two more figures for the intro

intro.tex

@@ -103,16 +103,55 @@ \chapter{Introduction to the topic of this book}
 
 The setup we adopt, ``HoTT'' or ``univalent foundations'', seems custom-built for such an approach.
 
-In practice, one of the most important things is to be able to \emph{compare} symmetries of ``thing 1'' and ``thing 2''.  In our case this amounts to nothing but a function the takes thing 1 to thing 2.
-\begin{quote}
-  Picture of the function as in the original Goodnotes, putting some names to things.
-\end{quote}
+In practice, one of the most important things is to be able to \emph{compare} symmetries of ``thing~1'' and ``thing~2''.  In our case this amounts to nothing but a function, $f$, that takes thing~1, $x_1$, to thing~2, $x_2$.
+\begin{center}
+\begin{tikzpicture}
+  \begin{scope}[scale=0.8]
+    \node (X) at (1,2) {$X_1$};
+    \node[dot,label=left:$x_1$] (x1) at (0,0) {};
+    \draw (0,-2)
+    .. controls ++(150:-1) and ++(180:1) .. (3,-2)
+    .. controls ++(180:-1) and ++(-100:1.3) .. (4.5,0)
+    .. controls ++(-100:-1.3) and ++(-10:2) .. (2,1.5)
+    .. controls ++(-10:-2) and ++(90:1) .. (-1,0)
+    .. controls ++(90:-1) and ++(150:1) .. (0,-2);
+    \draw[->] (x1) .. controls ++(80:-1) and ++(170:1) .. (.5,-1.5)
+    .. controls ++(170:-1) and ++(200:1) .. (3,-1.4)
+    .. controls ++(200:-1) and ++(-80:.5) .. (3.8,0)
+    .. controls ++(-80:-.5) and ++(-10:.3) .. (3,1)
+    .. controls ++(-10:-.3) and ++(80:1) .. (x1);
+    \draw (1,-1) arc(210:330:.8 and .5);
+    \draw (2.09,-1.18) arc(60:120:.8 and .7);
+    \draw (1.5,0) arc(210:330:.8 and .5);
+    \draw (2.59,-0.18) arc(60:120:.8 and .7);
+    \draw[->] (4.8,0) -- node[auto] {$f$} (6.3,0);
+  \end{scope}
+  \begin{scope}[xshift=6cm,scale=0.8]
+    \node (X) at (1,2) {$X_2$};
+    \draw (0,-1)
+    .. controls ++(200:-1) and ++(180:1) .. (2,-2)
+    .. controls ++(180:-1) and ++(270:1) .. (4,0)
+    .. controls ++(270:-1) and ++(20:2)   .. (2,2)
+    .. controls ++(20:-2)   and ++(90:1)  .. (-1,0)
+    .. controls ++(90:-1)  and ++(200:1) .. (0,-1);
+    \node[dot,label=below:$x_2$] (x2) at (0,0) {};
+    \draw[->] (x2) .. controls ++(-20:1.5) and ++(170:1) ..
+      (2,-1) .. controls ++(170:-1) and ++(-70:1) ..
+      (3.1,0) .. controls ++(-70:-1) and ++(90:.5) ..
+      (3.5,0) .. controls ++(90:-.5) and ++(-120:2) ..
+      (3,1) .. controls ++(-120:-2) and ++(-20:-1.5) .. (x2);
+    \draw (1,0) arc(210:330:.8 and .5);
+    \draw (2.09,-.18) arc(60:120:.8 and .7);
+  \end{scope}
+\end{tikzpicture}
+\end{center}
 While such comparisons of symmetries are traditionally handled by something called a ``group homomorphism'' which is a function satisfying a rather long list of axioms, in our case the only thing we need to know of the function is that it really does take thing 1 to thing 2 -- everything else then follows naturally.
 
-Some important examples have provocatively simple representations in this framework.  For instance, consider the circle
-\begin{quote}
-  picture of the pointed circle
-\end{quote}
+Some important examples have provocatively simple representations in this framework.  For instance, consider the circle shown in the margin.\marginnote{%
+  \begin{tikzpicture}
+    \node[dot,label=right:$x$] (base) at (1,0) {};
+    \draw (0,0) circle (1);
+  \end{tikzpicture}}
 Since symmetries are interpreted as loops, you see that you have a loop for every integer -- the number $7$ can be represented by looping seven times counterclockwise.  As we shall see, in our setup any loop is naturally identified with a unique integer (the ``winding number'' if you will).  Everything you can wish to know about the structure of the ``group of integers'' is encoded in the circle.
 
 Another example is the ``free group of words in two letters $a$ and $b$''.  This is represented by