).  His construction method relies on a result discussed very briefly in that text, which is reformulated here as the \emph{Configuration Construction Lemma}. In this paper, we extend this construction method
 to develop a purely ruler-and-compass technique (given a starting $m$-gon) to produce a number of different classes of chirally and dihedrally symmetric 3-configurations. Although these configurations could be constructed purely algebraically, using linear algebra, say, to determine numerical constraints to force the last three points of a particular construction sequence to lie on a single line, developing a geometric construction process turns out to be quite useful, experimentally, in finding new construction methods to produce new and interesting examples, such as developing infinite classes of more highly incident configurations that have symmetric 3-configurations as building blocks, where the linear-algebraic techniques might fail, e.g., due to the inability to force more than three points at a time to be collinear. For example, the single known $(20_{4})$ configuration, which was found by Gr\"{u}nbaum in 2008 \cite{Gru2008a}, may be analyzed as being formed from two copies (shown in red and black in Figure \ref{20-4}) of an astral 3-configuration (shown in Figure \ref{10-3ex}) whose construction is discussed in section \ref{chiralAstral}; the construction may be further generalized (with quite a lot of work) to produce an infinite family of $((2^{k-2}\cdot m)_{k})$ configurations \cite{BerFau2013}.
 
 \begin{figure}[htbp]
\begin{center}
\includegraphics[width=.4\linewidth]{20-4}
\caption{A $(20_{4})$ configuration, formed from two nested copies of the symmetric 3-configuration shown in Figure \ref{10-3ex}.}
\label{20-4}
\end{center}
\end{figure}

 \section{Double-arcs and reduced Levi graphs}
The configurations whose construction is described in this paper typically have the property that there are at least two symmetry classes of points, say $u$ and $w$, and two symmetry classes of lines, say $L$ and $N$, with the property that two points labelled $u$  lie on a line labelled $L$, and two points from $w$ lie on a line labelled $N$; we say that such a configuration has the \emph{double-arc} property. An example of a small configuration with this property, an astral $(10_{3})$ configuration, is shown in Figure \ref{10-3ex}; the configuration in Figure \ref{example} also has this property.  Note that there are symmetric geometric configurations which do not have this property. For example, the version of the $(9_{3})$ Pappus configuration shown in Figure \ref{papEx} has the property that each point contains one line from each of three symmetry classes of points, and similarly, each line contains one point from each of three symmetry classes of lines.


% original figure 3 location


Given a configuration, it is possible to construct an associated bipartite graph called a Levi graph, which has a black node for each vertex of the configuration and a white node for each line of the configuration, and two nodes are connected with an arc in the Levi graph if and only if the corresponding vertex and line are incident in the configuration. 
In their article \emph{Polycyclic Configurations} \cite{BobPis2003}, Marko Boben and Toma\v{z} Pisanski presented a way of analyzing the (rotational) symmetry of configurations via a graph-theoretic object called a \emph{voltage graph} (for a general discussion of voltage graphs, see \cite{GroTuc1987}). In their presentation, the voltage graph is a bipartite quotient, with respect to a cyclic group, of the Levi graph of a combinatorial configuration, and they asked whether it is possible to determine a geometric rotational realization of the configuration. 

\newpage

 \begin{figure}[htbp]
\begin{center}
\ffigbox{
\begin{subfloatrow*}
\subfloat[A $(10_{3})$ configuation with the ``double-arc'' property.]{\label{10-3ex}\includegraphics[width=.45\linewidth]{10-3ex}}
%
\subfloat[A chiral representation of the Pappus configuration, which has no double-arcs]{\label{papEx}\includegraphics[width=.45\linewidth]{PappusEx}}
\end{subfloatrow*}}{
\caption{(a) A $(10_{3})$ configuration with two symmetry classes of points and lines, with two points from one symmetry class and one point from a second symmetry class on lines labelled $L$ and $M$, and two lines from each of two symmetry classes of lines passing through points labelled $u$ and $w$. (b) A $(9_{3})$ configuration with three symmetry classes of points and lines, with one point from each of three symmetry classes of points on each line, and one line from each of three symmetry classes of lines through each point. In both figures, symmetry classes are indicated by color.}
\label{firstEx}}
\end{center}
\end{figure}

In this article, we modify their presentation of a voltage graph to emphasize the connection between the Levi graph and the voltage graph, to help develop new infinite families of configurations: that is, we consider construction techniques to produce infinite families of configurations for which the underlying unlabelled reduced Levi graph is the same.
We use the labeling conventions from the notion of a \emph{reduced Levi graph} introduced in Gr\"unbaum's monograph \cite[Section 1.6]{Gru2009b} (note that his object is different from the ``reduced Levi graph'' discussed by Artzy; see \cite[p. 150, exercise 6]{Gru2009b}) and the use of a directed multigraph from the definition of a voltage graph \cite{BobPis2003} to produce a modified reduced Levi graph (henceforth, simply \emph{reduced Levi graph}) which encapsulates significant structural properties of classes of symmetric configurations. 

In what follows, we use the terms \emph{nodes} and \emph{arcs} to refer to graph-theoretic objects, and we reserve the words \emph{points} and \emph{lines} for elements of a configuration.

Beginning with a particular polycyclic geometric configuration with $\mathbb{Z}_{m}$ cyclic symmetry (which may or may not be the full symmetry group of the configuration), the construction and interpretation of the reduced Levi graph  is as follows. 

%We begin with an $(h_{1}, h_{2})$-astral $(q,k)$-configuration (which by definition has $h_{1}$ symmetry classes of lines and $h_{2}$ symmetry classes of points) with symmetry group  $\mathbb{Z}_{m}$.

Suppose the configuration has $h_{1}$ symmetry classes of lines and $h_{2}$ symmetry classes of points under the rotational symmetries. We denote the classes of lines as $L_{0}, L_{1}, \ldots, L_{h_{1}-1}$ and the classes of points as  $v_{0}, v_{1}, \ldots, v_{h_{2}-1}$, and we assume each symmetry class contains $m$ elements. Within a symmetry class, say within class $v_{0}$, the elements of the symmetry class are labelled cyclically, so that point class $v_{0}$ contains points $(v_{0})_{0}, (v_{0})_{1}, \ldots, (v_{0})_{m-1}$ which form the vertices of a regular convex $m$-gon. (Note that if the number of symmetry classes is small, the symmetry classes may be labelled so as to avoid the double-indexing; for example, in Example \ref{Example} below, the symmetry classes of vertices are labelled as $u$, $v$, $w$ and the symmetry classes of lines are labelled $L$, $M$, $N$.)

We define the corresponding reduced Levi graph to be an edge-labelled bipartite multigraph (we are allowing doubled edges), with $h_{1}$ large nodes, each corresponding to a symmetry class of lines, and $h_{2}$ small nodes, each corresponding to a symmetry class of points. 

We define incidence between nodes according to the incidence between the points and lines in the configuration. In particular, we construct a directed arc from node $L_{k}$ to node $v_{j}$ with label $a$  precisely when line $(L_{k})_{0}$ passes through vertex $(v_{j})_{a}$, or alternately, when the line labelled $(L_{k})_{-a}$ intersects the node $(v_{j})_{0}$. Note that this labeling convention is opposite to the labeling scheme described for voltage graphs in \cite{BobPis2003}. In many cases, we will need the label of $0$, meaning that vertex $(v_{j})_{0}$ lies on line $(L_{j})_{0}$; to lessen the clutter of the diagram we will {\bf suppress the 0 label and draw such lines thick, unlabeled, and without arrows.}

If two lines $P$ and $Q$ intersect, we sometimes will refer to the intersection point as $P\wedge Q$, and if two points $v$ and $w$ are connected by a line, then we may call the line $v \vee w$. A double-arc 
%
%\begin{center} 
\tikz{ 
\node[vtx] (v) at (0:1){$u$};
\node[ fill=black!20,line] (M) at (180:1) {$L$};
\draw (v) edge [<-, bend left=20] node[midway, auto]{$a$} (M);
\draw[ultra thick] (M)edge [bend left=20] node[midway, auto]{} (v);
}
%\end{center}
%
 in a reduced Levi graph connecting line class $L$ and vertex class $u$ with labels $0$ and $a$ corresponds to the situation where line $L_{i}$ is incident with vertex $u_{i}$ and $u_{i+a}$; that is, 
$ L_{i} = u_{i} \vee u_{i+a}$
(or conversely, $u_{i} = L_{i} \wedge L_{i-a}$)
and the lines $L_{i}$ are said to be \emph{of span $a$}. (More generally, a double-arc is any pair of parallel arcs in a reduced Levi graph, but it is convenient to work with double-arcs in which one label is 0.)

\begin{example}\label{Example} Constructing a reduced Levi graph. \end{example}
We illustrate the construction of a reduced Levi graph using the $(9_{3})$ configuration shown in Figure \ref{cfg}. It has 3-fold rotational symmetry,  three classes of lines, called $L$, $M$ and $N$, and three classes of points $u$, $v$, $w$;  in the figures the $0$-th element of each symmetry class is shown larger. Incidences between the points and lines are shown in Table \ref{exTable}; in the table, all indices are taken modulo 3.

\begin{table}[htbp]
\caption{Incidences between points and lines of the $(9_{3})$ configuration shown in Figure 2.}
\label{exTable}
\begin{tabular}{c| c c c || c |c c c}
Point & \multicolumn{3}{c||}{incident to} & Line & \multicolumn{3}{c}{incident to}\\
\hline
$u_{i} $& $L_{i}$&$ L_{i+2}$&$ N_{i+1}$ & $L_{i}$ & $u_{i}$&$ u_{i+1}$&$ v_{i}$\\
$v_{i}$ & $L_{i}$&$ M_{i}$&$ N_{i}$ & $M_{i}$ & $v_{i}$&$ w_{i}$&$ w_{i+1}$\\
$w_{i}$ & $M_{i}$&$M_{i+2}$&$ N_{i+1}$ & $N_{i}$ & $u_{i+2}$&$ v_{i}$&$ w_{i+2}$
\end{tabular}
\end{table}

 \begin{figure}[htbp]
\begin{center}
\ffigbox{
\begin{subfloatrow*}
\subfloat[A $(9_{3})$ configuration]{\label{cfg} \includegraphics[width=.4\linewidth]{example}}
%
\subfloat[The corresponding reduced Levi graph ]{\label{VG} 
 \begin{tikzpicture}[baseline=(current bounding box.south)]
\node[vtx,white, fill=red] (u) at (0:2) {$u$};
\node[vtx,white, fill=blue] (v) at (120:2) {$v$};
\node[vtx, white,fill=orange] (w) at (240:2) {$w$};
\node[line,white,fill=magenta!90!blue] (L) at (0+60:2) {$L$};
\node[line,white, fill=cyan] (M) at (120+60:2) {$M$};
\node[line,white, fill=green!60!black] (N) at (240+60:2) {$N$};
%
%\draw[ultra thick] (L) edge[bend left=30]  (v);
%\draw (u) edge [bend left=30] node[midway, auto]{$0$} (L);
%\draw (L)edge [bend left=30] node[midway, auto]{$a$} (v);
\draw (u) edge [<-,bend left=30] node[midway, auto]{$1$} (L);
\draw[ultra thick] (L)edge [bend left=30] node[midway, auto]{} (u);
\draw[ultra thick] (v)edge  node[midway, auto]{} (L);

\draw[ultra thick] (v)edge  node[midway, auto]{} (N);
\draw (w)edge[<-]  node[midway, auto]{$2$} (N);
\draw (u)edge[<-]  node[midway, auto]{$2$} (N);


\draw[ultra thick] (w) edge [bend left=30] node[midway, auto]{} (M);
\draw (M)edge [->, bend left=30] node[midway, auto]{$1$} (w);
\draw[ultra thick] (M)edge  node[midway, auto]{} (v);

\end{tikzpicture}$\mathbb{Z}_{3}$}

\end{subfloatrow*}}{
\caption{A 3-configuration and its corresponding reduced Levi graph. Note that the thick lines indicate that the labelling on the arc is 0, and nodes corresponding to lines are shown larger. For example, since point $v_{i}$ lies on line $L_{i}$, the arc between node $v$ and node $L$ is drawn thick, but is unlabelled. On the other hand, since point $w_{i+2}$ lies on line $N_{i}$, the arc between nodes $w$ and $N$ is labelled with a 2.}
\label{example}}
\end{center}
\end{figure}




A useful fact is that given a (labelled) reduced Levi graph, by ``rotating labels'' about consecutive nodes appropriately, it is possible to zero-out labels along any given spanning tree of the graph.  To see this, consider the underlying multigraph for the reduced Levi graph shown in Figure \ref{VG}. In Figure \ref{niceTree}, we have chosen and highlighted a particularly desirable spanning tree for the graph, namely, a Hamiltonian path that begins and ends at a double-arc.%The ``standard version'' of a reduced Levi graph has such a spanning tree. 

\begin{figure}[htbp]
 \begin{tikzpicture}{scale=1/2}[baseline=(current bounding box.center)]
\node[vtx,white, fill=red] (u) at (0:2) {$u$};
\node[vtx,white, fill=blue] (v) at (120:2) {$v$};
\node[vtx,white, fill=orange] (w) at (240:2) {$w$};
\node[line,white,fill=magenta] (L) at (0+60:2) {$L$};
\node[line,white, fill=cyan] (M) at (120+60:2) {$M$};
\node[line,white, fill=green!60!black] (N) at (240+60:2) {$N$};
%
%\draw[ultra thick] (L) edge[bend left=30]  (v);
%\draw (u) edge [bend left=30] node[midway, auto]{$0$} (L);
%\draw (L)edge [bend left=30] node[midway, auto]{$a$} (v);
\draw (u) edge [<-,bend left=30] node[midway, auto]{} (L);
\draw[ultra thick, dotted] (L)edge [bend left=30] node[midway, auto]{} (u);
\draw[ultra thick,dotted] (v)edge  node[midway, auto]{} (L);

\draw[ultra thick, dotted] (v)edge  node[midway, auto]{} (N);
\draw [ultra thick, dotted](w)edge  node[midway, auto]{} (N);
\draw (u)edge[<-]  node[midway, auto]{} (N);


\draw[ultra thick,dotted ] (w) edge [bend left=30] node[midway, auto]{} (M);
\draw (M)edge [<-,bend left=30] node[midway, auto]{} (w);
\draw (M)edge[->]  node[midway, auto]{} (v);

\end{tikzpicture}
\caption{A nice spanning tree, indicated on an unlabeled underlying reduced Levi graph.}
\label{niceTree}
\end{figure}

How can we change the labelling on the original configuration so that in the reduced Levi graph, we have a spanning tree (in this case, a spanning path) of zero-labelled edges? We already have the path $u - L - v - N$ all labelled with $0$, so we need to continue the path by making the arc from $N$ to $w$ be labelled $0$. To do this, we add $-2$ to every arc incident with $w$, with addition done modulo 3 since the symmetry group of the configuration is $\mathbb{Z}_{3}$. This is equivalent to subtracting 2 from every index of the vertex labels associated with $w$ in the labelling of the configuration (that is, rotating the labels assigned to the elements of the symmetry class $w$ by two steps backwards). Next, we want to make one of the arcs between $w$ and $M$ have label 0, so we add 2 to every label incident with $M$ (both on the node, and for the labels of the lines in the configuration). We now have achieved the desired 0-labelled spanning path; the sequence of operations is shown in Figure \ref{niceTreeSequence}.

\begin{figure}[htbp]
 \begin{tikzpicture}[scale=.9,baseline=(current bounding box.center)]
\node[vtx, white,fill=red] (u) at (0:2) {$u$};
\node[vtx, white,fill=blue] (v) at (120:2) {$v$};
\node[vtx, white,fill=orange] (w) at (240:2) {$w$};
\node[line,white,fill=magenta] (L) at (0+60:2) {$L$};
\node[line,white, fill=cyan] (M) at (120+60:2) {$M$};
\node[line, white,fill=green!60!black] (N) at (240+60:2) {$N$};
%
%\draw[ultra thick] (L) edge[bend left=30]  (v);
%\draw (u) edge [bend left=30] node[midway, auto]{$0$} (L);
%\draw (L)edge [bend left=30] node[midway, auto]{$a$} (v);
\draw (u) edge [<-,bend left=30] node[midway, auto]{$1$} (L);
\draw[ultra thick] (L)edge [bend left=30] node[midway, auto]{} (u);
\draw[ultra thick] (v)edge  node[midway, auto]{} (L);

\draw[ultra thick] (v)edge  node[midway, auto]{} (N);
\draw (w)edge[<-]  node[midway, auto]{$2$} (N);
\draw (u)edge[<-]  node[midway, auto]{$2$} (N);


\draw[ultra thick] (w) edge [bend left=30] node[midway, auto]{} (M);
\draw (M)edge [->,bend left=30] node[midway, auto]{$1$} (w);
\draw[ultra thick] (M)edge  node[midway, auto]{} (v);

\end{tikzpicture} {\LARGE $\to$}  \begin{tikzpicture}[scale=.9,baseline=(current bounding box.center)]
\node[vtx,white, fill=red] (u) at (0:2) {$u$};
\node[vtx, white,fill=blue] (v) at (120:2) {$v$};
\node[vtx,white, fill=orange] (w) at (240:2) {$w$};
\node[line,white,fill=magenta] (L) at (0+60:2) {$L$};
\node[line, white,fill=cyan] (M) at (120+60:2) {$M$};
\node[line, white,fill=green!60!black] (N) at (240+60:2) {$N$};
%
%\draw[ultra thick] (L) edge[bend left=30]  (v);
%\draw (u) edge [bend left=30] node[midway, auto]{$0$} (L);
%\draw (L)edge [bend left=30] node[midway, auto]{$a$} (v);
\draw (u) edge [<-,bend left=30] node[midway, auto]{$1$} (L);
\draw[ultra thick] (L)edge [bend left=30] node[midway, auto]{} (u);
\draw[ultra thick] (v)edge  node[midway, auto]{} (L);

\draw[ultra thick] (v)edge  node[midway, auto]{} (N);
\draw[ultra thick] (w)edge  node[midway, auto]{$0$} (N);
\draw (u)edge[<-] node[midway, auto]{$2$} (N);


\draw (w) edge [<-,bend left=30] node[midway, auto]{$-2$} (M);
\draw (M)edge [bend left=30, ->] node[midway, auto]{$-1$} (w);
\draw[ultra thick] (M)edge  node[midway, auto]{} (v);
%rotate around $w$
\end{tikzpicture} {\LARGE $\to $}
 \begin{tikzpicture}[scale=.9,,baseline=(current bounding box.center)]
\node[vtx,white, fill=red] (u) at (0:2) {$u$};
\node[vtx,white, fill=blue] (v) at (120:2) {$v$};
\node[vtx, white,fill=orange] (w) at (240:2) {$w$};
\node[line,white,fill=magenta] (L) at (0+60:2) {$L$};
\node[line,white, fill=cyan] (M) at (120+60:2) {$M$};
\node[line, white,fill=green!60!black] (N) at (240+60:2) {$N$};
%
%\draw[ultra thick] (L) edge[bend left=30]  (v);
%\draw (u) edge [bend left=30] node[midway, auto]{$0$} (L);
%\draw (L)edge [bend left=30] node[midway, auto]{$a$} (v);
\draw (u) edge [<-,bend left=30] node[midway, auto]{$1$} (L);
\draw[ultra thick] (L)edge [bend left=30] node[midway, auto]{} (u);
\draw[ultra thick] (v)edge  node[midway, auto]{} (L);

\draw[ultra thick] (v)edge  node[midway, auto]{} (N);
\draw[ultra thick] (w)edge  node[midway, auto]{} (N);
\draw (u)edge[<-]  node[midway, auto]{$2$} (N);


\draw[ultra thick] (w) edge [bend left=30] node[midway, auto]{} (M);
\draw (M)edge [->,bend left=30] node[midway, auto]{$1$} (w);
\draw (M)edge[->]  node[midway, auto]{$2$} (v);

\end{tikzpicture}

$\vcenter{\hbox{\includegraphics[width=.28\linewidth]{smallEx1.pdf}}}$ {\LARGE $\to$ }
$\vcenter{\hbox{\includegraphics[width=.28\linewidth]{smallEx2.pdf} }}$ {\LARGE $\to $} 
$\vcenter{\hbox{\includegraphics[width=.28\linewidth]{smallEx3.pdf}}}$

\caption{Achieving a desired spanning tree. Initial graph: the original reduced Levi graph. The middle graph is obtained by ``rotating about $w$'' by addiing $-2$ to each label for arcs incident with $w$ in the original graph. The right-hand graph is obtained from the middle graph by ``rotating about $M$'' by adding $2$ to each label for arcs incident with $M$.}
\label{niceTreeSequence}
\end{figure}

A useful feature of the reduced Levi graph is that since every arc in the graph corresponds to a point-line incidence in the configuration, the degree of the vertex-nodes and the line-nodes of the reduced Levi graph is equal to the type of configuration the graph corresponds to. For instance, in the example shown in Figure \ref{example}, every node in the graph is of degree 3, and the configuration is a 3-configuration. In general, a $(q,k)$-configuration corresponds to a reduced Levi graph with $q$-valent line-nodes and $k$-valent vertex-nodes.  A $(q,k)$-configuration where $q = k$ is said to be \emph{balanced}; if $q \neq k$ the configuration is \emph{unbalanced}. (In the literature (e.g., \cite{BetBriPis2000}), balanced combinatorial configurations have often been called symmetric, but that terminology clashes with our desire to use \emph{symmetric} to refer to a geometric property of a configuration. In this language---as in so much else---we follow Gr\"{unbaum}; see \cite[p. 16]{Gru2009b} for his discussion of the issue.)  An unbalanced (3,4)-configuration and its corresponding reduced Levi graph is shown in Figure \ref{unbalancedEx}. Note that under the cyclic symmetry group $\mathbb{Z}_{7}$, it has three symmetry classes of lines and four symmetry classes of points (indicated by color); each of the vertex-nodes is 3-valent, while each of the line-nodes is 4-valent. (On the other hand, if the configuration is viewed as having dihedral symmetry, it has two symmetry classes of points and two symmetry classes of lines.)

\begin{figure}[htbp]
\begin{center}
\ffigbox{
\begin{subfloatrow*}
\subfloat[]{
\includegraphics[width=.5\linewidth]{3-4ex}}
%\hspace{1in}
\subfloat[]{
 \begin{tikzpicture}[scale=1,baseline=(current bounding box.south)]
\node[draw, circle, fill=orange,inner sep=3pt] (o) at (0+30:2) {};
\node[draw, circle, fill=black,inner sep=3pt] (b) at (120+30:2) {};
\node[draw, circle, fill=red,inner sep=6pt] (R) at (240+30:2) {};
\node[draw, circle, fill=green,inner sep=6pt] (G) at (0+60+30:2) {};
\node[draw, circle, fill=blue,inner sep=3pt] (bl) at (120+60+30:2) {};
\node[draw, circle, fill=magenta,inner sep=3pt] (m) at (240+60+30:2) {};
\node[draw, circle, fill=cyan!50!white,inner sep=6pt] (C) at (0:0) {};
%
%\draw[ultra thick] (L) edge[bend left=30]  (v);
%\draw (u) edge [bend left=30] node[midway, auto]{$0$} (L);
%\draw (L)edge [bend left=30] node[midway, auto]{$a$} (v);
\draw[ultra thick]  (o) edge [bend left=30] node[midway, auto]{} (G);
\draw(G)edge [->,bend left=30] node[midway, auto]{1} (o);

\draw (b) edge [<-,bend left=30] node[midway, auto]{$2$} (G);
\draw[ultra thick] (G)edge [bend left=30] node[midway, auto]{} (b);

\draw (R) edge [->,bend left=30] node[midway, auto]{$2$} (bl);
\draw[ultra thick] (bl)edge [bend left=30] node[midway, auto]{} (R);

\draw (m) edge [<-,bend left=30] node[midway, auto]{$1$} (R);
\draw[ultra thick] (R)edge [bend left=30] node[midway, auto]{} (m);

\draw[ultra thick] (C)edge  node[midway, auto]{} (b);
\draw (bl)edge[<-]  node[midway, auto]{3} (C);
\draw(o)edge[<-]  node[midway, auto]{2} (C);
\draw[ultra thick] (C)edge  node[midway, auto]{} (m);
\end{tikzpicture}$\mathbb{Z}_{7}$
}
\end{subfloatrow*}}{
\caption{(a) An unbalanced $(3,4)$-configuration; (b) the corresponding reduced Levi graph. In the reduced Levi graph, the large nodes correspond to the lines and the smaller nodes to the vertices of the configuration; symmetry classes are coded by color. In the configuration, the $0$-th element of each symmetry class is shown larger, and subsequent elements are labelled cyclically, counterclockwise.}
\label{unbalancedEx}}
\end{center}
\end{figure}
\section{The Configuration Construction Lemma}\label{LemmaSec}

The configuration construction lemma is very easy to state and prove, but it provides a broadly useful method for a purely geometric construction for a number of symmetric configurations. In particular, it lets us precisely construct geometric 3-configurations whose reduced Levi graphs contain two non-adjacent pairs of double-arcs.

A variant of this lemma was presented by Gr\"{u}nbaum  in \cite[Section 2.7]{Gru2009b} in the context of constructing a single, highly restricted class of 3-configurations, the astral 3-configurations with cyclic symmetry.

\begin{lemma}[Configuration Construction Lemma] 
%Let $R$ and $R'$ be two points on a circle centered at $O$, and let $\angle ROR' = \alpha$. If $Q$ is any point on the circumcircle $ \mc{C}(R,O,R')$ and $Q'$ is on $C_{O}(Q)$ so that $\angle QOQ' = \alpha$, then $\overline{QQ'}$ passes through $R'$. Conversely, if $Q$ and $Q'$ are any two points on a circle centered at $O$ so that $\angle QOQ' = \alpha$ and $Q, R', Q'$ are all collinear, then $Q$ lies on the circumcircle  $\mc{C}(R,O,R')$.

%Let $v_{0}, v_{1}, \ldots, v_{m-1}$ form the vertices of a regular $m$-gon with center $\mc{O}$. Let $\mc{C}$ be the circle passing through the points $\mc{O}, v_{d-c}, v_{c}$. Let $L_{0}$ be a line passing through $v_{0}$, and define $w_{0}$ to be the intersection of $L_{0}$ with $\mc{C}$. Let $w_{i}$ be the rotation of $w_{0}$ by $\frac{2 \pi i}{m}$ about $\mc{O}$. Then $w_{0}$, $w_{c}$, $v_{d}$ are all collinear.
Let $v_{0}, v_{1}, \ldots, v_{m-1}$ form the vertices of a regular $m$-gon with center $\mc{O}$. Let $\mc{C}$ be the circle passing through the points $\mc{O}, v_{d}, v_{d-c}$, and let $\angle v_{d}\mc{O}v_{d-c} = \frac{2 \pi c}{m}$. Let  $w_{0}$ be any point on $\mc{C}$, and define $w_{i}$ to be the rotation of $w_{0}$ by $\frac{2 \pi i}{m}$ about $\mc{O}$. Then $w_{0}$, $w_{c}$, $v_{d}$ are all collinear.
\end{lemma}

\begin{figure}[htbp]
\begin{center}
\ffigbox{
\begin{subfloatrow*}
\subfloat[Lemma statement]{\includegraphics[width=.4\linewidth]{CCL1}}\subfloat[proving the lemma]{\label{CCLproof}\includegraphics[width=.4\linewidth]{CCL2}}
\end{subfloatrow*}}{
\caption{Illustrating the configuration construction lemma.}
\label{geomConstr}}
\end{center}
\end{figure}


\begin{proof}
%Note that by construction, $\angle v_{d-c}\mc{O}v_{d} = \alpha =  \frac{2 \pi c}{m}$. 
Construct a point $X$ on the line $\overline{v_{d-c}{w_{0}}}$ so that the oriented angle $\angle w_{0}\mc{O} X = -\alpha$. (See Figure \ref{CCLproof}.) Note that both $\angle \mc{O}v_{d} v_{d-c}$ and $\angle \mc{O}w_{0} v_{d-c}$ are subtended by chord $\mc{O}v_{d-c}$ of circle $\mc{C}$, so they are equal. Moreover, since $X$ lies on $\overline{v_{d-c}{w_{0}}}$, $\angle \mc{O}w_{0} X = \angle \mc{O}v_{d} v_{d-c}$. Therefore, triangles $\triangle w_{0}\mc{O} X$ and $\triangle v_{d} \mc{O}v_{d-c}$, which have two pairs of equal angles, are similar, so because $\triangle v_{d} \mc{O} v_{d-c}$ is isosceles by construction, it follows that $\triangle w_{0}  \mc{O} X$ is also isosceles. That is, $X$ and $w_{0}$ lie on the same circle centered at $\mc{O}$. Because $\angle w_{0}\mc{O} X = -\frac{2 \pi c}{m}$, it follows that $X = w_{-c}$, by the definition of the $w_{i}$. 

If we define $N_{-c}$ to be the line passing through $w_{0}$ and $v_{d-c}$, we have just shown that the point $X = w_{-c}$ also lies on $N_{-c}$. Defining $N_{i-c}$ to be the rotation of $N_{-c}$ through $\frac{ 2 \pi i}{m}$, it follows that $N_{i-c}$ contains the points  $w_{0+i}$ and $v_{d-c+i}$, and $w_{-c+i}$. That is, the line $N_{0} = N_{c-c}$ contains the points $w_{c}$, $v_{d}$, and $w_{0}$, as was to be shown.
\end{proof}

If we consider the effect of the configuration construction lemma on corresponding nodes and arcs of a partial reduced Levi graph  under the action of the group $\mathbb{Z}_{m}$, %corresponding to a subconfiguration whose vertices and lines $v_{i}, w_{i}$, $L_{i}, N_{i}$ are constructed via the lemma %then the lemma says: beginning with a node labelled $v$ and a node labelled $L$ (where $L_{0}$ passes through $v_{0}$ as in the statement of the lemma and $L_{i}$ is the rotation of $L_{0}$ through $\frac{2 \pi i}{m}$ about the center of the convex $m$-gon), then $v$ and $L$ are connected by an arc labelled $0$. The point $w_{0}$ lies on $L_{0}$ 
then we arrive at the following ``gadget'' in the reduced Levi graph, shown in the left-hand side of Figure \ref{CCLvolt}, where the dashing between $N$ and $v$ indicates that the collinearity is forced. If in addition $w_{0}$ is the intersection of $\mc{C}$ with some other line $L_{0}$ which is an element of a symmetry class $L$ --- which is the case in all of the subsequent constructions --- then we have in the reduced Levi graph the gadget shown in the right-hand side of Figure \ref{CCLvolt}.

\begin{figure}[htbp]
\centering
\ffigbox{
\begin{subfloatrow*}[2]
\subfloat[ $w_{0}$ is on $\mc{C}$...]{
\begin{tikzpicture}[scale=2]
\node[line, fill=red!20] (N) at (-1,0) {$N$};
\node[vtx, fill=red!20] (w) at (0,0) {$w$};
\node[vtx] (v) at (-1.5,1) {$v$};
\draw (N) edge[->,bend left = 30] node[auto, midway]{$c$} (w);
\draw[ultra thick] (w) edge[bend left=30, ultra thick] (N);
\draw[dashed, ->] (N) -- node[auto, midway]{$d$} (v);
\end{tikzpicture}}
%
%
%
\subfloat[ ...and also on $L_{0}$]{\begin{tikzpicture}[scale=2]
\node[line, fill=red!20] (N) at (-1,0) {$N$};
\node[vtx, fill=red!20] (w) at (0,0) {$w$};
\node[vtx] (v) at (-1.5,1) {$v$};
\node[line] (L) at (.5, 1) {$L$};
\draw (N) edge[->,bend left = 30] node[auto, midway]{$c$} (w);
\draw[ultra thick] (w) edge[bend left=30, ultra thick] (N);
\draw[dashed, ->] (N) -- node[auto, midway]{$d$} (v);
\draw[ultra thick] (L) -- (w);
\end{tikzpicture}}
\end{subfloatrow*}}{
\caption{``Gadgets'' in reduced Levi graphs found as a consequence of using the construction in the configuration construction lemma. (a) The point $w_{0}$ lies on the circle $\mc{C}$ passing through $\mc{O}, v_{d}, v_{d-c}$, and $N_{0}$ is a line of span $c$ through $w_{0}$; (b) In addition, $w_{0}$ also lies on some line $L_{0}$.  }
\label{CCLvolt}}
\end{figure}

\subsection{Chiral astral 3-configurations}\label{chiralAstral}
We first apply the Configuration Construction Lemma to construct a chiral astral 3-configuration, which has two symmetry classes of points and lines, and whose reduced Levi graph is shown in Figure \ref{astralVolt}; since there are three discrete parameters in the reduced Levi graph, this type of configuration has been represented by the symbol $m\#(a,b; d)$ (for example, in Section 2.7 of \cite{Gru2009b} as $m\#(b,c;d)$). The smallest example of such a configuration is shown in Figure \ref{10-3ex}, with symmetry group $\mathbb{Z}_{5}$. The method presented here  for constructing chiral astral 3-configurations is a variant of the method described in \cite[Section 2.7]{Gru2009b} but is significantly more straightforward.

\begin{figure}[htbp]
\begin{center}
\begin{tikzpicture}[scale=2]
\node[ fill=black!20,line] (L) at (-45:1) {$L$};
\node[vtx] (v) at (45:1){$v$};
\node[fill=black!20,line] (M) at (135:1) {$N$};
\node[vtx] (w) at (-135:1){$w$};
\draw[ultra thick] (L) edge[bend left=30]  (v);
\draw[<-] (v) edge [bend left=30] node[midway, auto]{$a$} (L);
\draw[ultra thick] (M) edge[bend left=30]  (w);
\draw[<-] (w) edge [bend left=30] node[midway, auto]{$b$} (M);
\draw[ultra thick] (L)--(w);
\draw[->] (M)--(v) node[midway, auto] {$d$};
\end{tikzpicture} $\mathbb{Z}_{m}$
\caption{The reduced Levi graph for a cyclic astral 3-configuration}
\label{astralVolt}
\end{center}
\end{figure}

The existence of two pairs of double-arcs in the reduced Levi graph shown in Figure \ref{astralVolt} mean that we need a class of lines $L$ of span $a$ with respect to some points $v$ and a class of lines $N$ of span $b$ with respect to some points $w$. The 0-labelled line between $L$ and $w$ indicates that point $w_{0}$ must lie somewhere on line $L_{0}$. Finally, the \tikz{ 
\node[vtx] (v) at (45:1){$v$};\node[line, fill=black!20] (M) at (135:1) {$N$};
\draw[->] (M)--(v) node[midway, auto] {$d$};
} arc indicates that line $N_{0}$ must also contain $v_{d}$ (as well as $w_{0}$ and $w_{b}$, since $N_{0}$ is span $b$ with respect to the $w_{i}$). A simple application of the configuration construction lemma allows us to construct such a configuration. First, we construct the points $v_{i}$ as the vertices of a regular convex $m$-gon, and next we construct the lines $L_{i}$ as the lines of span $a$ with respect to the $v_{i}$. We then define $w_{0}$ to be an intersection of line $L_{0}$ with  the circle $\mc{C}$ passing through $\mc{O}$, $v_{d}$ and $v_{b-d}$; the notation $d'$ in the symbol $m\#(a,b; d')$ indicates we should choose the leftmost intersection, while $m\#(a,b; d)$ indicates we should choose the rightmost intersection. Finally, the configuration construction lemma lets us conclude that the points $w_{0}, w_{b}, v_{d}$ are collinear, so we label the line through those three points as $N_{0}$, and define $N_{i}$ to pass through $w_{i}, w_{i+b}, v_{i+d}$. Note that with the exception, possibly, of constructing the original vertices of a regular $m$-gon, the entire construction is achieved using only a straightedge and compass.

 As an example, consider the construction of $7\#(3,2;1')$, shown in Figure \ref{astralExFig}.%; note in the application of the configuration construction lemma, $c = 2$ and $d = 1$.

\begin{example}\label{astralEx} Constructing an astral 3-configuration $7\#(3,2;1')$ \end{example}
\be
\item Construct the points $v_{i} = \left(\cos\left(\frac{2 \pi i}{7}\right), \sin\left(\frac{2 \pi i}{7} \right)\right)$.
\item Construct lines $L_{i} = v_{i} \wedge v_{i+3}$.
\item Construct the circle $\mc{C}$ passing through $\mc{O}$, $v_{1}$ and $v_{1-2} = v_{-1}$.
\item Let $w_{0}$ be an intersection of line $L_{0}$ with $\mc{C}$. In this case, we choose the leftmost intersection, because of the $1'$ (instead of 1) in the symbol description.
\item Construct the lines $M_{i}$ connecting $w_{i}$ and $w_{i+2}$ and note that by the configuration construction lemma, $M_{i}$  also passes through $v_{i+1}$.
\ee

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=.5\linewidth]{astralEx.pdf}
\caption{Constructing an astral configuration $7\#(3,2;1')$. The 0th element of each symmetry class is shown larger, and elements are cyclically labelled counterclockwise; the leftmost intersection of $\mc{C}$ with $L_{0}$ is chosen to define $w_{0}$.}
\label{astralExFig}
\end{center}
\end{figure}

In general, the circumcircle $\mc{C}$ intersects line $L_{0}$ in two places. In particular, suppose the line $L_{0}$ is parametrized as $(1-t) v_{0} + t (v_{a})$ and $\mc{C}$ intersects $L_{0}$ at two points with parameter values $t = \gamma$ and $t = \gamma'$, where $\gamma\leq \gamma'$. By writing $m\#(a,b;d)$ we mean that we choose the intersection point with $t = \gamma$, that is, the rightmost intersection, and $m\#(a,b; d')$ means that we choose the intersection  with $t = \gamma'$, or the leftmost intersection. Figure \ref{bothIsect} shows the two configurations $9\#(4,1;3)$ and $9\#(4,1;3')$.

\begin{figure}[htbp]
\centering
\ffigbox{
\begin{subfloatrow*}
\subfloat[The configuration $9\#(4,1;3)$;  $w_{0}$ uses the rightmost intersection of $\mc{C}$ with $L_{0}.$]{
\includegraphics[width=.45\linewidth]{CompletedAstralEx1.pdf}
}
\subfloat[The configuration $9\#(4,1;3')$: $w_{0}$ uses the leftmost intersection.]{
\includegraphics[width=.45\linewidth]{CompletedAstralEx2.pdf}}
\end{subfloatrow*}}{
\caption{Two chiral astral 3-configurations with $m = 9$, $a = 4$, $b = 3$, $t =2$. In both configurations, the points $v_{i}$ and lines $L_{i}$ are black and the points $w_{i}$ and lines $N_{i}$ are red.}
\label{bothIsect}}
\end{figure}

%
%The general construction for astral 3-configurations with symbol $m\#(a,b; t)$ begins with the vertices $v_{0}, v_{1}, \ldots, v_{m-1}$ which form the vertices of a regular convex $m$-gon; the reduced Levi graph is shown in Figure \ref{astralVolt}. Because of the double-arc connecting $v$ and $L$, with labels $0$ and $a$, the lines $L$ are span $a$ with respect to the vertices $v$. That is, 
%we define
%\[L_{i} = v_{i} \wedge v_{i+a}\]
%and geometrically we construct each line $L_{i}$ by drawing a line through the points labelled $v_{i}$ and $v_{i+a}$. The thick arc between $L$ and $w$ indicates that point $w_{0}$ should lie on line $L_{0}$, and then the double-arc between $w$ and $M$ indicates that the lines $M$ should be span $b$ with respect to points $w$. Finally, the arc labelled $t$ connecting $M$ and $v$ means that the line $M_{-t}$ should pass through vertex $v_{0}$, or alternately, that line $M_{0}$ passes through $v_{t}$.
%
%To determine where, precisely, the point $w_{0}$ should be placed on line $L_{0}$, we apply the Configuration Construction Lemma. Because we want $w_{0}$, $w_{b}$ and $v_{t}$ to all be collinear on line $N_{0}$, the Configuration Construction Lemma says that if we define $w_{0}$ to be the intersection of the circumcircle $\mc{C}$ passing through $v_{t}$ and $v_{t-b}$ with line $L_{0}$, then the desired collinearity of $w_{0}$, $w_{b}$ and $v_{t}$ is achieved. 




\section{Construction of infinite classes of symmetric 3-configurations}

Other infinite classes of symmetric 3-configurations may be constructed geometrically using the configuration construction lemma.


\subsection{Multilateral 3-configurations} These configurations have a reduced Levi graph which alternates double-arcs and single arcs, shown in Figure \ref{multilateralVolt}; the configuration itself consists of mutually inscribed and circumscribed (possibly star) $m$-gons, where the inscribed $m$-gons need not be the same type. For example, Figure \ref{multilateralEx} shows two multilateral $4$-astral 3-configurations with $m = 5$ and $h$ = 4, along with the associated reduced Levi graphs; one uses only convex 5-gons, and the other uses a mixture of pentagrams and convex pentagons. 

\begin{figure}[htbp]
\begin{center}
\begin{tikzpicture}[auto, scale = 2.5]
\tikzstyle{every node}=[inner sep=1pt];%draw, shape=circle];
\foreach \i in {0, ..., 3}
	{
	\path  (\i*72:1) node[draw, shape=circle] (v\i) {$v_\i$};
	\path (\i*72 + 36:1) node[draw, shape=circle, fill=black!20] (L\i){$L_\i$};	
	
	%\draw (a) to node {1} node [swap] {1Õ} (b)
	\draw (v\i) 
	%to 
	edge[<-,bend right=30] 
	node[above, auto,swap, bend right=30] {$a_\i$} 
	%edge[bend right=30] 
	(L\i);
	\draw (v\i) edge[bend left=30, ultra thick] (L\i);
	%\pgfmathparse{mod(\i+1,4)} \let\ii\pgfmathresult;
	}
	\path (4*72:1) node[draw, shape=circle] (vh) {$v_{h-1}$};
	\path (4*72+36:1) node[draw, shape=circle,fill=black!20] (Lh) {$L_{h-1}$};
	\draw (vh) edge[<-,bend right=30] node[above, auto, swap, bend right=30] {$a_{h-1}$} (Lh);
	\draw (vh) edge[ultra thick,bend left=30] (Lh);
\draw(L0) edge[ultra thick] (v1);
\draw(L1) edge[ultra thick] (v2);
\draw(L2) edge[ultra thick] (v3);
\draw(L3) edge[dotted, ultra thick] (vh);
%\draw [->] (\from) -- (\to) node[midway,fill=red!20] {\from--\to};
\draw [->] (Lh)-- (v0) node[midway] {$t$};
\end{tikzpicture}$\mathbb{Z}_{m}$
\caption{The reduced Levi graph for a multilateral $h$-astral $3$-configuration}
\label{multilateralVolt}
\end{center}
\end{figure}


Multilateral 3-configurations were investigated carefully from an algebraic viewpoint (although not using that name) in  Boben and Pisanski's article \emph{Polycyclic Configurations} \cite{BobPis2003}, where they were referred to as $\mc{C}_{3}(k, (p_0, p_{1}, \ldots, p_{n}), t)$, where $n$ is the number of symmetry classes of points and lines and the symmetry group of the configuration is $\mathbb{Z}_{k}$. They are also discussed in \cite[Section 5.2]{Gru2009b}.
   
   
%The first generalization of astral 3-configurations is to ``multilateral'' 3-confgurations, whose reduced Levi graph alternates single arcs and double-arcs. If the group of the reduced Levi graph is $\mathbb{Z}_{m}$, the  corresponding construction consists of mutually inscribed and circumscribed  regular and/or star $m$-gons.

The construction for $h$-astral multilateral 3-configurations with symmetry group $\mathbb{Z}_{m}$ is as follows.

\be
\item Begin with an arbitrary set of vertices $v_{0}$ forming the vertices of a regular $m$-gon. Typically, 
\[(v_{0})_{i} = \left( \cos\left( \frac{2 \pi i}{m}\right), \sin \left( \frac{2 \pi i}{m} \right) \right).\]
\item The first double-arc indicates that the first set of lines $(L_{0})_{i}$ should be of span $a_{0}$ with respect to the points $v_{0}$. That is, construct $(L_{0})_{i} = (v_{0})_{i} \wedge (v_{0})_{i+a_{0}}$.
\item The next, zero-labelled single arc means that we can place $(v_{1})_{i}$ arbitrarily on line $(L_{0})_{i}$. That is, we choose a value for a continuous parameter $\lambda_{1}$ (with $\lambda_{1} \neq 0,1$ to prevent degeneracy) so that
\[ (v_{1})_{i} = (1-\lambda_{1})(v_{0})_{i} + \lambda_{1}(v_{0})_{i+a_{0}}.\] 

\newpage
\begin{figure}[htbp]
\centering
\ffigbox{
\begin{subfloatrow*}
\subfloat{\includegraphics[width=.4\linewidth]{multilatEx1.pdf}}%\hspace{.3in}%
\subfloat{\begin{tikzpicture}[ scale = 2]
\tikzstyle{vtx}=[inner sep=1pt,draw, shape=circle];
\tikzstyle{line}=[inner sep=3pt,draw, shape=circle];

\node[vtx,white, fill=black] (v0) at (0:1) {$v_{0}$};
\node[vtx, white,fill=green!40!black!90!white]  (v1) at (1*90:1) {$v_{1}$};
\node[vtx, fill=green!60!black!40!white] (v2) at (2*90:1) {$v_{2}$};
\node[vtx, fill=orange!70!white] (v3) at (3*90:1) {$v_{3}$};

\node[line,white, fill=black] (L0) at (0*90+45:1) {$L_{0}$};
\node[line,white, fill=green!40!black!90!white] (L1) at (1*90+45:1) {$L_{1}$};
\node[line, fill=green!60!black!40!white] (L2) at (2*90+45:1) {$L_{2}$};
\node[line,white,fill=red] (L3) at (3*90+45:1) {$L_{3}$};

\foreach \i in {0,1,2,3}{
\draw[ultra thick] (v\i) edge[bend left = 30] (L\i);
\draw[->] (L\i)edge[bend left = 30]  node[auto, midway] {1}  (v\i) ;
}
\foreach \i/\j in {0/1,1/2,2/3}{
\draw[ultra thick](L\i) -- (v\j);}

\draw[->] (L3)--(v0) node[midway, auto, swap] {1}; 
\end{tikzpicture}$\mathbb{Z}_{5}$}
\end{subfloatrow*}

\begin{subfloatrow*}\subfloat{
\includegraphics[width=.4\linewidth]{multilatEx2.pdf}}
%\hspace{.3in}%
\subfloat{\begin{tikzpicture}[ scale = 2,baseline=(current bounding box.south)]
\tikzstyle{vtx}=[inner sep=1pt,draw, shape=circle];
\tikzstyle{line}=[inner sep=3pt,draw, shape=circle];

\node[vtx,white, fill=black] (v0) at (0:1) {$v_{0}$};
\node[vtx, white,fill=green!40!black!90!white]  (v1) at (1*90:1) {$v_{1}$};
\node[vtx, fill=green!60!black!40!white] (v2) at (2*90:1) {$v_{2}$};
\node[vtx, fill=orange!70!white] (v3) at (3*90:1) {$v_{3}$};

\node[line,white, fill=black] (L0) at (0*90+45:1) {$L_{0}$};
\node[line,white, fill=green!40!black!90!white] (L1) at (1*90+45:1) {$L_{1}$};
\node[line, fill=green!60!black!40!white] (L2) at (2*90+45:1) {$L_{2}$};
\node[line,white,fill=red] (L3) at (3*90+45:1) {$L_{3}$};

\foreach \i in {0,1,2,3}{
\draw[ultra thick] (v\i) edge[bend left = 30] (L\i);
}

\foreach \i/\j in {0/1,1/2,2/3}{
\draw[ultra thick](L\i) -- (v\j);}

\draw[] (L0)edge[->,bend left = 30]  node[midway, auto] {1}(v0); 
\draw[] (L1)edge[->,bend left = 30] node[midway, auto] {2}(v1) ; 
\draw[] (L2)edge[->,bend left = 30] node[midway, auto] {2}(v2) ; 
\draw[] (L3)edge[->,bend left = 30]node[midway, auto] {1} (v3) ; 
\draw[->] (L3)--(v0) node[midway, auto, swap] {1}; 
\end{tikzpicture}$\mathbb{Z}_{5}$
}\end{subfloatrow*}}
{
\caption{Two multilateral 3-configurations, with $m = 5$. In both configurations, the first  vertex class $v_{0}$ is colored black, and the last vertex class $v_{3}$ is colored orange; the first line class $L_{0}$ is black and the last line class $L_{3}$ is red. Top: $\lambda_{0} = \lambda_{1} = \lambda_{2} = .4$; Bottom: $\lambda_{0} = .4, \lambda_{1} = \lambda_{2} = .3$.}
\label{multilateralEx}}
\end{figure}





\item This process continues iteratively: for $1 \leq j \leq h-2$,
\[ (v_{j})_{i} =   (1-\lambda_{j})(v_{j-1})_{i} + \lambda_{j}(v_{j-1})_{i+a_{j-1}}\] and
\[ (L_{j})_{i} = (v_{j})_{i} \wedge (v_{j})_{i+a_{j}}\]

\item Finally, to construct the $h$-th orbit of points and lines (that is, the lines $L_{h-1}$ and the points $v_{h-1}$), note that in the reduced Levi graph we have a double-arc between $L_{h-1}$ and $v_{h-1}$, labelled $a_{h-1}$ and $0$, and a single arc labelled $t$ between $L_{h-1}$ and $v_{0}$. This labelling indicates that the lines $L_{h-1}$ should be of span $a_{h-1}$ with respect to  the vertices $v_{h-1}$, and moreover, line $(L_{h-1})_{0}$ must pass through point $(v_{0})_{t}$. We use the configuration construction lemma and the lines $L_{h-1}$ to construct the required points $v_{h-1}$: point $(v_{h-1})_{0}$ is an intersection of the  line $(L_{h-1})_{0}$ with the circumcircle $\mc{C}$ passing through $\mc{O}$, $(v_{0})_{t}$ and $(v_{0})_{t-a_{h-1}}$. The rest of the points $(v_{h-1})_{i} $ are the rotated images, and  $(L_{h-1})_{i} := (v_{h-1})_{i} \wedge (v_{h-1})_{i+a_{h-1}}$.
\ee

Note that for a given multilateral $h$-astral 3-configuration, if $h > 2$ then there are $h-2$ continuous parameters $\lambda_{1}, \ldots, \lambda_{h-2}$: non-astral multilateral configurations are movable! Also, if $h = 2$ then the construction produces the chiral astral 3-configurations already described.





\subsection{Caterpillar 3-configurations} These have the reduced Levi graphs of the form shown in Figure \ref{snake}. Note that there is a ``wiggle'' of zero-labels along the center of the configuration.

\begin{figure}[htbp]
\begin{center}
\begin{tikzpicture}[scale=1.5]
	\path (0,0) node[vtx] (v0) {$v_{0}$};
	\path (0,1) node[line,fill=black!20] (L0) {$L_{0}$};
	\path (1,0) node (L1)[line, fill=black!20] {$L_{1}$};
	\path (1,1) node (v1)[vtx] {$v_{1}$};
	\path (2,0) node (v2)[vtx] {$v_{2}$};
	\path (2,1) node (L2)[line,fill=black!20] {$L_{2}$};
	\path (3,0) node (L3)[line,fill=black!20] {$L_{3}$};
	\path (3,1) node (v3)[vtx] {$v_{3}$};
	\path (4,1) node (L4)[line,fill=black!20] {$L_{h-2}$};
	\path (4,0) node (v4)[vtx] {$v_{h-2}$};
	\path (5,0) node (Lh)[line,fill=black!20] {$L_{h-1}$};
	\path (5,1) node (vh)[vtx] {$v_{h-1}$};
	
	\foreach \i in {1, ..., 4}
	{\draw[ultra thick] (L\i) -- (v\i);}
	
	\draw[ultra thick] (L0) -- (v1) ;
	\draw[ultra thick] (L1) -- (v2);
	\draw[ultra thick] (L2) -- (v3);
	\draw[ thick] (L3) edge[dotted] (v4);
	\draw[ultra thick] (L4) -- (vh)node[midway, below=2pt] {};
	
	\draw[<-] (v0) -- (L1) node[midway, above=2pt] {$a_{1}$} ;
	\draw[<-] (v1) -- (L2)node[midway, above=2pt] {$a_{2}$};
	\draw[<-] (v2) -- (L3)node[midway, above=2pt] {$a_{3}$};
	\draw (v3) edge[dotted, thick] (L4);
	\draw[<-] (v4) -- (Lh)node[midway, below=2pt]{$t$};
	
	\draw (v0) edge[bend right=30, ultra thick] (L0);
	\draw (v0) edge[<-,bend left=30]  node[midway, left] {$a_{0}$} (L0);
	
	\draw (vh) edge[bend right=30, ultra thick] (Lh);
	\draw (vh) edge[<-,bend left=30]node[midway, right] {$a_{h-1}$} (Lh);

	\end{tikzpicture}$\mathbb{Z}_{m}$
\caption{A reduced Levi graph for a caterpillar 3-configuration}
\label{snake}
\end{center}
\end{figure}

To construct a caterpillar configuration:

\begin{enumerate}
\item Construct vertices $(v_{0})_{i}$ as the vertices of a regular convex $m$-gon. Construct lines $(L_{0})_{i}$ as lines of span $a_{0}$; that is, construct $(L_{0})_{i} := (v_{0})_{i} \vee (v_{0})_{i+a_{0}}$.
\item For each $i = 1, 2, \ldots, h-2$, place vertex $(v_{j})_{0}$ arbitrarily on line $(L_{j-1})_{0}$, and construct the rest of the vertices $(v_{j})_{i}$ as the image under rotation by $\frac{2 \pi i}{m}$ of vertex $(v_{j})_{0}$. Then construct lines $(L_{j})_{i}$ as
$(L_{j})_{i} := (v_{j})_{i} \vee (v_{j-1})_{i+a_{j}}$. Note that these lines $(L_{j})_{i}$ for $j>1$ connect vertices from two consecutive symmetry classes!
More precisely, there are continuous parameters $\lambda_{1}, \ldots, \lambda_{h-2}$ so that $(v_{1})_{0} = (1-\lambda_{1}) (v_{0})_{0} + \lambda_{1} (v_{0})_{a_{0}}$, while if $j = 2, \ldots, h-2$,  $(v_{j})_{i} = (1-\lambda_{j}) (v_{j-1})_{i} + \lambda_{j} (v_{j-2})_{i+a_{j}}$.
\item To complete the configuration, we need to construct a class of vertices $(v_{h-1})_{i}$ and a class of lines $(L_{h-1})_{i}$ so that $(v_{h-1})_{i}$ lies on $(L_{h-2})_{i}$, $(L_{h-1})_{i} = (v_{h-2})_{i} \vee (v_{h-2})_{i+a_{h-1}}$, and $(L_{h-1})_{i}$ passes through vertex $(v_{h-2})_{t}$. To do so, we apply the configuration construction lemma. Construct the circumcircle passing through $(v_{h-2})_{t}$, $(v_{h-2})_{t-a_{h-1}}$, and the center of the configuration. Define $(v_{h-1})_{0}$ to be one of the two intersections of this circle with line $(L_{h-2})_{0}$. The configuration construction lemma says that if we now construct  line $(L_{h-1})_{0} := (v_{h-1})_{0} \vee (v_{h-1})_{a_{h-1}}$, then this line passes through $(v_{h})_{t}$; by symmetry, if line $(L_{h})_{i}$ is the image under rotation by $\frac{2 \pi i}{m}$, then each line $(L_{h})_{i}$ passes through $(v_{h})_{i}$, $(v_{h})_{i+a_{h-1}}$, and $(v_{h-1})_{t}$, as desired.
\end{enumerate}


%There are some obvious constraints on the choices for the $a_{i}$ and $t$.
%\begin{itemize}
%\item $1 \leq a_{0}, a_{h} < \frac{m}{2}$, because by convention, we use the smallest possible value for a span $a$ line.
%\item 

%\end{itemize}

An interesting example of a caterpillar configuration is shown in Figure \ref{caterpillarEx}; it is a geometric realization of a  famous $(15_{3})$ configuration known as the Cremona-Richmond configuration (compare with the drawing in \cite[p. 2]{Gru2009b}).

Note that an $h$-astral caterpillar configuration has $h$ discrete parameters, corresponding to the labels $a_{0}, a_{1}, \ldots, a_{h-1}, t$ in the reduced Levi graph, and $h-2$ \emph{continuous} parameters, corresponding to the arbitrary placement of vertex $(v_{j})_{0}$ on line $(L_{j-1})_{0}$ for $j = 1, 2, \ldots, h-2$ in the construction of the configuration. If $h = 2$, the caterpillar construction produces again a chiral astral 3-configuration, which has no continuous parameters.

\begin{figure}[htbp]
\centering
\ffigbox{
\begin{subfloatrow*}
\subfloat{\includegraphics[width=.5\linewidth]{cremona}}
\subfloat{
\begin{tikzpicture}[scale = 2]
\node[vtx, white,fill=black] (u) at (0,1) {$u$};
\node[line,white, fill =black] (L) at (0,0) {$L$};
\node[line,white, fill = blue] (M) at ($(u) + (1,0)$) {$M$};
\node[vtx, white,fill = blue] (v) at ($(L) + (1,0)$) {$v$};
\node[line, white,fill = red] (N) at ($(v) + (1,0)$) {$N$};
\node[vtx,white, fill = red] (w) at ($(M) + (1,0)$) {$w$};
\draw[ultra thick] (N) edge[bend left=30]  (w);
\draw (w) edge [<-,bend left=30] node[midway, right]{$1$} (N);
\draw[ultra thick] (L) edge[bend right=30]  (u);
\draw (u) edge [<-,bend right=30] node[midway, left]{$2$} (L);
\draw[ultra thick](L)--(v) --(M) -- (w);
\draw[<-] (u) -- node[auto]{1} (M);
\draw[<-] (v) -- node[auto]{3} (N);% ++(.3,-.3) node {$\mathbb{Z}_{5}$};

\end{tikzpicture}$\mathbb{Z}_{5}$
}
\end{subfloatrow*}
}{
\caption{The Cremona-Richmond configuration analyzed as a caterpillar configuration, along with the corresponding reduced Levi graph.}
\label{caterpillarEx}}
\end{figure}


\section{Other symmetric 3-configurations}


There is a single underlying reduced Levi graph that corresponds to chiral 3-configurations with two orbits of points and lines, which we discussed in Section \ref{LemmaSec}, shown in Figure \ref{astralVolt}. In \emph{Polycyclic Configurations} \cite[Figures 8 and 9]{BobPis2003}, Boben and Pisanski present all possible (unlabelled) 3-valent multigraphs which might be the underlying graph for a reduced Levi graph of  3-configurations with 3 and 4 orbits of points and lines; their figures are reproduced in Figures \ref{volt3orbits} and \ref{volt4orbits}. Of these, seven have at least one double-arc, and using variants of the techniques discussed previously, it is possible to develop geometric constructions to produce configurations with these as their underlying reduced Levi graphs. 

\newpage
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=.6\linewidth]{3Orbit.pdf}
\caption{The  possible reduced Levi graphs corresponding to 3-configurations with three orbits of points and of lines (image taken from \cite{BobPis2003}). Note that the middle graph corresponds to 3-astral multilateral configurations and the rightmost graph corresponds to 3-astral caterpillar configurations.}
\label{volt3orbits}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=.6\linewidth]{4Orbit.pdf}
\caption{The possible reduced Levi graphs corresponding to 3-configurations with four orbits of points and of lines (image taken from \cite{BobPis2003}), i.e., 4-astral 3-configurations. The graph in the (1,1)-entry corresponds to a multilateral configuration and the (2,2)-entry to a caterpillar configuration. Configurations corresponding to the graphs in the $(1,3)$-,  $(2,1)$-, and $(2,3)$-entries may be constructed using variants of the construction techniques described in this article, with examples shown in Figures \ref{weirdPic}a,b and Figure \ref{WeirdEx3} respectively. The construction methods used to produce these configurations are discussed in Examples \ref{weird} and \ref{weirder}.}
\label{volt4orbits}
\end{center}
\end{figure}

The single reduced Levi graph with 3 orbits which has no double-arcs  corresponds to ``pappus-like'' configurations (Figure \ref{paplikePic}); the Pappus configuration has a realization with 3-fold rotational symmetry with such a graph as its underlying reduced Levi graph, shown in Figure \ref{pappusPic}.  Similar $h$-astral configurations for odd $h$ (which Boben and Pisanski mentioned briefly \cite{BobPis2003} as configurations which had realizations over odd M\"{o}bius graphs) are straightforward to construct using algebraic techniques but no purely geometric construction technique is yet known. It is also possible using algebraic techniques to construct $h$-astral 3-configurations for even $h$ in which the corresponding reduced Levi graphs have no double-arcs and are even prism graphs instead of odd M\"{o}bius graphs (since the reduced Levi graphs must be bipartite). Figure \ref{cubical} shows a configuration whose reduced Levi graph is the cubical (4-prism) graph, i.e., the (1,2)-entry in Figure \ref{volt4orbits}, but again, no purely geometric construction is known.




\begin{figure}[htbp]
\centering
\ffigbox{
\begin{subfloatrow*}
\subfloat[A chiral realization of the Pappus configuration]{\label{pappusPic} \includegraphics[width=.4\linewidth]{pappus2}}%generalizedPappusLikeN3[3, 0, 1, {1, 1}, {.0, .2, .2}, .7, 1] 
\subfloat[A ``pappus-like'' configuration, with $m = 6$]{\label{paplikePic} \includegraphics[width=.4\linewidth]{pappuslike2}}%generalizedPappusLikeN3[6, 0, 2, {1, 1}, {.0, .2, .2}, 2, 1] 
\end{subfloatrow*}

\begin{subfloatrow*}
\subfloat[A configuration whose unlabelled reduced Levi graph is the 5-M\"{o}bius graph, over $\mathbb{Z}_{5}$]{\label{moebius}\includegraphics[width=.4\linewidth]{moebius2}}%generalizedPappusLikeN3[5, 0, 2, {1, 1, 2, 1}, {.3, .6, .2}, .4, 1]
\subfloat[A configuration whose unlabelled reduced Levi graph is the cubical graph, over $\mathbb{Z}_{4}$]{\label{cubical} \includegraphics[width=.4\linewidth]{cubical2}} %generalizedPappusLikeN3[4, 0, 1, {1, 1, 2}, {.3, .6, .2}, .8, 1] 
\end{subfloatrow*}}{
\caption{Various small 3-configurations whose associated reduced Levi graphs contain no double-arcs.}
\label{pappus}}
\end{figure}


\begin{example}\label{weird}Constructing  unusual 4-astral 3-configurations with two double-arcs in the reduced Levi graph. \end{example}To construct configurations whose underlying reduced Levi graph has two double-arcs, first identify a spanning path in the reduced Levi graph that begins at one double-arc and ends at the other. Assign labels to edges that don't participate in the spanning path, and use the path to determine the construction order. (Of course, in general finding Hamiltonian paths in graphs is a hard problem. However, for small  graphs, it is straightforward.) Figure \ref{weirdPic} shows two examples, whose construction is described below.

\begin{figure}[!htbp]
\begin{center}
\subfloat[$m = 5$ and $(a_{0}, a_{1}, a_{2}, a_{3}, t) = (2,1,1,2,1)$. Important vertices and lines are labelled.]{\label{ex3}
\includegraphics[width=.4\linewidth]{Weird4orbitalEx.pdf}
\hspace{.1in}
\begin{tikzpicture}[scale=2]
\node[draw, circle, inner sep = 2pt,white, fill=blue] (v1) at (0,0) {$v_{1}$};
\node[draw, circle, inner sep = 6pt,white,  fill=black] (L0) at (1,1) {$L_{0}$};
\node[draw, circle, inner sep = 6pt,white, fill=blue] (L1) at (1,0) {$L_{1}$};
\node[draw, circle, inner sep = 6pt,  fill=green] (L3) at (1,-1) {$L_{3}$};
\node[draw, circle,inner sep = 2pt, white, fill=black] (v0) at (2,1) {$v_{0}$};
\node[draw, circle,inner sep = 2pt,white, fill=red] (v2) at (2,0) {$v_{2}$};
\node[draw, circle,inner sep = 2pt, fill=green] (v3) at (2,-1) {$v_{3}$};
\node[draw, circle, inner sep = 6pt,white,  fill=red] (L2) at (3,0) {$L_{2}$};

\draw[ultra thick] (v0) edge[bend right=30] (L0);
\draw[ultra thick] (L0)--(v1)--(L1);
\draw[ultra thick] (v2) edge[bend right=30] (L1);
\draw[ultra thick] (v2)--(L2)--(v3);
\draw[ultra thick] (v3) edge[bend right=30] (L3);

\draw  (v0) edge[<-,bend left=30] node[midway, below]{$a_{0}$} (L0) ;
\draw  (v2) edge[<-,bend left=30] node[below,midway]{$a_{1}$} (L1) ;

\draw  (v3) edge[<-,bend left=30] node[below, midway]{$a_{3}$} (L3) ;
\draw  (v0) edge[<-] node[right ,midway]{$a_{2}$} (L2) ;
\draw  (L3) edge[->] node[ auto, midway]{$t$} (v1) ;
\end{tikzpicture}$\mathbb{Z}_{5}$
}

\vspace*{-1ex}
\subfloat[Here, $(a_{0}, a_{1}, a_{2}, a_{3}, t) = (1,2,1,1,1)$ and $m = 4$. The 0th element of each symmetry class is shown larger; colors between the configuration and the reduced Levi graph correspond.]{\includegraphics[width=.4\linewidth]{WeirdEx2.pdf}
\hspace{.1in}
\begin{tikzpicture}[scale=2.5]
\node[draw, circle, inner sep = 2pt, white, fill=black] (v0) at (22.5+5*45:1) {$v_{0}$};
\node[draw, circle, inner sep = 6pt, white, fill=black] (L0) at  (22.5+6*45:1) {$L_{0}$};
\node[draw, circle, inner sep = 2pt,white,  fill=blue] (v1) at (22.5+7*45:1) {$v_{1}$};
\node[draw, circle, inner sep = 6pt, white, fill=blue] (L1) at  (22.5+4*45:1) {$L_{1}$};
\node[draw, circle, inner sep = 2pt,white, fill=red] (v2) at (22.5+3*45:1) {$v_{2}$};
\node[draw, circle, inner sep = 6pt,white,  fill=red] (L2) at  (22.5+2*45:1) {$L_{2}$};
\node[draw, circle, inner sep = 2pt, fill=green] (v3) at (22.5+1*45:1) {$v_{3}$};
\node[draw, circle, inner sep = 6pt,  fill=green] (L3) at  (22.5+0*45:1) {$L_{3}$};


\draw[ultra thick] (v0) edge[bend right=30] (L0);
\draw[ultra thick] (L0)--(v1)--(L1)--(v2);
\draw[ultra thick] (v2) edge[bend right=30] (L2);
\draw[ultra thick] (L2)--(v3);
\draw[ultra thick] (v3) edge[bend right=30] (L3);

\draw  (v0) edge[<-,bend left=30] node[ midway, above]{$a_{0}$} (L0) ;
\draw  (v0) edge[<-] node[midway, below]{$a_{1}$} (L1) ;

\draw  (v3) edge[<-,bend left=30] node[midway, above]{$a_{3}$} (L3) ;
\draw  (v2) edge[<-,bend left=30] node[midway,above ]{$a_{2}$} (L2) ;
\draw  (L3) edge[->] node[ auto, midway]{$t$} (v1) ;
\end{tikzpicture}$\mathbb{Z}_{4}$

}

\caption{Two unusual 4-astral 3-configurations, and the corresponding general  reduced Levi graphs.}
\label{weirdPic}
\end{center}
\end{figure}

\newpage
As an example, consider the spanning path shown in the reduced Levi graph in Figure \ref{ex3}, with $m = 5$ and $(a_{0}, a_{1}, a_{2}, a_{3}, t) = (2,1,1,2,1)$. To construct the corresponding configuration:
\begin{enumerate}
\item Construct the vertices $(v_{0})_{i}$ as the vertices of a regular convex $5$-gon.
\item Following the thick path, we encounter a double-arc, so we construct lines $(L_{0})_{i}$ of span $a_{0} = 2$ with respect to the $(v_{0})_{i}$.
\item We encounter $(v_{1})$ in the reduced Levi graph, so we place $(v_{1})_{0}$ arbitrarily on line $(L_{0})_{0}$, and construct the rest of the $(v_{1})_{i}$ by cyclically rotating $(v_{1})_{0}$ (we ``spin the points around'').
\item We construct line $(L_{1})_{0}$ {\bf arbitrarily} (that is, at an arbitrary angle) through point $(v_{1})_{0}$, and construct the rest of the symmetry class by spinning the lines around.
\item Next, we encounter a double-arc, so we need to construct point $(v_{2})_{0}$ to lie on the intersection of lines $(L_{1})_{0}$ and $(L_{1})_{a_{1}} = (L_{1})_{1}$: that is, $(v_{2})_{0} = (L_{1})_{0} \wedge (L_{1})_{a_{1}}$. Construct the rest of the $(v_{2})_{i}$ by spinning.
\item Since $L_{2}$ in the reduced Levi graph is adjacent to both $v_{0}$ (with a label of $a_{2}$) and $v_{2}$ (with a label of $a_{1}$), we construct $(L_{2})_{0} = (v_{2})_{a_{1}} \wedge (v_{0})_{a_{2}}$.%Example 3, step 6 (on page 21). The label on the edge between L_2 and v_2 is 0 and not a_2, so (L_2)_0 = (v_2)_0 /\ (v_0)_{a_2}
\item Finally, we are close to our final double-arc. We need to construct the $(v_{3})_{i}$ and the $(L_{3})_{i}$ so that $(v_{3})_{0}$ lies on $(L_{2})_{0}$ and also so that a line through $(v_{3})_{0}$ and $(v_{3})_{a_{3}}$  passes through $(v_{1})_{t}$. To do so, we apply the configuration construction lemma and define $(v_{3})_{0}$ to be an intersection of the circumcircle through $(v_{1})_{t}$,  $(v_{1})_{t-a_{3}}$ and $\mc{O}$ with line $(L_{2})_{0}$. In the example in Figure \ref{weirdPic}a we constructed a circumcircle passing through $(v_{1})_{1}$, $\mc{O}$, and $(v_{1})_{1-2} = (v_{1})_{-1}$ and chose the farther intersection with $(L_{2})_{0}$ as the point $(v_{3})_{0}$. Construct the rest of the $(v_{3})_{i}$ as usual, and construct lines $(L_{3})_{i}$ to be of span $a_{3}$ with respect to these $(v_{3})_{i}$.
\end{enumerate}

The construction technique for the second type of configuration (shown in Figure \ref{weirdPic}b) is slightly simpler. Construct the $(v_{0})_{i}$ as the vertices of a regular convex $m$-gon. Following the thick path in the reduced Levi graph, construct the lines $(L_{0})_{i}$ as lines of span $a_{0}$. Construct the point $(v_{1})_{0}$ arbitrarily on line $(L_{0})_{0}$ and construct the rest of the $(v_{1})_{i}$ as rotated images. Construct $(L_{1})_{0} = (v_{1})_{0} \vee (v_{0})_{a_{1}}$ and the rest of the $(L_{1})_{i}$ as rotated images. Construct $(v_{2})_{0}$ arbitrarily on $(L_{1})_{0}$, and the $(L_{2})_{i}$ of span $a_{2}$. Finally, we need the lines $(L_{3})_{i}$ to be span $a_{3}$ with respect to the $(v_{3})_{i}$ and pass through $(v_{1})_{t}$, so we construct the circumcircle through $(v_{1})_{t}$, $(v_{1})_{t-a_{3}}$ and the center, and choose an intersection with $(L_{2})_{0}$ to construct $(v_{3})_{0}$; the configuration construction lemma says if we construct the $(v_{3})_{i}$ in this way, then the lines $(L_{3})_{i}$ constructed to be span $a_{3}$ with respect to the $(v_{3})_{i}$ pass through $(v_{1})_{i+t}$.

\begin{example}\label{weirder} Constructing a 3-configuration whose reduced Levi graph has only a single double-arc.\end{example}

While it is easier to construct configurations with a pair of non-adjacent double-arcs, in fact, it is possible to construct configurations with only a single double-arc in the reduced Levi graph, by choosing a 0-labelled path in the reduced Levi graph that \emph{ends} at the double-arc. An example of the construction technique for the configuration shown in Figure \ref{WeirdEx3} follows, with $m = 4$ and $(a_{0}, a_{1}, a_{2}, a_{3}, t) = (1,2,3,1,0)$. This is the final configuration whose underlying reduced Levi graph is listed in Figure \ref{volt4orbits} and contains a double-arc.


\begin{figure}
\centering
\ffigbox{
\begin{subfloatrow*}
\subfloat{
\includegraphics[width=.5\linewidth]{WeirdEx3.pdf}}
%\hspace{.1in}
\subfloat{\begin{tikzpicture}[scale=1.5]
\node[draw, circle, inner sep = 2pt, white, fill=black] (v0) at (3,2.3) {$v_{0}$};
\node[draw, circle, inner sep = 6pt,white,   fill=black] (L0) at  (3,0) {$L_{0}$};
\node[draw, circle, inner sep = 2pt, white, fill=blue] (v1) at (0,0) {$v_{1}$};
\node[draw, circle, inner sep = 6pt, white,  fill=blue] (L1) at  (1,1) {$L_{1}$};
\node[draw, circle, inner sep = 2pt,white,  fill=red] (v2) at (2,1) {$v_{2}$};
\node[draw, circle, inner sep = 6pt, white,  fill=red] (L2) at  (0,2.3) {$L_{2}$};
\node[draw, circle, inner sep = 2pt,  fill=green] (v3) at (1,3) {$v_{3}$};
\node[draw, circle, inner sep = 6pt,fill=green] (L3) at  (2,3) {$L_{3}$};


\draw[ultra thick] (v0)--(L0)--(v1)--(L1)--(v2)--(L2)--(v3);
\draw[ultra thick] (v3) edge[bend right=30] (L3);

\draw  (v0) edge[<-] node[ auto, midway]{$a_{0}$} (L1) ;
\draw  (v2) edge[<-] node[ auto, midway]{$a_{1}$} (L0) ;

\draw  (v3) edge[<-,bend left=30] node[ auto, midway]{$a_{3}$} (L3) ;
\draw  (v1) edge[<-] node[ auto, midway]{$a_{2}$} (L2) ;
\draw  (L3) edge[->] node[ auto, midway]{$t$} (v0) ;
\end{tikzpicture}$\mathbb{Z}_{4}$}
\end{subfloatrow*}}{

\caption{A 4-astral 3-configuration with a single double-arc in the reduced Levi graph. Here, $m = 4$ and $(a_{0}, a_{1}, a_{2}, a_{3}, t) = (1,2,3,1,0)$. The $0$-th element of each symmetry class is shown larger, and colors indicate correspondence between the configuration and the reduced Levi graph.}
\label{WeirdEx3}}
\end{figure}

\begin{enumerate} 
\item Begin by constructing the vertices $(v_{0})_{i}$ as the vertices of a regular convex $m$-gon. Construct line $(L_{0})_{0}$ {\bf arbitrarily} through vertex $(v_{0})_{0}$ and construct the rest of the $(L_{0})_{i}$.
\item Construct $(v_{1})_{0}$ arbitrarily on line $(L_{0})_{0}$, and construct the rest of the $(v_{1})_{i}$.
\item Next, construct $(L_{1})_{i} = (v_{1})_{i} \vee (v_{0})_{i+a_{0}}$.
\item Then, construct the points $(v_{2})_{i}$ as the intersection  $(L_{1})_{i} \wedge (L_{0})_{i+a_{1}}$.
\item Construct $(L_{2})_{i} = (v_{2})_{i} \vee (v_{1})_{i+a_{2}}$.
\item Finally, we use the circumcircle construction lemma to construct $(v_{3})_{i}$ and $(L_{3})_{i}$ so that $(L_{3})_{i}$ are lines of span $a_{3}$ with respect to the points $(v_{3})$ that pass through the points $(v_{0})_{i+t}$, by constructing the circumcircle through the center of the $m$-gon, $(v_{0})_{t}$, and $(v_{0})_{t-a_{3}}$ and taking the intersection of the circumcircle and line $(L_{2})_{0}$ to be point $(v_{3})_{0}$.
\end{enumerate}
Note each of the classes of configurations described in Examples \ref{weird} and \ref{weirder}, as well as the 4-astral multilateral and caterpillar 3-configurations,  have two continuous parameters, corresponding to arbitrary placement of certain points and/or lines. 
%
%\subsection{``Pappus-like'' and ``cubical'' configurations}\label{pappus}
%
%To construct a configuration whose labelled voltage graph is shown in Figure \ref{pappusVoltage}, we can use the following algorithm. This is not a purely geometric construction, as it requires solving a determinant equation for a good choice of radius. Note that this is the first case of configurations with odd M\"{o}bius graph as voltage graph; in \cite{BobPis2003} they indicate that all odd M\"{o}bius graphs have rotational realization but do not provide any details of the construction.
%
%\begin{figure}[htbp]
%\begin{center}
%
%\caption{Voltage graph for ``Pappus-like'' configurations.}
%\label{pappusVoltage}
%\end{center}
%\end{figure}
%
%
%The algorithm:
%\begin{small}
%\begin{verbatim}
%u[i_] := {Cos[2 Pi i/m], Sin[2 Pi i/m]};
%(* L[0][i_, t_]:= (1-t) u[i]+ t*u[i+a]; *)
%
%v[i_, r_] := r* {Cos[2 Pi i/m + \[Theta]], Sin[2 Pi i/m + \[Theta]]};
%
%L[0][i_, t_, r_] := (1 - t) u[i] + t*v[i, r]; (*blue*)
%
%L[1][i_, t_, r_] := (1 - t) u[i] + t*v[i + c, r];(*red*)
%
%M[i_, t_, r_] := (1 - t)*u[i + b] + t*v[i + d, r];(*green*)
%
%w[0, r_] := 
% M[0, NSolve[M[0, t, r] == L[0][-a, q, r], {t, q}][[1, 1, 2]], r];
%w[i_, r_] := RotationTransform[2 Pi i/m][ w[0, r]];
%(*Print["this is w:",w[0,x]];*)
%
%(* constraint:M[0] =  v[d], w[0], u[b] collinear*)
%constr = 
% Map[Chop, FullSimplify[Det[{
%      Append[w[0, r], 1],
%      Append[v[c, r], 1],
%      Append[u[0], 1]
%      }
%     ], Infinity]] == 0;
%(*Print[constr];*)
%If[constr === True,
% Print["Any radius will do; we'll choose 1/2"];
% s = 1/2,
% (*else*)
% ans = NSolve[constr, r];           
% s = Table[ans[[i, 1, 2]], {i, 1, Length[ans]}][[chooseMe]];
% ];
% \end{verbatim}
%\end{small}

\section{Dihedrally symmetric 3-configurations}
Any of the reduced Levi graphs used to construct configurations with chiral symmetry discussed in the previous sections may also be used to construct configurations with dihedral symmetry, by a simple modification of the construction technique; we say that we ``dihedralize'' the original configuration. Suppose you have a chirally symmetric configuration whose reduced Levi graph contains a double-arc at the end of a 0-labelled path whose nodes, say $w$ and $N$, are attached by single arcs to two other nodes in the reduced Levi graph, say $L$ and $v$ respectively. Delete $w$ and $N$ and the attached arcs from the reduced Levi graph, and duplicate the remaining subgraph, attaching superscripts of $+$ to all the node labels in one copy and $-$ to all the node labels in the other copy (see Figure \ref{dihedralVolt}).

 We reflect this labeling in the reduced Levi graph within the original configuration by first constructing a subconfiguration consisting of the vertices and lines corresponding to the $+$ superscripted sub-reduced-Levi graph, and consider the labels of the vertices and lines to be superscripted with $+$ in the obvious way. Next, choose an arbitrary mirror $\mc{M}$ passing through the center of the subconfiguration. Finally, reflect the vertices and lines over $\mc{M}$, and superscript the reflected images with $-$ (see Figures \ref{subcfgs} and \ref{subcfg1} for examples).
    
Now construct new nodes $w^{+}$ and $N^{+}$ and new arcs in the reduced Levi graph, by using the configuration construction lemma in the configuration, by defining $w_{0}^{+}$ to be the intersection of the circumcircle through $v_{d}^{+}$, $\mc{O}$ and $v_{d-c}^{+}$ with line $L_{0}^{-}$, and then define $N_{0}^{+}$ to be the line through $w_{0}^{+}, w_{c}^{+}$, and $v_{d}^{+}$; construct the other vertices $w_{i}^{+}$ and the lines $N_{i}^{+}$ in the configuration in the appropriate way. Finally, reflect all the lines $N^{+}$ and points $w^{+}$ over $\mc{M}$ to construct $N^{-}$ and $w^{-}$; this mirroring is indicated by green dashing in  Figure \ref{dihedralVolt}. By construction, line $N_{0}^{-}$ contains points $w_{0}^{-}$, $w_{c}^{-}$ and $v_{d}^{-}$, and point $w_{0}^{-}$ lies on lines $L_{0}^{+}$, $N_{0}^{-}$, and $N_{c}^{-}$.


 \begin{figure}[htbp]
\begin{center}
%
{\begin{subfloatrow*}[3]
%\subfloat[a reduced Levi graph]{
\subfloat{
\begin{tikzpicture}[scale=.8]
\node[name=h,cloud, draw, fill=gray!20, aspect=.6] at (0,0) {\parbox{.2in}{\phantom{Levi}}}; 
\node[line, fill=white] (L) at (h.north east) {$L$};
\node[vtx, fill=white] (v) at (h.south east) {$v$};
\node[vtx] (w) at ($(L) + (2,0)$) {$w$};
\node[line,fill=white] (N) at ($(v) + (2,0)$) {$N$};
\draw[ultra thick] (L) -- (w) ;
\draw[<-] (v) -- node[auto] {$d$} (N) ;
\draw (w) edge[<-, bend left=30] node[midway, right] {$c$} (N);
\draw[ultra thick] (w) edge[bend right=30] (N);
%\draw[baseline=(current bounding box.center), snake, ->] ($(w) + (.5, .5)$) --+(1,0);
\end{tikzpicture}}
%
\subfloat{\begin{tikzpicture}%[baseline=current bounding box.south]
\draw[->, ultra thick] (0,0) -- (.5,0);
\end{tikzpicture}}
%
%\subfloat[Converting the reduced Levi graph of a chirally symmetric configuration to the reduced Levi graph of a dihedrally symmetric  configuration]{
\subfloat{
\begin{tikzpicture}[scale=1.7]
\node[name=h,cloud, draw, fill=gray!20, aspect=.6] at (0,0) {\parbox{.5in}{
\center{\tiny{reduced Levi subgraph} $+$}}
}; 
\node[name=j,cloud, draw, fill=gray!20, aspect=.6] at (4,0) {\parbox{.54in}{\center{\tiny{reduced Levi subgraph} $-$}}}; 
\node[line, fill=white] (L) at (h.north east) {$L^{+}$};
\node[vtx, fill=white] (v) at (h.south east) {$v^{+}$};
%
\node[line, fill=white] (L-) at (j.north west) {$L^{-}$};
\node[vtx, fill=white] (v-) at (j.south west) {$v^{-}$};
%\node[vtx] (w) at ($(L) + (1,0)$) {$w^{+}$};
%
\node[line,fill=white] (N) at ($(v) + (.7,.3)$) {$N^{+}$};
\node[vtx] (w+) at ($(L-) - (.7,.3)$) {$w^{+}$};
\draw[ultra thick] (w+) -- (L-);
\draw (w+) edge[<-, bend left=20] node[near end, below] {$c$} (N);
\draw[ultra thick] (w+) edge[bend right=20] (N);
\draw[<-] (v) -- node[midway, below] {$d$} (N) ;
%
\node[line, green!70!black] (N-) at ($(v-) - (.7,-.3)$) {$N^{-}$};
\node[vtx, green!70!black] (w-) at ($(L) +(.7,-.3)$) {$w^{-}$};
\draw[ultra thick, dashed, green!70!black] (w-) -- (L);
\draw[<-,dashed, thick, green!70!black] (w-) edge[bend right=20] node[near end, below] {$c$} (N-);
\draw[ultra thick,  dashed, green!70!black] (w-) edge[bend left=20] (N-);
\draw[<-, dashed,  green!70!black]  (v-) -- node[midway, below left] {$d$} (N-) ;
\end{tikzpicture}
}
\end{subfloatrow*}}

\vspace*{-6ex}
\caption{Beginning with a reduced Levi graph for a chirally symmetric 3-configuration that includes a double-arc, it is possible to convert the reduced Levi graph to a reduced Levi graph for a dihedrally symmetric configuration.% Delete from the reduced Levi graph the double-arc and two attached arcs, and duplicate the remaining subgraph, attaching $+$ to the labels from one side and $-$ to the other. Using the configuration construction lemma, construct the set of vertices $w^{+}$ %by defining $w_{0}^{+}$ to be the intersection of the circumcircle through $v_{d}^{+}$ $\mc{O}$ and $v_{d-c}^{+}$ with line $L_{0}^{-}$, and then define $N_{0}^{+}$ to be the line through $w_{0}^{+}, w_{c}^{+}$, and $v_{d}^{+}$; construct the other vertices $w^{+}$ 
%and the lines $N^{+}$ as described in the text. Finally, mirror all the lines $N^{+}$ and $w^{+}$ to be $N^{-}$ and $w^{-}$; 
The final mirroring is indicated by green dashing in the reduced Levi graph.}
\label{dihedralVolt}
\end{center}
\end{figure}


As indicated by the reduced Levi graphs, if the chirally symmetric 3-configurations have $h$ symmetry classes of points and lines, then the dihedralized versions have $2h$ symmetry classes of points and lines under the symmetry group $\mathbb{Z}_{m}$. However, by construction, the elements $w^{+}$ and $w^{-}$ lie in the same symmetry class under the symmetry group $D_{m}$, since by construction $w_{0}^{+}$ reflects to $w_{0}^{-}$, etc.  
As explicit examples, we dihedralize a chiral astral 3-configuration, to produce a dihedrally symmetric astral 3-configuration, which gives a new geometric construction for the dihedral astral DD configurations discussed in \cite[Section 2.8]{Gru2009b} that does not rely on solving some quadratic equation. We also dihedralize the Cremona-Richmond configuration shown in Figure \ref{caterpillarEx}.

\begin{example} Constructing a dihedral astral 3-configuration \end{example}

The dihedralized chiral astral 3-configuration has the reduced Levi graph shown in Figure \ref{astralDihVolt}.
\begin{figure}[!b]
\begin{center}

\begin{tikzpicture}[scale=1.5]
%\node[name=h,cloud, draw, fill=gray!20, aspect=1] at (0,0) {\parbox{.7in}{voltage graph $+$}}; 
%\node[name=j,cloud, draw, fill=gray!20, aspect=1] at (6,0) {\parbox{.7in}{voltage graph $-$}}; 
\node[line, white, fill=black] (L) at (0,2) {$L^{+}$};
\node[vtx, white, fill=black] (v) at (0,0) {$v^{+}$};
\draw[ultra thick] (L) edge[bend left = 30] (v); \draw[->] (L) edge[bend right = 30] node[midway, left] {$a$} (v);

\node[line, fill=gray!40] (L-) at (6,2) {$L^{-}$};
\node[vtx, fill=gray!40] (v-) at (6,0) {$v^{-}$};
\draw[ultra thick] (L-) edge[bend right = 30] (v-); \draw[->] (L-) edge[bend left = 30] node[midway, right] {$a$} (v-);
%\node[vtx] (w) at ($(L) + (1,0)$) {$w^{+}$};

\node[line,white, fill=red] (N) at ($(v) + (1.2,.3)$) {$N^{+}$};
\node[vtx, white, fill=red] (w+) at ($(L-) - (1.2,.3)$) {$w^{+}$};
\draw[ultra thick] (w+) -- (L-);
\draw (w+) edge[<-,bend left=20] node[midway, above] {$b$} (N);
\draw[ultra thick] (w+) edge[bend right=20] (N);
\draw[<-] (v) -- node[midway, below] {$d$} (N) ;

\node[line, green!70!black, fill=magenta] (N-) at ($(v-) - (1.2,-.3)$) {$N^{-}$};
\node[vtx, green!70!black, fill=magenta] (w-) at ($(L) +(1.2,-.3)$) {$w^{-}$};
\draw[ultra thick, dashed, green!70!black] (w-) -- (L);
\draw[<-,dashed, thick, green!70!black] (w-) edge[bend right=20] node[midway, above] {$b$} (N-);
\draw[ultra thick,  dashed, green!70!black] (w-) edge[bend left=20] (N-);
\draw[<-,dashed, green!70!black]  (v-) -- node[auto] {$d$} (N-) ;

\end{tikzpicture}$\mathbb{Z}_{3}$

\caption{The reduced Levi graph corresponding to the dihedralized chiral astral 3-configuration whose reduced Levi graph is shown in Figure \ref{astralVolt}. The node colors correspond to the colors of the points and lines in the  configuration shown in Figure \ref{astralDih} (in which $m = 3$, $a = b = d = 1$).}
\label{astralDihVolt}
\end{center}
\end{figure}
%
%\newpage
Figure \ref{astralDih} shows four steps in the construction of the corresponding configuration, in the case where $m = 3$ and $a = b = t = 1$. Begin by constructing vertices $v_{i}^{+}$ and their mirror images $v_{i}^{-}$ and the span $a$ lines $L_{i}^{+}$ and $L_{i}^{-}$. Next,  construct the circle $\mc{O}$ through $v_{t}^{+}$, $v_{t-b}^{+}$ and $\mc{O}$, and intersect that with line $L_{0}^{-}$ to form the point $w_{0}^{+}$;  construct the rest of the points $w_{i}^{+}$ as rotated images. The lines $N_{i}^{+}$ are constructed as span $b$ lines with respect to the $w_{i}$. Finally, reflect the elements $w_{i}^{+}$ and $N_{i}^{+}$ over the mirror to form $w_{i}^{-}$ and $N_{i}^{-}$ finish the construction.

\newpage
\begin{figure}[!htbp]
\centering
\ffigbox{
\begin{subfloatrow*}\subfloat[The signed subconfigurations, which consist of vertices $v^{+}$ and $v^{-}$ and lines $L^{+}$ and $L^{-}$]{\label{subcfgs}\includegraphics[width=.4\linewidth]{astralDih1.pdf}}\subfloat[The point $w_{0}^{+}$ is the intersection of $L_{0}^{-}$ and $\mc{C}$.]{ \includegraphics[width=.4\linewidth]{astralDih2.pdf}}
\end{subfloatrow*}

\begin{subfloatrow*}
\subfloat[Constructing the rest of the $w_{i}^{+}$ and the lines $N_{i}^{+}$]{\includegraphics[width=.4\linewidth]{astralDih3.pdf} }\subfloat[Mirroring to construct the  $w_{i}^{-}$ and $N_{i}^{-}$]{\includegraphics[width=.4\linewidth]{astralDih4.pdf}}
\end{subfloatrow*}}{
\caption{Constructing a dihedrally symmetric astral  $(12_{3})$ configuration. The green dashed line is the mirror $\mc{M}$. The initial subconfiguration in black is the $+$ and the one in gray is the $-$; the new points $w^{+}$ and new lines $N^{+}$ are red, and their reflected images are in magenta.}
\label{astralDih}}
\end{figure}

Unlike the chiral astral 3-configurations, these dihedral astral 3-configurations are movable: they have a continuous parameter corresponding to the angle between $\overline{\mc{O}v_{0}}$ and $\mc{M}$, as well as three discrete parameters.

\newpage
\begin{example} Constructing a dihedral version of a caterpillar configuration. \end{example}
Here, we will construct a dihedralized version of the Cremona-Richmond configuration, which was shown previously in Figure \ref{caterpillarEx}. The reduced Levi graph is shown in Figure \ref{dihCatVolt}, and the various steps of the construction are shown in Figure \ref{dihCat}. We begin by constructing the subconfiguration corresponding to vertex nodes $u$ and $v$ and line nodes $L$ and $M$, shown in black and blue, and reflect them over the mirror $\mc{M}$ shown in green; the reflected subconfiguration is shown in Figure \ref{subcfg1} in lighter colors (gray and cyan, respectively). Next, we need to construct new nodes $w^{+}$, which should be 0-adjacent to $L^{-}$, and $N^{+}$ in the reduced Levi graph so that $N^{+}$ is of span $1$ with respect to the $w_{i}^{+}$ and adjacent to $v^{+}$ with a label of $3$. To do so, we apply the configuration construction lemma, which says that if we construct $w_{0}^{+}$ to be the intersection of the circumcircle $\mc{C}$ through $v_{3}^{+}$, $\mc{O}$, and $v_{(3-1)}^{+} = v_{2}^{+}$ with line $M_{0}^{-}$ (note that you may need to vary the placement of the mirror so that this intersection point exists), then the points $w_{0}^{+}, w_{1}^{+}$ and $v_{3}^{+}$ will all be collinear (and we call that line $N_{0}^{+}$), while $w_{0}^{+}$ will lie on $M_{0}^{-}$. Finally, after constructing all the points $w_{i}^{+}$ and lines $N_{i}^{+}$, we construct the reflected images $w_{i}^{-}$ and $N_{i}^{-}$ to complete the configuration.

\begin{figure}[htbp]
\begin{center}
\begin{tikzpicture}[scale=1.5]
%\node[name=h,cloud, draw, fill=gray!20, aspect=1] at (0,0) {\parbox{.7in}{voltage graph $+$}}; 
%\node[name=j,cloud, draw, fill=gray!20, aspect=1] at (6,0) {\parbox{.7in}{voltage graph $-$}}; 
\node[line, white, fill=black] (L) at (0,0) {$L^{+}$};
\node[vtx,white,  fill=black] (u) at (0,2) {$u^{+}$};
\draw[ultra thick] (L) edge[bend right = 30] (u); \draw[->] (L) edge[bend left = 30] node[auto, swap] {$2$} (u);
\node[vtx, white, fill=blue] (v) at ($(L) + (1,0)$) {$v^{+}$};
\node[line, white, fill=blue] (M) at ($(u) + (1,0)$) {$M^{+}$};


\node[line, fill=gray!40] (L-) at (6,0) {$L^{-}$};
\node[vtx, fill=gray!40] (u-) at (6,2) {$u^{-}$};
\draw[ultra thick] (L-) edge[bend left = 30] (u-); \draw[->] (L-) edge[bend right = 30] node[auto] {$2$} (u-);
\node[vtx, white, fill=cyan!90!white] (v-) at ($(L-) - (1,0)$) {$v^{-}$};
\node[line, white, fill=cyan] (M-) at ($(u-) - (1,0)$) {$M^{-}$};



\node[line, white, fill=red] (N) at ($(v) + (1.2,.3)$) {$N^{+}$};
\node[vtx, white,fill=red] (w+) at ($(M-) - (1.2,.3)$) {$w^{+}$};
\draw[ultra thick] (w+) -- (M-);
\draw[<-] (w+) edge[bend left=20] node[near end, below] {$1$} (N);
\draw[ultra thick] (w+) edge[bend right=20] (N);
\draw[<-] (v) -- node[auto] {$3$} (N) ;

\node[line, green!80!black, fill=magenta] (N-) at ($(v-) + (-1.2,.3)$) {$N^{-}$};
\node[vtx, green!80!black, fill=magenta] (w-) at ($(M) +(1.2,-.3)$) {$w^{-}$};
\draw[ultra thick, dashed, green!70!black] (w-) -- (M);
\draw[<-,dashed, thick, green!70!black] (w-) edge[bend right=20] node[near end, below] {$1$} (N-);
\draw[ultra thick,  dashed, green!70!black] (w-) edge[bend left=20] (N-);
\draw[<-,dashed, green!70!black]  (v-) -- node[auto, swap] {$3$} (N-) ;

\draw[ultra thick] (M) -- (v) --(L) (M-)--(v-)--(L-);
\draw[<-] (u) edge[<-] node[auto]{1} (M) (u-) -- node[auto, swap]{1} (M-);

\end{tikzpicture}$\mathbb{Z}_{5}$

\caption{The reduced Levi graph for the dihedralized Cremona-Richmond configuration.}
\label{dihCatVolt}
\end{center}
\end{figure}


\section{Constraints on labels; expansion of the constructions}
In the previous sections, the constructions have all been based on the notion that we are working with a reduced Levi graph that has valid labels; that is, following the construction techniques using the given labels, a geometric configuration is produced. However, we have been silent on what allowable labels are. Because the reduced Levi graphs must correspond to geometric configurations, there are some immediate constraints on allowable labels. For example, if $a$ and $0$ are labels on a double-arc in a reduced Levi graph over $\mathbb{Z}_{m}$, then $0 < a < \frac{m}{2}$, since we do not allow lines of span $a$ to be diameters of a configuration (and because the Levi graph of the configuration (which the reduced Levi graph covers) can contain no 4-cycles), and by convention we assume that we take the smallest possible span measurement. 

\begin{figure}[H]
\centering
%\vspace*{-4ex}
\ffigbox{
\begin{subfloatrow*}
\subfloat[The subconfiguration corresponding to deleting the nodes $w$ and $N$ and the adjacent arcs, plus its mirror image (shown with lighter colors).]{\label{subcfg1} \includegraphics[width=.45\linewidth]{catDih1} }
\subfloat[Constructing $w_{0}$ as the intersection of $\mc{C}$ with $M_{0}^{-}$ (thick and gray), and $N_{0}$.]{ \includegraphics[width=.45\linewidth]{catDih2}}
\end{subfloatrow*}

\begin{subfloatrow*}
\subfloat[Constructing the rest of the $w_{i}$ and $N_{i}$]{\includegraphics[width=.45\linewidth]{catDih3} }
\subfloat[Mirroring constructs the $w_{i}^{-}$ and $N_{i}^{-}$]{\includegraphics[width=.45\linewidth]{catDih4}}
\end{subfloatrow*}}{
\caption{Constructing a  $(30_{3})$ configuration as a dihedralized version of the Cremona-Richmond configuration.}
\label{dihCat}}
\end{figure}

%\newpage

However, in general, fully describing allowable labels for these construction methods for 3-configurations is beyond the scope of this paper and will be discussed in a subsequent work.

In addition to possible choices of labels that would lead to reduced Levi graphs that do not correspond to actual constructable configurations (perhaps the circumcircle does not intersect the necessary line, for example), it is also possible for the construction methods to fail because they produce too many incidences. 
For example, Figure \ref{oops} shows a configuration corresponding to $12\#(4,4;1)$, a chiral astral 3-configuration with $m = 12, a = 4, b = 4, d = 1$ which is a 4-configuration rather than a 3-configuration. A complete classification of when and why these unintended incidences occur in the case of chiral astral 3-configurations will also be addressed in a subsequent work.





\begin{figure}[htbp]
\begin{center}

\ffigbox{
%
\begin{subfloatrow*}
\subfloat{
\includegraphics[width=.5\linewidth]{oops}
}
%
\subfloat{
\begin{tikzpicture}[scale=2]
\node[line, white, fill=black] (L) at (-45:1) {$L$};
\node[vtx,  white,fill=black] (v) at (45:1){$v$};
\node[line, white,fill=red] (M) at (135:1) {$N$};
\node[vtx,  white,fill=red] (w) at (-135:1){$w$};
\draw[ultra thick] (L) edge[bend left=30]  (v);
\draw (v) edge [<-,bend left=30] node[midway, auto]{$4$} (L);
\draw[ultra thick] (M) edge[bend left=30]  (w);
\draw (w) edge [<-,bend left=30] node[midway, auto]{$4$} (M);
\draw[ultra thick] (L)--(w);
\draw[->] (M)--(v) node[midway, auto] {$1$};
\end{tikzpicture} $\mathbb{Z}_{12}$
}
\end{subfloatrow*}}
{
\caption{The reduced Levi graph for  $12\#(4,4;1)$ and the corresponding configuration, which has too many incidences to be a 3-configuration.}
\label{oops}}
\end{center}
\end{figure}


It is also possible to use the configuration construction lemma to produce geometric constructions for more highly incident configurations. For example, it is possible to repeat the construction of astral chiral 3-configurations multiple times with the same initial polygon to construct $k$-configurations for arbitrary $k$. This construction is discussed in more detail in \cite{BerFau2013}.

\section{Open Questions}

\begin{question} What are allowable labels for the caterpillar graphs? Are there classifications of labels that generate configurations with too many incidences? \end{question}
\begin{question} Are there other interesting infinite families of 3-configurations, similar to the multilateral and caterpillar 3-configurations?\end{question}
\begin{question} It is possible to use the construction of chiral astral 3-configurations multiple times, with an iterative process, to produce more highly incident configurations (e.g., 4-, 5- and 6-configurations); this construction is discussed in \cite{BerFau2013}. Can other 3-configurations be used to construct highly incident configurations?\end{question}
\begin{question} Given two chirally symmetric 3-configurations and their associated reduced Levi graph, is it possible to connect the reduced Levi graphs together to produce a new chiral configuration, similar to how the dihedralization technique produced a new configuration beginning with two copies of a single reduced Levi graph?\end{question} 
\begin{question}Is there a purely geometric construction method for Pappus-like configurations or configurations whose reduced Levi graphs are the even prism graphs or the odd M\"{o}bius graphs?\end{question}



\section*{Acknowledgements} The author thanks Jill Faudree and Toma\v{z} Pisanski for helpful comments, and the students in her Fall 2010 \emph{Math 665: Configurations} graduate topics course for stimulating some of the initial ideas.


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\end{document}