--- a/sandbox.tex	Mon Dec 12 21:05:33 2011 -0800
+++ b/sandbox.tex	Mon Dec 12 22:11:55 2011 -0800
@@ -12,45 +12,4 @@
 \newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}}
 
 
-\begin{tikzpicture}
-\node[circle,fill=black,inner sep=1pt] (A) at (1.73,0) {};
-\node[circle,fill=black,inner sep=1pt] (B) at (-1.73,0) {};
-\draw[dashed] (A) -- (B);
-\node[circle,fill=black,inner sep=1pt] (C) at (0,0) {};
-\node[circle,fill=black,inner sep=1pt] (D) at (0.8,0) {};
-\begin{scope}[yshift=-1cm]
-\path[clip] (0,0) circle (2);
-\begin{scope}[yshift=2cm]
-\draw (0,0) circle (2);
-\node[circle,fill=black,inner sep=1pt] (L2) at (-90:2) {};
-\node[circle,fill=black,inner sep=1pt] (L1) at (-120:2) {};
-\end{scope}
-\end{scope}
-\begin{scope}[yshift=1cm]
-\path[clip] (0,0) circle (2);
-\begin{scope}[yshift=-2cm]
-\draw (0,0) circle (2);
-\node[circle,fill=black,inner sep=1pt] (U) at (90:2) {};
-\end{scope}
-\end{scope}
-\begin{scope}
-\path[clip] (0,1) circle (2);
-\path[clip] (0,-1) circle (2);
-\foreach \n in {1,...,6} {
-	\path (B) to[out=0,in=-180] node[coordinate,pos=\n/6] (BC\n) {} (C);
-	\draw[green!50!brown] (BC\n) -- +(0,1);
-	\draw[green!50!brown] (BC\n) -- +(-1.5,-1);
-}
-\foreach \n in {0,...,6} {
-	\path (C) to[out=0,in=-180] node[coordinate,pos=\n/6] (CD\n) {} (D);
-	\path (L1) to[out=-30,in=-180] node[coordinate,pos=\n/6] (L1L2\n) {} (L2);
-	\draw[blue] (CD\n) -- (L1L2\n);
-}
-\foreach \n in {0,...,6} {
-	\path (L1) to[out=40,in=-140] node[coordinate,pos=\n/6] (L1C\n) {} (C);
-	\path (L2) to[out=50,in=-130] node[coordinate,pos=\n/6] (L2D\n) {} (D);
-	\draw[blue] (L1C\n) -- (L2D\n);
-}
-\end{scope}
-\end{tikzpicture}
 \end{document}

--- a/text/appendixes/comparing_defs.tex	Mon Dec 12 21:05:33 2011 -0800
+++ b/text/appendixes/comparing_defs.tex	Mon Dec 12 22:11:55 2011 -0800
@@ -1033,7 +1033,55 @@
 \begin{figure}[t]
 \begin{align*}
 \mathfig{0.4}{triangle/triangle4f} \\
-\mathfig{0.4}{triangle/triangle4f_i}
+\begin{tikzpicture}
+\node[circle,fill=black,inner sep=1pt] (A) at (1.73,0) {};
+\node[circle,fill=black,inner sep=1pt] (B) at (-1.73,0) {};
+\draw[dashed] (A) -- (B);
+\node[circle,fill=black,inner sep=1pt] (C) at (0,0) {};
+\node[circle,fill=black,inner sep=1pt] (D) at (0.8,0) {};
+\begin{scope}[yshift=-1cm]
+\path[clip] (0,0) circle (2);
+\begin{scope}[yshift=2cm]
+\draw (0,0) circle (2);
+\node[circle,fill=black,inner sep=1pt] (L2) at (-90:2) {};
+\node[circle,fill=black,inner sep=1pt] (L1) at (-120:2) {};
+\end{scope}
+\end{scope}
+\begin{scope}[yshift=1cm]
+\path[clip] (0,0) circle (2);
+\begin{scope}[yshift=-2cm]
+\draw (0,0) circle (2);
+\node[circle,fill=black,inner sep=1pt] (U) at (90:2) {};
+\end{scope}
+\end{scope}
+\begin{scope}
+\path[clip] (0,1) circle (2);
+\path[clip] (0,-1) circle (2);
+\foreach \n in {1,...,6} {
+	\path (B) to[out=0,in=-180] node[coordinate,pos=\n/6] (BC\n) {} (C);
+	\draw[green!50!brown] (BC\n) -- +(0,1);
+	\draw[green!50!brown] (BC\n) -- +(-1.5,-1);
+}
+\foreach \n in {0,...,5} {
+	\path (C) to[out=0,in=-180] node[coordinate,pos=\n/5] (CD\n) {} (D);
+	\path (L1) to[out=-30,in=-180] node[coordinate,pos=\n/5] (L1L2\n) {} (L2);
+	\draw[blue] (CD\n) -- (L1L2\n);
+}
+\foreach \n in {0,...,5} {
+	\path (L1) to[out=40,in=-140] node[coordinate,pos=\n/5] (L1C\n) {} (C);
+	\path (L2) to[out=50,in=-130] node[coordinate,pos=\n/5] (L2D\n) {} (D);
+	\draw[blue] (L1C\n) -- (L2D\n);
+}
+\foreach \n in {1,3,...,9} {
+	\path(C) to[out=30,in=150] node[coordinate,pos=\n/10] (CA\n) {} (A);
+	\path(D) to[out=0,in=180] node[coordinate,pos=\n/10] (DA\n) {} (A);
+	\path(L2) to[out=0,in=-120] node[coordinate,pos=\n/10] (L2A\n) {} (A);
+	\draw[brown] (CA\n) -- +(0,1);
+	\draw[brown] (CA\n) to[out=0] (DA\n) -- (L2A\n);
+}
+\draw[dashed] (U) -- (C) -- (L1) (L2) -- (D);
+\end{scope}
+\end{tikzpicture}
 \end{align*}
 \caption{Vertical composition in the triangle axiom.}
 \label{fig:vertical-composition}