\chapter{Introduction}

\hrule

Things to do:

\begin{itemize}
\item More examples
\item Exercises
\item Finish proof
\item Reading out mental image state
\item Four ladies chain (3/4)
\item Split circ 1 1/2
\item Boys circ, girls circ
\item Split the outsides, separate, etc.
\item Six-couple dancing
\item Non-sex-symmetric IS possible, but many nice properties go away
\end{itemize}

\hrule

This book will describe a ``mental image'' system of square dance
calling.  Such systems do something that might at first glance seem to
be impossible: enable you to call correctly resolved symmetric
sequences that are not in any way memorized without looking at a
square.  Of course, if you could keep track of eight dancers in your
head (there actually are a few people who can do this) you could do
this directly.  But most people can't keep track of eight dancers
mentally.  The mental image system requires you to keep track of only
{\em one\/} dancer, plus an additional amount of information that is
so small that you can keep track of it by inconspicuously counting on
the fingers of one hand.

That it is mathematically possible to encode the state of an entire
square for resolution purposes in the position of one dancer plus a
tiny amount of information is rather remarkable.  That it is possible
for an ordinary human caller to manipulate this information
effectively, while calling, is even more so.  With the mental image
system, you can call while blindfolded!

\section{Why Bother?}

Before spending the time required to become familiar with the mental
image system, you should ask ``Why bother?''  This is a difficult
question to answer.  Let us first look at the tools callers use to
produce choreography:

\begin{itemize}
\item Reading pre-writen sequences;
\item Calling complete memorized sequences;
\item Calling memorized sequences enhanced on-the-fly by substitution
of equivalents or insertion of zeros;
\item Chaining memorized modules that transform among known
setups of dancers;
\item Mental image generation of sequences;
\item Ad-lib calling followed by sight resolution; and
\item Intuition, that is, methods that a caller can't really describe
effectively.  As callers become more expert, they often use this more.
\end{itemize}

These methods interact in subtle ways inside a caller's consciousness.
Since generation of correct sequences is such an important part of the
caller's art, it is not surprising that experienced and accomplished
callers use a mixture of these methods, often without consciously
thinking about it or being able to explain it logically.

The very algorithmic methods that newer callers use (``bring the
primary couple together in two-faced lines and then\ldots'') are
generally not used by very experienced callers.  With a great deal of
experience, callers often just know what to do.  When asked to explain
afterward how they made a sequence resolve, they may say things like
``I had people with partners in boy-boy-girl-girl waves. It was
obvious that \call{out roll circulate} to a \call{right and left
grand} would work,'' or ``the boys were out of sequence and the girls
were in sequence: \call{boys trade, extend, right and left grand} was
the getout'', or ``that was clearly the \call{recycle, left allemande}
position.''  It is for this reason that we have listed ``intuition''
in the above list.  In fact, the holistic ways that experienced
callers use often lead to frustration when they try to explain things
to newer callers who need to be fed in smaller, logically organized,
doses.

Because of the subtle ways that methods interact in the thinking of
experienced callers, it is easier to explain how a mental image
system fits into the overall scheme of things in terms of the methods
familiar to less experienced callers.  So we will explain that first.

The mental image system is a very powerful method for keeping track of
the mathematical arrangement of the dancers.  It is vastly more
powerful, for a given amount of memorization, than the
transformation-of-known-states method.  For example, one could
memorize the fact that split circulate goes from F1p to Q1z, but what
if the setup isn't F1p?  The mental image system can deal with split
circulate from any kind of boxes with one sex looking in and the other
looking out.  There is just {\em one\/} rule to learn for this call.
It fits in naturally with the rules for all other calls.  It is not
necessary to memorize the various choreographic relationships such as
L1o and W1c and $\frac{1}{2}$B1r.

With the mental image system, you can make up zeros and equivalents on
the fly.  There is simply no need ever to memorize them.

If you want to feature \call{spin the top, right and left grand}
getouts in some tip, you can do so in an extremely general way, rather
than by memorizing equivalents or formulas for choreographic
relationships.

If you want to add \call{plan ahead, double pass thru, stack the line,
right and left grand} to you repertoire, you can do so by just
learning the mental image formulation of the individual calls
involved.

When sight resolving, you can read out the mental image state by
looking at the key couples, and proceed from there.  This gives you
the full power of the mental image system for your getouts while
allowing you full flexibility while calling.

Perhaps the most extraordinary aspect of the mental image system is
that you can practice it at home.  Of course, the generation of
correctly resolved sequences is only one part of what a caller needs
to do well, and many of the other aspects of calling can't be
practiced without live dancers, so we have to keep this in
perspective.  Practicing on live dancers is best, and is absolutely
necessary for improving one's timing, presentation, and showmanship.
However, we believe that the ability to practice one's choreography at
home will lead to more effective use of one's practice time with live
dancers.  The number of blunders that your dancers will be exposed to
will be reduced.

For expert callers, the usefulness of the mental image system is a
less straightforward case.  In some ways, it's similar to the
rationale for learning Latin: even if you don't speak the language it
helps you to understand and think clearly about languages.

The mental image system can be a useful part of your collection of methods
for creating sequences.  You can use it directly when you consider it
appropriate to do so.  Furthermore, the knowledge of the system helps in
the understanding of such topics as ``technical zeros'' and ``technical
equivalents.''  Finally, the knowledge of mental image is useful in
managing formations and arrangements.  When you are an experienced
mental image practitioner, you will have a better feel for what gender
arrangement will result from a given call, even when you are doing ad-lib
sight calling.

\section{Background}

Mental image calling has existed in various forms for many years.  The
current system can trace its roots back to the book {\em The
Fundamentals of Hash Calling\/} by Dr.\ Julius King.  This book was
used and improved upon by Don Beck to produce the book {\em Out of
Sight\/}, which has been the classic text on mental image calling for
more than a decade. 

The book you're reading now modifies and builds upon Beck's work in a
number of ways.  First of all, it significantly reduces the amount of
information you have to manipulate in real-time.  For example, Beck
requires the memorization of eight ``allemande spots'' while we
require only four, thus allowing them to be represented by the fingers
on one hand (freeing your brain for more important tasks).  We also
simplify the way one moves the allemande spot, and expand the system
to handle more calls and a few additional formations (like diamonds).
Finally, we have attempted to present the information in a technically
precise manner.  You should always know exactly what is legal to call
and why.  Taking this to extremes, we have even provided a correctness
proof of the mental image system.  Most people will happily ignore
this section, but it's there for the die-hard mathematicians.

\section{About this Book}

Square dance calls will be indicated as follows: \call{right and left
thru}.  While the mental image system is most suited for Mainstream
and Plus calls (where it meshes nicely with the ``standard
applications'' idea), we have provided information on a variety of
calls from the Advanced and Challenge lists.  People not familiar with
these levels can simply ignore references to calls they don't know.
But if you do dance or call these levels, we hope they will provide
additional insight.

In the same vein, this book is meant for a variety of audiences: those
who want to use mental image in their day-to-day calling but don't
really care how it works, those who have some interest in how it
works, and those who have a deep desire to understand it.  We will
frequently diverge into discussions of esoteric topics --- you are
free to ignore them.  \note{Perhaps we should mark our diversions.}
The most important rules of the mental image system that you must know
are marked like this:

\highlight{You should really know this!}

In this book, we use the following symbols to indicate dancers:

\displaytwo{ \bdancer n }{boy}{ \gdancer n }{girl}

The dot indicates the dancer's nose and hence his/her facing
direction.  It is assumed that the caller is standing at the bottom of
the page.

The following terms will be used throughout this book.

\begin{itemize} 

\item {\em Formation\/} is the footprints the dancers are standing on
and the directions they are facing.  Possible formations are facing
lines, right-handed waves, right-handed diamonds, left-handed
hourglasses, and right-handed $\frac{1}{4}$-tag.

\item {\em Arrangement\/} is the location of boys and girls within a
given formation.  For any symmetric eight-person formation, there are
exactly six possible arrangements.  For example, from facing lines we
can have (with their Callerlab designations):

\displaythree
{ \gdancer s & \bdancer s & \gdancer s & \bdancer s \\
  \bdancer n & \gdancer n & \bdancer n & \gdancer n }
{\#0 lines}
{ \bdancer s & \gdancer s & \bdancer s & \gdancer s \\
  \gdancer n & \bdancer n & \gdancer n & \bdancer n }
{\#$\frac{1}{2}$ lines}
{ \gdancer s & \gdancer s & \bdancer s & \bdancer s \\
  \bdancer n & \bdancer n & \gdancer n & \gdancer n }
{\#1 lines}

\displaythree
{ \bdancer s & \bdancer s & \gdancer s & \gdancer s \\
  \gdancer n & \gdancer n & \bdancer n & \bdancer n }
{\#2 lines}
{ \bdancer s & \gdancer s & \gdancer s & \bdancer s \\
  \bdancer n & \gdancer n & \gdancer n & \bdancer n }
{\#3 lines}
{ \gdancer s & \bdancer s & \bdancer s & \gdancer s \\
  \gdancer n & \bdancer n & \bdancer n & \gdancer n }
{\#4 lines}

Formation and setup are the ``F'' and ``A'' of the Callerlab ``FASR''
(formation, arrangement, sequence, relationship) nomenclature.  We
will not use the sequence or relationship parts, because the mental
image system subsumes them.  In fact, we will generally not use the
Callerlab names at all.  You can use whatever method you like for
thinking about formations and arrangements.  The correspondence
between our notation and the Callerlab notation will be discussed in
Chapter~\ref{chap:insight}.

\item {\em Setup\/} is the combination of Formation and Arrangement.

\end{itemize}

\section{Scope}

As mentioned earlier, the mental image system restricts the setups and
calls that can be used.  This section describes the restrictions on
setups.  Individual calls will be treated throughout the rest of the
book.

The mental image system deals principally with four-person
sex-symmetric setups: four-person setups with two boys and two girls
where each person's diagonal opposite is of the same sex and is facing
in the opposite direction.

Here are some acceptable four-person setups:

\displayfour
{ \bdancer n & \gdancer s & \gdancer n & \bdancer s }
{Waves}
{ \gdancer s & \bdancer s \\
  \bdancer n & \gdancer n }
{Facing couples}
{ \bdancer e & \gdancer e \\
  \gdancer w & \bdancer w }
{Box}
{ \bedancer e \\
  \gdancer n & \gdancer s \\
  \bedancer w }
{Diamond}

Here are some setups that are not acceptable:

\displaythree
{ \bdancer n & \gdancer s & \bdancer n & \gdancer s}
{Opposite is wrong sex}
{ \bdancer e & \bdancer e \\
  \gdancer w & \gdancer w }
{Opposite is wrong sex}
{ \bdancer n & \gdancer n & \gdancer n & \bdancer n}
{Opposite is facing same direction}

Many eight-person formations can be looked at multiple ways.  For
example, normal facing lines can be looked at as lines of four or two
boxes of four with facing couples.  In the mental image system, it is
only legal to call calls that look at the formation in a sex-symmetric
way.  For example, a single line of four people facing the same way
is not symmetric.
However, facing couples are.  So from facing lines, only calls that
treat the formation as facing couples may be called.

However, it is possible to use other calls from these formations under
certain circumstances.  These will be dealt with individually later.
Also, it is a mistake to think that the mental image system can {\em
only\/} handle four-person calls.  There are many common eight-person
calls that it can handle as well.  However, in general four-person
calls are easy to handle, while eight-person calls must be treated as
special cases.

\section{Organization of This Book}

Now that we have discussed the scope of the mental image system, the
rest of the book will be devoted to explaining how to use it.  The
result will be a powerful method that can be used with few
restrictions to call just about anything on the programs up through
Plus, and some of the calls on A-1, A-2, and higher.  The plan is as
follows:

In Chapter~\ref{chap:centers} we will discuss how to call to the
heads or the sides in the centers of the square.

In Chapter~\ref{chap:four-person-side} we will allow calling
four-person sex-symmetric material on each side of the set, so that
all eight people can dance.

In Chapter~\ref{chap:eight-person} we will introduce a number of
``staple'' eight-person calls and the techniques necessary to use
them.  At this point it will be possible to call high-quality
sequences at actual dances.

In Chapter~\ref{chap:something} we will show how to ``read out'' the
designee and allemande spot from a live square.  This will make it
possible to sight call with complete freedom, and use the mental image
system only for resolving.

In Chapter~\ref{chap:something} we will give an extensive list of the
eight-person calls that the system can handle.  This list will be far more
extensive than anyone is likely to want to use, but if you really
want to call cast a shadow with the mental image system, you can.

In Appendix~\ref{app:lists} we give a complete list of calls thru Plus
and some calls from A-1, A-2, and higher as a quick reference.

In Appendix~\ref{app:proof} we give the mathematical proof that
the system works, in case you are interested.