\chapter{Eight-Person Calls and Fictitious Moves}
\label{chap:eight-person}

In this chapter we introduce some of the sophisticated operations that
make the mental image system very powerful.  The calls we are
particularly interested in are \call{trade by} and \call{all 8
circulate} or \call{couples circulate}, along with the Advanced calls
\call{cross-over circulate} and \call{trade circulate}, since these
are the most commonly used.

\section{\call{trade by}}

Let's try something a little more daring than what we've been doing in
the previous chapter.

\begin{sequence}
heads square thru 4\\
square thru 3
\end{sequence}

We would like to bring the designee over to the other side and dance
with the other people for a while.  The obvious call is trade by, but
that's an eight-person call, so we don't know how to use it.  Instead, we
can call the following, using only four-person calls:

\begin{sequence}
partner trade \\
centers trade \\
centers pass thru
\end{sequence}

The overall setup is now an 8-chain, so we have a sex-symmetric setup
(couples facing) on each side.  Therefore we can legitimately start
calling to each side, with the designee working with the other side
dancers.  Now the three call module above is equivalent to the call
\call{trade by}, so we have established the legitimacy of our first
8-person call:

\highlight{
If the designee is in the center, you can call a \call{trade by}.
Just follow the designee as he crosses to the other side of the set.
Do not change or move the allemande spot.
}

The restriction that the designee be in the center will be removed
later, when we develop much better facility with eight-person calls.
For now, we have to live with this restriction.  If we called a
\call{trade by} when the designee was on the end, we would effectively
be calling to the center four when the designee is not in the center.
This is forbidden.

{\em Never call to the centers, unless the designee is one of the centers.
The change-of-designee theorem, described below, will be able to free
us from the restriction on our choreography that this rule might seem
to impose.\/}

With our new-found freedom to call \call{trade by}, and our awareness
that the designee must be in the center, we can call lots of material.
In fact, we have freedom to use \x\ calls anywhere.

\begin{sequence}
heads square thru 4 \\
slide thru \\
right and left thru \\
flutter wheel (\x) \\
square thru 2 \\
trade by \\
swing thru (\x) \\
boys run (\x) \\
bend the line (\x) \\
right and left thru \\
slide thru \\
allemande left
\end{sequence}

The system is now sufficiently powerful that, in small doses, you
could use it for an occasional sequence on real dancers at a real
dance.

Make up more sequences like this that involve calling \x\ calls in
various places.  Practice, practice, practice.

\section{The Change-of-Designee Theorem}

Change of designee is an operation we can perform when we need the
designee to be someone other than who he currently is.  We saw earlier
that we can perform the call \call{trade by} by just doing it, as long
as the designee is in the center.  If the designee is on the end,
extra work is needed.

The change-of-designee operation changes the designee to the
corresponding person in the other four-person setup.  \note{Just what
does corresponding person mean? Give examples.} The two setups are
always identical formations, so anyone's counterpart in the other
setup is always of the same sex, and is facing the same way.  The
theorem is as follows:

\highlight{We can change the designee to the corresponding person at
any time when we have identical formations on each side.  If the
allemande spot is an \x, don't do anything to it.  If it is an \o,
move it by 2 positions.}

We have identical formations on each side at any time when we aren't
calling specifically to the centers.  If we start with \call{heads
pass the ocean}, the designee has no counterpart anywhere else.  Of
course, we wouldn't want to change the designee in that setup anyway,
because we need him in the center.

When we move an \o\ allemande spot by 2, we could think of it as
moving in whichever direction won't fall of the end, or we could
simply say to move it in either direction, wrapping around as needed.
What we want is this:

\allemandeaO $\Leftrightarrow$ \allemandecO

\allemandebO $\Leftrightarrow$ \allemandedO

It's simpler to think in terms of wrapping around, since

\highlight{All spot motions are ``modulo 4,'' that is, they wrap
around if they reach the edge.  There are absolutely no exceptions to
this rule.}

We will not prove this theorem here.  A proof is in the appendix.  You
should already be well accustomed to the wraparound, from your
practice with \x\ calls on each side while the allemande spot is in
various places.

To see that it is plausible, suppose the dancers are able to do an
\call{allemande left}.  The setup might be:

\displayone
{ \bdancer e & \gdancer w & \bdancer e & \gdancer w \\
  \gdancer e & \dbdancer w & \gdancer e & \bdancer w }
{\allemandebO}

If we change designee, we get:

\displayone
{ \bdancer e & \gdancer w & \bdancer e & \gdancer w \\
  \gdancer e & \bdancer w & \gdancer e & \dbdancer w }
{\allemandedO}

Moving an \o\ spot by 2 is the right thing to do to preserve the fact
that the square is resolved.  

Now consider parallel waves in which the dancers could do
\call{recycle, allemande left}.  The setup might be:

\displayone
{ \dbdancer e & \bdancer e \\
  \gdancer w & \gdancer w \\
  \gdancer e & \gdancer e \\
  \bdancer w & \bdancer w }
{\allemandecX}

If we change designee, we get:

\displayone
{ \bdancer e & \dbdancer e \\
  \gdancer w & \gdancer w \\
  \gdancer e & \gdancer e \\
  \bdancer w & \bdancer w }
{\allemandecX}

Not moving an \x\ spot is the right thing.  When we \call{recycle}
from here, the designee will pull the spot to the right, and the
\call{allemande left} will still work.

You should not need to perform an explicit change of designee unless
you do some of the more esoteric calls in
Chapter~\ref{chap:something}.  Instead, we will use it to work out
very simple and direct ways to do various interesting calls.

\section{\call{trade by} Revisited}
\label{sec:tradeby}

As we saw earlier, this staple of calling can be done simply by doing
it, as long as the designee is in the center.  What if he or she
isn't?  We have, for example, this:

\displayone
{ \gdancer w & \bdancer e & \gdancer w & \bdancer e \\
  \dbdancer w & \gdancer e & \bdancer w & \gdancer e }
{}

We do a change of designee:

\displayone
{ \gdancer w & \bdancer e & \gdancer w & \bdancer e \\
  \bdancer w & \gdancer e & \dbdancer w & \gdancer e }
{}

and then do the call:

\displayone
{ \bdancer e & \gdancer w & \bdancer e & \gdancer w \\
  \gdancer e & \dbdancer w & \gdancer e & \bdancer w }
{}

Now it is slightly burdensome to deal with the \o\ spot move from the
designee change.  There is a simple fix for that: change the designee
again!  This will move the \o\ spot back where it was, so you never
need to worry about changing the allemande spot at all.

Now we have:

\displayone
{ \bdancer e & \gdancer w & \bdancer e & \gdancer w \\
  \gdancer e & \bdancer w & \gdancer e & \dbdancer w }
{}

So what happened?  The designee moved back by three positions (but the
real dancer moved forward one position to the far side of the
square!).  This, like many eight-person calls, is a ``fictitious
move.''  The designee moves mentally in a way that is different from
the actual dancer.  When you use calls from this and later chapters,
don't get too attached to any specific person as your designee.  You
can use the actual squares for timing, but their live designees will
trick you by dancing many eight-person calls differently.

Instead of moving the designee back by three, it is actually easier to
think of him as going forward by one, and ``wrapping around.''  As
with allemande spot motions, there is a lot of wrapping around in
designee motions, and you should get used to it.

The wrapping around is in the designee's own column of four, either
the four spots close to you or the spots far from you:

\note{Picture from page 44}

So the final rule for \call{trade by} is:

\highlight{Move the designee forward ({\em i.e.}, the way he is
facing) by one position, wrapping around in his column if necessary.}

That's all there is to it!

\call{trade by} and \call{pass thru} have a symbiotic relationship
with each other.  Each of them moves the designee forward one position
with wraparound.  So we can use them (use the correct one!) to move
the designee wherever we want in his column.  If the designee is in
front and the spot is an \o, we can use them (or the call \call{eight
chain $\langle n\rangle$}, which is a convenient abbreviation for
them) to get to an \call{allemande left} or \call{right and left
grand}.

Example: What do we do from here?

\displayone
{ \bdancer w & \gdancer e & \bdancer w & \gdancer e \\
  \gdancer w & \dbdancer e & \gdancer w & \bdancer e }
{\allemandedO}

Answer: We are two spots away.  \call{trade by, pass thru, right and
left grand}.

What do we do from here?

\displayone
{ \bdancer e & \gdancer w & \bdancer e & \gdancer w \\
  \gdancer e & \dbdancer w & \gdancer e & \bdancer w }
{\allemandecO}

Answer: We are three spots away.  \call{eight chain 3, allemande
left}.

By the way, you may have noticed that having the designee in the half
of the set near us, and an \o\ allemande spot, is a good setup for a
resolve, requiring only some appropriate number of \call{pass thru},
\call{trade by}, \call{all 8 circulate}, or \call{extend} calls to finish
the job.  This is because, when in waves, 8-chain, trade by, lines
facing, or lines back-to-back, we have the following:

\highlight{If the designee is in front, his or her own sex dancers are
in sequence.  If in back, they are out of sequence.  If the allemande
spot is an \o, the men and women are in the same sequence as each
other (both in or both out).  If the spot is an \x, they are in
different sequence.}

\highlight{Warning: The preceding ``fact'' is sometimes misquoted as
being true in all cases.  It is true for lines facing in or out,
8-chain, trade by, or waves.  It is {\em not\/} true in a two-faced
line.}

Clearly, having the designee in front and in an \o\ means that
everyone is in sequence, and all that is needed is to rotate them
around to their partners or corners.

Consideration of sequence is not something you need to use.  You just
let the designee and the allemande spot do their work.  But if you
like to think in terms of sequence, this might be useful.

\section{\call{all 8 circulate} and \call{couples circulate}}
\label{sec:all8circ}

This is another call that turns out to have an incredibly simple
formulation once we work it out.  It is fortunate that such common and
important eight-person calls have this property.  Many other
eight-person calls don't fare nearly as well.

From waves, \call{all 8 circulate} is equivalent to rear back from
your wave, \call{pass thru, trade by}, and step to a wave of the
original handedness.

That means that, once again, we perform no manipulations at all on the
allemande spot.  We have the designee rear back to an 8-chain and move
forward 2 spots with wraparound.  Moving forward 2 spots with
wraparound just means moving to his counterpart on the other side.

After he steps into a new wave, all that has happened is that he has
moved to the corresponding place in the other wave.

So, the final rule for \call{all 8 circulate} is:

\highlight{To do \call{all 8 circulate}, just move the designee
forward or backward, as appropriate, to the other wave or 2-faced
line.}

That's all there is to it!

\displaythree
{ \bdancer e & \dbdancer e \\
  \gdancer w & \gdancer w \\
  \gdancer e & \gdancer e \\
  \bdancer w & \bdancer w }
{Before}
{ \dbdancer e & \bdancer e \\
  \gdancer w & \gdancer w \\
  \gdancer e & \gdancer e \\
  \bdancer w & \bdancer w }
{After \call{all 8 circulate}}
{ \bdancer e & \dbdancer e \\
  \gdancer w & \gdancer w \\
  \gdancer e & \gdancer e \\
  \bdancer w & \bdancer w }
{After second \call{all 8 circulate}}

If the designee moves forward, the move is not fictitious.  If he
moves backwards, the move is fictitious.

This means that \call{circulate twice} gets to the same mental image
state, so it must be a zero.  Is \call{circulate twice} really a zero?
Well, sort of.  It rotates the whole setup by 180 degrees, which will
not affect resolution.  This makes an important point:

\highlight{The mental image system only guarantees resolution.  It
does not keep track of set rotation.}

Of course, if you avoid all fictitious moves (which gets increasingly
difficult as you use more advanced techniques) then the designee will
always be the right person, and the \call{allemande left} will happen
exactly where you think it is happening.  You need to be aware of this
when making up singing call figures.

We still have to do \call{couples circulate}.  It is equivalent to
\call{slither}, \call{all 8 circulate}, and \call{slither}.  Since
\call{slither} is an \o\, we can just move the designee forward or
backward the same as for \call{all 8 circulate}.

By the way, moving the designee forward or backward is just the
change-of-designee operation.  So instead of moving the designee, we
could just leave him where he is and do the spot transformation: If the
spot is an \o\ move it by 2, and if it is an \x\ do nothing at all.
(This means that, if the spot is an \x, \call{all 8 circulate} and
\call{couples circulate} are zeroes.  Specifically, they are
``\x-conditional zeroes.'')  You will probably find it easier,
however, to move the person and leave the spot alone.

\section{\call{cross-over circulate} and \call{trade circulate}}
\label{sec:xovercirc}
\label{sec:tradecirc}

These calls are in the Advanced program, but we might as well mention
them here, since they fall out naturally from the nice behavior of
\call{circulate}.

\call{cross-over circulate}, from 2-faced lines, is just \call{couples
circulate} and \call{half sashay}.  \call{half sashay} is an \o\ call,
so just move the designee forward or back with a \call{half sashay}.
From waves, \call{cross-over circulate}, or its more common name
\call{trade circulate}, is just \call{all 8 circulate} and
\call{spread}.  So, since \call{spread} is an \o\ call, it is handled
the same way.

\displayone
{ \bdancer e & \bdancer e \\
  \gdancer w & \gdancer w \\
  \gdancer e & \gdancer e \\
  \dbdancer w & \bdancer w }
{\allemandedO}
\call{trade circulate, LA}

From 2-faced lines, \call{trade circulate} is just \call{couples
circulate} and \call{as couples spread}.  (By \call{spread} we simply mean
the two-person call ``slide nose-to-nose with the other dancer''.)
\call{as couples spread} is an \o.  So the rule here is to have the designee
do the peculiar ``diagonal'' circulate either forward or backward.

\displaytwo
{ \bdancer e & \dbdancer e \\
  \gdancer e & \gdancer e \\
  \gdancer w & \gdancer w \\
  \bdancer w & \bdancer w }
{Before}
{ \gdancer w & \gdancer w \\
  \bdancer w & \bdancer w \\
  \dbdancer e & \bdancer e \\
  \gdancer e & \gdancer e }
{After \call{trade circulate}}

At this point, the system that we have developed is quite powerful.
Congratulations!  You can call a lot of really presentable material
now (if you do enough practice, that is).  Don't do too much of this
at real dances, though, or the dancers will notice that you never use
columns, diamonds, or \call{bend the line} or \call{wheel and deal}
from lines back-to-back.

\section{Reorienting Calls}

\subsection{\call{bend the line}}
\label{sec:bendthelinebb}

Now that you have practiced calling with both orientations and are
equally skillful with both, we can do several more calls.  The
most important of these is \call{bend the line} from lines
back-to-back.  From a two-faced line, \call{bend the line} is a simple
four-person \x\ call.  We are now going to do the much more difficult
lines back-to-back version.  The rule is that it is an
``\o-conditional equivalent'' to \call{trade}.  That means that we do
a \call{trade} for the designee, and, if the allemande spot is an \o,
do not change it.  If the spot is an \x, move it by 2.

Check this by moving your checkers through:

\begin{sequence}
heads square thru 4\\
slide thru\\
pass thru\\
bend the line {\rm (spot was an \o, so don't change it)}\\
slide thru\\
allemande left
\end{sequence}

Similarly,

\begin{sequence}
heads lead right\\
square thru 2\\
bend the line {\rm (spot was an \x, so you must move it by 2)}\\
pass the ocean\\
recycle\\
eight chain 3\\
allemande left
\end{sequence}

There is a proof of this in Appendix~\ref{app:proof}, if you are
interested.

There is just one problem: \call{bend the line} changes the setup from
a left-to-right orientation to a front-to-back orientation and vice
versa.

\note{Picture?}

Now, if you plan to call with your eyes closed and not worry about the
correlation between the actual set and your mental image of it, you
don't need to worry about this.  (Of course, if you call that way, you
didn't need to read the section on calling to the sides.)

If you want your mental image to match the real formation, just rotate
your hand and your mental image, including the location of the
designee, by 90 degrees.  Which way should you rotate them?  Whichever
way feels comfortable for your wrist.  There are only two orientations
that you ever use: the ``heads'' orientation (back of the hand toward
the dancers) and one or the other of the ``sides'' orientations (back
of the thumb toward the dancers), depending on which hand you use.
When you call \call{bend the line}, just rotate naturally from the
orientation you are in to the other orientation that is natural for
you.  Be sure to rotate your mental image of where the designee is in
the same direction as you rotated your hand.

You should get accustomed to rotating your hand and your mental image
back and forth together.  For example, if you use the left hand, you
do this:

\note{Picture}

\note{Example page 57/58/59}

We have described \call{bend the line} from lines facing out, because
we consider it not to be a smooth call from lines facing in.  If you
want to call it from lines facing in, it is an ``\o-conditional zero''
with reorientation.  So you don't do the mental trade.  Just move an
\x\ spot by 2 and re-orient the setup.

Many of the more esoteric calls described in
Chapter~\ref{chap:something} involve reorientation.  This means that,
after you do whatever else the directions say to do, you will find the
dancers oriented 90 degrees off of your mental image.  Reorient your
hand and your image.

Example: \call{checkmate the column} (A2) is described as \call{trail
off}, take the designee's opposite, and reorient.

\note{Example}

While many of the esoteric calls involve reorientation, very few of
the ordinary calls that we will consider do so.  Aside from \call{bend
the line}, the calls that we will consider are \call{cloverleaf},
\call{wheel and deal} from lines back-to-back, \call{first couple
left, next couple right}, and \call{circle to a line}.

\subsection{\call{circle to a line} and 1P2P Openings}
\label{sec:circletoaline}

The call \call{circle to a line} is a fairly difficult call.  It is
not expected that you actually learn the full mental image formulation
for it.  In case you are interested, here it is:

\highlight{\begin{itemize}
\item Starting setup is an 8-chain.
\item Circle the designee counterclockwise $\frac{1}{4}$.
\end{itemize}}

\displaytwo
{ \dbdancer e & \gdancer w \\
  \gdancer e & \bdancer w }
{Before}
{ \gdancer s & \bdancer s \\
  \dbdancer n & \gdancer n }
{After}

\displaytwo
{ \bdancer e & \gdancer w \\
  \dgdancer e & \bdancer w }
{Before}
{ \gdancer s & \bdancer s \\
  \bdancer n & \dgdancer n }
{After}

\highlight{\begin{itemize}
\item Change the spot from \o\ to \x\ or \x\ to \o, and pull it toward the
designee in either case.  Note that this is unusual.  Normally the
spot is pulled toward the designee if it changes from \x\ to \o\ and
pushed away if it changes from \o\ to \x.  In this case, pull it
toward the designee no matter what.  Mental image formulations of
eight-person calls often have strange quirks like that.  Of course, this
pulling, like all spot manipulations, wraps around.  If the spot is
already at the designee's extreme end, push it to the other end.
\item Finally, reorient the setup.
\end{itemize}}

This is hardly worth memorizing, especially since this call is
practically never used except after a \call{heads/sides lead right} or
\call{wheel thru}.  So, instead, you might as well memorize the entire
module \call{lead right, circle to a line}.  It is this:

\newcaption{With a male designee.}
\displaytwo
{ \gdancer s & \bdancer s & \gdancer s & \bdancer s \\
  \dbdancer n & \gdancer n & \bdancer n & \gdancer n }
{\ctablebox{\allemandeaO \\ After \call{sides lead right} and
\call{circle to a line}}}
{ \gdancer s & \bdancer s & \gdancer s & \bdancer s \\
  \bdancer n & \gdancer n & \dbdancer n & \gdancer n }
{\ctablebox{\allemandecO \\
...or this}}

\displaytwo
{ \bdancer e & \gdancer w \\
  \gdancer e & \bdancer w \\
  \bdancer e & \gdancer w \\
  \gdancer e & \dbdancer w }
{After \call{heads lead right} and \call{circle to a line}}
{ \bdancer e & \gdancer w \\
  \gdancer e & \dbdancer w \\
  \bdancer e & \gdancer w \\
  \gdancer e & \bdancer w }
{...or this}

\newcaption{With a female designee.}
\displaytwo
{ \gdancer s & \bdancer s & \gdancer s & \bdancer s \\
  \bdancer n & \dgdancer n & \bdancer n & \gdancer n }
{\ctablebox{\allemandebO \\ After \call{sides lead right} and
\call{circle to a line}}}
{ \gdancer s & \bdancer s & \gdancer s & \bdancer s \\
  \bdancer n & \gdancer n & \bdancer n & \dgdancer n }
{\ctablebox{\allemandedO \\
...or this}}

\displaytwo
{ \bdancer e & \gdancer w \\
  \gdancer e & \bdancer w \\
  \bdancer e & \dgdancer w \\
  \gdancer e & \bdancer w }
{After \call{heads lead right} and \call{circle to a line}}
{ \bdancer e & \dgdancer w \\
  \gdancer e & \bdancer w \\
  \bdancer e & \gdancer w \\
  \gdancer e & \bdancer w }
{...or this}

We have shown two choices in each case, since the change-of-designee
operation lets us put the designee in either place.

The important and remarkable thing to notice is that in all cases the
designee is standing on his or her allemande spot.  This means that everyone
is in correct sequence for an \call{allemande left} or \call{right and
left grand} if you have them \call{quarter in} or \call{quarter out},
respectively.  Of course, the \call{lead right, circle to a line}
opening is well known for creating that situation, which has the
famous name ``L1p'' (or 1P2P in older notation).

So instead of memorizing the formulation of \call{circle to a line},
just remember that the L1p setup is this:

\displayone
{ \gdancer s & \bdancer s & \gdancer s & \bdancer s \\
  \dbdancer n & \gdancer n & \bdancer n & \gdancer n }
{\allemandeaO}

\noindent or its reoriented equivalent.  Remember that if you have the
\call{heads lead right...}, you must orient the setup as if you had
started with the sides, and vice versa.

\subsection{\call{wheel and deal} from Lines Back-to-Back}
\label{sec:wheelanddealbb}

\call{wheel and deal} from lines back-to-back is one of the harder
calls.  Perhaps it belongs in the chapter on esoteric calls, but it
really is an important call at Mainstream, so you should probably
learn it.

\highlight{\begin{itemize}
\item Be sure the designee is in the left-side couple.  Use
change-of-designee if necessary to make this so.
\item Rotate the designee's couple (including the designee, of course)
clockwise 90 degrees around in its box of 4.
\item Change an \o\ to an \x\ or \x\ to an \o, and push the spot away
from the designee in either case.
\item The designee is now in the center of a DPT setup.
\item Reorient.
\end{itemize}}

A little explanation is required:

You must first make sure that the designee is in the left-side couple,
that is, one of these spots:

\displayone
{ \dbdancer n & \dbdancer n & \bdancer n & \bdancer n \\
  \bdancer s & \bdancer s & \dbdancer s & \dbdancer s }
{}
  
Of course, if your designee is a man, and the couples are normal,
it will be one of these spots:

\displayone
{ \dbdancer n & \gdancer n & \bdancer n & \gdancer n \\
  \gdancer s & \bdancer s & \gdancer s & \dbdancer s }
{}

Next, rotate the designee's group of four clockwise, for example,
like this:

\displaytwo
{ \dbdancer n & \gdancer n \\
  \gdancer s & \bdancer s }
{Before}
{ \gdancer w & \dbdancer e \\
  \bdancer w & \dgdancer e }
{After}

This will leave the designee in the center if everything was done
correctly.  For example, we might have this:

\displaytwo
{ \dbdancer n & \gdancer n & \pbdancer n & \pgdancer n \\
  \pgdancer s & \pbdancer s & \pgdancer s & \pbdancer s }
{\ctablebox{\allemandedX \\ Before}}
{ \gdancer e & \dbdancer e & \pgdancer w & \pgdancer w \\
  \pbdancer e & \gdancer e & \pbdancer w & \pbdancer w }
{\ctablebox{\allemandeaO \\ After}}

We then note that we are in a DPT setup, like this:

\displayone
{ \bdancer e & \dbdancer e & \gdancer w & \gdancer w \\
  \gdancer e & \gdancer e & \bdancer w & \bdancer w }
{\allemandeaO}

Finally, we have to reorient.

Example:

\begin{sequence}
heads square thru 4\\
everyone square thru 4
\end{sequence}

\displayone
{ \bdancer n & \gdancer n & \bdancer n & \gdancer n \\
  \gdancer s & \dbdancer s & \gdancer s & \bdancer s }
{\allemandebO}

Before we can call \call{wheel and deal}, we have to change
designee:

\displayone
{ \bdancer n & \gdancer n & \bdancer n & \gdancer n \\
  \gdancer s & \bdancer s & \gdancer s & \dbdancer s }
{\allemandedO}

Then we do the call:

\displayone
{ \bdancer e & \bdancer e & \gdancer w & \gdancer w \\
  \gdancer e & \gdancer e & \dbdancer w & \bdancer w }
{\allemandecX}

and reorient:

\displayone
{ \gdancer s & \bdancer s \\
  \gdancer s & \dbdancer s \\
  \bdancer n & \gdancer n \\
  \bdancer n & \gdancer n }
{}

Then we can call

\begin{sequence}
centers pass thru\\
touch 1/4\\
walk and dodge\\
bend the line\\
star thru\\
allemande left
\end{sequence}

\section{Double Pass Thru and Completed DPT Formations}

\note{Should treat ZOOM in here somewhere.}

The double pass thru (DPT) and completed double pass thru (CDPT)
formations violate the rules of having a symmetric setup on each side,
so it seems that we can't deal with them.  However, we already know
how to have the \call{heads} or \call{sides star thru} and those four
do some calls in the center.  The trick was that, as long as we call
only to the centers, the ends don't need to match them.  We only need
to match the centers and ends when we want to start calling to
everyone.

We make the following formulation:

\highlight{A DPT setup is legal only if the designee is in the center.
It is equivalent to the Trade By setup that would be obtained if we
had the ends trade.  It follows that legal DPT setups have same sex in
tandem with each other.}

Like so many aspects of the mental image system, this sounds more
forbidding that it really is.  In practice, you don't have to worry
about that trade on the outside.  Things work out pretty well.  You
{\em do\/} need to worry about making sure the designee is in the
center.  Change of designee comes in handy when you are about to
give a call that goes into a DPT or CDPT formation.

Now we need to figure out how to represent a completed double pass
thru (CDPT) setup.  Strange as it may seem, we use a formulation that
has the designee on the {\em outside\/}, not in the center.  It
follows that you must never call things to the centers (e.g.
\call{chase right, boys run}) while in a CDPT.  The only things you may
do are the few things we will discuss here.

\note{Why is this?  If the designee is in the middle, you should be
able to call to just the centers.}

\note{Rob:  This formulation makes it easier to call things that one
actually uses in practice, like double pass thru and cloverleaf, at
the expense of things that would be dangerous and difficult to call,
and that one wouldn't do very often in practice anyway.  Should this
be mentioned in the text?}

\note{Bill: Is it not legal to use the change-of-designee here to
dance around in the middle, and then change it back to do 8-person
calls?}

Here is the formulation:

\highlight{A CDPT setup is legal only if the designee is on the
outside.  It is equivalent to the Trade By setup that would be
obtained by having the centers \call{trade}.  A CDPT must have the
same sex in tandem with each other.}

\subsection{\call{dive thru} and \call{pass to the center}}
\label{sec:divethru}
\label{sec:passtothecenter}

These calls can be done from an 8-chain.  What if we had this?

\displayone
{ \bdancer e & \gdancer w & \dbdancer e & \gdancer w \\
  \gdancer e & \bdancer w & \gdancer e & \bdancer w }
{}

The designee would go to the outside of the resulting DPT setup, which
is not allowed.  We can fix the problem by doing a change-of-designee
operation.  Simply change the designee to this one:

\displayone
{ \dbdancer e & \gdancer w & \bdancer e & \gdancer w \\
  \gdancer e & \bdancer w & \gdancer e & \bdancer w }
{}

\noindent and move the spot by 2 if it is an \o.

\call{dive thru} and \call{pass to the center} are just equivalent to
\call{pass thru} and \call{ends trade}.  We have defined the DPT setup
as being equivalent to having the ends \call{trade}, or in this case
undo the \call{trade} that they did, so we simply don't worry about
them.

The rule for these two calls is therefore just this:

\begin{itemize}
\item Be sure the designee is on the outside.  Use the
change-of-designee if necessary.
\item Have the designee \call{pass thru}.
\item The setup is now a DPT, with the designee in the center, as required.
\end{itemize}

\call{heads lead right}

\displayone
{ \bdancer e & \gdancer w & \dbdancer e & \gdancer w \\
  \gdancer e & \bdancer w & \gdancer e & \bdancer w }
{\allemandebX}

(change designee; spot is \x\ so it doesn't move)

\displayone
{ \dbdancer e & \gdancer w & \bdancer e & \gdancer w \\
  \gdancer e & \bdancer w & \gdancer e & \bdancer w }
{\allemandebX}

\call{pass to the center}

\displayone
{ \bdancer e & \dbdancer e & \gdancer w & \gdancer w \\
  \gdancer e & \gdancer e & \bdancer w & \bdancer w }
{\allemandebX}

That's pretty simple for an eight-person call.  Of course there are subtle
differences between the two calls -- \call{dive thru} is considered
tasteless if people are sashayed, and \call{pass to the center} can be
done from waves, since it starts with a \call{pass thru}.  You know
all that.

So what can we do in a DPT setup?  We will discuss the call
\call{double pass thru} shortly.  Other than that, we are limited to
having the four people in the center do things.  When they get into a
back-to-back setup among themselves, so that they are facing the
outsides, we can notice that we have an 8-chain setup, and start
calling to everyone.  The simplest thing to do along those lines is
\call{pass thru}.  We can also have the centers \call{square thru 3}.
In fact, many other things are possible, including \x\ calls (like
\call{swap around}) and complex things that temporarily go into other
formations (\call{touch 1/4, box circulate, quarter thru, extend}).
The thing we have to be careful about is not to get them sashayed
relative to the outsides.  Common culprits in this regard are
\call{swing thru} and \call{spin the top}.  If we called a
\call{centers swing thru} from a DPT, we couldn't then have them
\call{extend} to the outsides.  (Remember that \call{extend} is
analyzed, for the purposes of mental image, as \call{centers step
thru} to form an 8-chain, and then everyone \call{step to a wave}.  The
dancers don't know that.  To them it's just dancing.)

We could fix this problem by having the centers \call{swing thru}
again before the \call{extend}.  Another thing we can do is tell the
outsides to \call{half sashay}.  If you are calling to the centers, and you
see that they are going to come out sashayed relative to the outsides,
just call \call{outsides roll away} while they are waiting.  (To see
that this works, observe that it is equivalent to having the centers
\call{half sashay}, which would give a legal 8-chain setup, followed by
everyone \call{half sashay}, which is a legal call in each box.  Note also
that \call{half sashay} is an \o.)

\call{heads spin the top while the sides half sashay}
\displayone
{ \idancer & \gdancer e & \idancer \\
  \gdancer e & \dbdancer w & \bdancer w \\
  \bdancer e & \bdancer e & \gdancer w \\
  \idancer & \gdancer w & \idancer }
{\allemandebX}

\call{extend}
\displayone
{ \gdancer e & \gdancer e \\
  \dbdancer w & \bdancer w \\
  \bdancer e & \bdancer e \\
  \gdancer w & \gdancer w }
{\allemandebX}
\call{boys run, wheel and deal, pass thru, allemande left}

\subsection{\call{ferris wheel}}
\label{sec:ferriswheel}

Another simple call we can do involving DPT setups is \call{ferris
wheel}.

\highlight{\begin{itemize}
\item Be sure the designee is looking in.  Use change-of-designee if
necessary to make this so.
\item Just do the call.  The designee is now in the center of a DPT
setup.
\end{itemize}}

\subsection{\call{double pass thru}}
\label{sec:doublepassthru}

The call \call{double pass thru} is somewhat peculiar, but it is well
worth learning because it is very useful.

\highlight{Starting in a DPT setup with the designee in the center,
mentally change to the designee's local opposite on his side of the
set, and have him turn 180 degrees.  The allemande spot does not
change.  The setup is now a CDPT with the designee on the outside.}

The mental action is like a \call{trade} and \call{pass thru}:

\displaytwo
{ \bdancer e & \dbdancer e & \gdancer w & \gdancer w \\
  \gdancer e & \gdancer e & \bdancer w & \bdancer w}
{\call{heads star thru}}
{ \gdancer w & \gdancer w & \bdancer e & \bdancer e \\
  \dbdancer w & \bdancer w & \gdancer e & \gdancer e}
{After \call{double pass thru}}

\section{Calls from a CDPT Setup}

We are only going to discuss four things you can do from a completed
double pass thru setup.  There are a few others in the chapter on
esoteric calls.  Remember that the designee is always on the outside.

\subsection{\call{ends trade}}
\label{sec:endstradecdpt}

This is simple.  Just do it.  You will then have an 8-chain setup.

\subsection{\call{cloverleaf}}
\label{sec:cloverleaf}

Do the opposite of the mental action you do for \call{double pass
thru}.  This is effectively a \call{trade} and \call{pass thru}.  Then
move the spot by 2.  Finally, reorient the set.

Example:

\call{heads star thru, double pass thru}

\displayone
{ \gdancer w & \gdancer w & \bdancer e & \bdancer e \\
  \dbdancer w & \bdancer w & \gdancer e & \gdancer e }
{\allemandebO}

\call{cloverleaf}

\displaytwo
{ \bdancer e & \dbdancer e & \gdancer w & \gdancer w \\
  \gdancer e & \gdancer e & \bdancer w & \bdancer w }
{\ctablebox{\allemandedO \\ before reorientation}}
{ \gdancer s & \bdancer s \\
  \gdancer s & \dbdancer s \\
  \bdancer n & \gdancer n \\
  \bdancer n & \gdancer n}
{after reorientation clockwise}

\call{centers pass thru, slide thru, slide thru, allemande left}.

\subsection{\call{track 2}}
\label{sec:track2}

This is extraordinarily simple.  It is an eight-person call, with all
the potential for complexity that eight-person calls have, but the
complexities happen to cancel each other.  Just have the designee do
the call.

\displayone
{ \gdancer w & \gdancer w & \bdancer e & \bdancer e \\
  \dbdancer w & \bdancer w & \gdancer e & \gdancer e }
{\allemandedX}

\call{track 2}

\displayone
{ \bdancer e & \dbdancer e \\
  \gdancer w & \gdancer w \\
  \gdancer e & \gdancer e \\
  \bdancer w & \bdancer w}
{\allemandedX}

\subsection{\call{first couple go left, next couple go right}}
\label{sec:firstleftnextright}

Have the designee's couple undo the action of a \call{lead left}.  (If
the designee is a man and the couples are normal, he just does a
\call{quarter right}.)  Change the spot from \o\ to \x\ or \x\ to \o,
and pull it toward the designee.  Finally, reorient the set.

\displayone
{ \gdancer w & \gdancer w & \bdancer e & \dbdancer e \\
  \bdancer w & \bdancer w & \gdancer e & \gdancer e }
{\allemandebO}

\call{first couple go left, next couple go right}

\displaytwo
{ \gdancer s & \bdancer s & \gdancer s & \dbdancer s \\
  \bdancer n & \gdancer n & \bdancer n & \gdancer n }
{\ctablebox{\allemandecX \\ before reorientation}}
{ \bdancer e & \gdancer w \\
  \gdancer e & \bdancer w \\
  \bdancer e & \gdancer w \\
  \gdancer e & \dbdancer w}
{after reorientation clockwise}

\section{Columns}
\label{sec:columns}
\label{sec:columncirculate}

\note{Move this section earlier}

Columns with alternating sex are just two sex-symmetric box setups,
and, as such, can be handled in a straightforward way by the mental
image system.  Any applicable four-person call can thus be called.  Among
the more obvious such calls are:

\begin{sequence}
\call{touch 1/4}  {\rm (Facing lines to columns)} \\
\call{boys/girls run}  {\rm (Columns to 8-chain or trade by)} \\
\call{split circulate}  {\rm (not recommended)} \\
\call{triple scoot}  {\rm (technically eight-person call but is just a
\call{trade})} \\
\call{hinge} {\rm (columns $\Leftrightarrow$ tidal wave)} \\
\call{crossfire} {\rm (2-faced lines to columns)}
\end{sequence}

The major problem is that \call{column circulate} is a difficult
eight-person call that is nearly indispensable.  So we just have to deal
with it.

\highlight{\begin{itemize}
\item If the designee is a trailer in his box of four (that is, number
2 or 4 in the column), move the allemande spot in the direction that
he faces.  Do not change the \x\ or \o\ nature of the spot.  The spot
motion wraps around if necessary.  If the designee is a leader in his
box (number 1 or 3 in the column), move the spot in the opposite
direction from his facing direction.
\item In any case, move the designee backwards, wrapping around if
needed.  That is, if the designee was initially number 4 in the
column, the wraparound effectively pushes him forward 3 places to be
number 1.
\end{itemize}
}

The spot motion could be described as ``move the spot in the same
direction as the direction from the designee towards his tandem
partner.''  A person's ``tandem partner'' is the person he is
naturally paired with in tandem.  This is analogous to the normal
``lateral'' partner, which is the person someone is naturally paired
with laterally.  So if the designee is a leader, his tandem partner is
behind him, so the spot moves ``backward'' relative to the way he
faces.  If the designee is a trailer, his tandem partner is in front
of him, so the spot moves ``forward'' relative to his facing direction.

It should be emphasized that, unlike many other calls, the spot motion
is {\em not\/} characterized as being ``toward'' or ``away from'' the
designee.  Which side of the set the designee is on doesn't matter.
If the designee is in either of these two places:

\displayone
{ \dbdancer e & \gdancer e & \dbdancer e & \gdancer e \\
  \gdancer w & \bdancer w & \gdancer w & \bdancer w }
{}

\noindent the spot motion is left to right (as seen by the caller).
If the designee is in either of these two places:

\displayone
{ \gdancer e & \dbdancer e & \gdancer e & \dbdancer e \\
  \bdancer w & \gdancer w & \bdancer w & \gdancer w }
{}

\noindent the spot motion is right to left.

\note{What about if the designee is the other two boys?}

\note{Insert example from page 78/79.}

There is an important simplification we can make, so that we don't
have to deal with the complexity of the two very different rules for
which way the spot goes and which way the designee goes.  Most column
circulates start with women in the lead and men as trailers.  If you
restrict yourself to that arrangement only, you will still be able to
call reasonable figures with mental image.  So, if your designee is a
man:

\highlight{The designee must be a trailer.  The spot goes forward and
he goes backward.}

\noindent and don't bother learning the next rule.  If your designee
is a women, don't bother with the preceding rule.  Learn this one
instead:

\highlight{The designee must be a leader.  Both she and the spot move
backward.}

\subsection{\call{coordinate}}
\label{sec:coordinate}

Once we have mastered \call{column circulate}, \call{coordinate} is
easy.  It is equivalent to \call{column circulate} and \call{trail
off}.  Even if you don't call the A2 program, but call Plus, you
should learn \call{trail off}, just so you can do \call{coordinate}.
It is a simple \o\ call.  Remember that the men will be leads and the
women trailers at this point.  So, for a male designee, \call{trail
off} is just this:

\displaytwo
{ \gdancer e & \dbdancer e \\
  \bdancer w & \gdancer w }
{before}
{ \bdancer e \\
  \gdancer e \\
  \gdancer w \\
  \dbdancer w }
{after}

\noindent For a female designee, just learn this:

\displaytwo
{ \dgdancer e & \bdancer e \\
  \bdancer w & \gdancer w }
{before}
{ \bdancer e \\
  \gdancer e \\
  \dgdancer w \\
  \bdancer w }
{after}

\call{heads square thru 2, slide thru, touch 1/4}

\displayone
{ \bdancer e & \gdancer e & \bdancer e & \gdancer e \\
  \gdancer w & \bdancer w & \gdancer w & \dbdancer w }
{\allemandeaX}

\call{coordinate}

\displaytwo
{ \gdancer e & \bdancer e & \gdancer e & \bdancer e \\
  \dbdancer w & \gdancer w & \bdancer w & \gdancer w }
{\ctablebox{\allemandedX \\ after \call{circulate}}}
{ \dbdancer e & \bdancer e \\
  \gdancer e & \gdancer e \\
  \gdancer w & \gdancer w \\
  \bdancer w & \bdancer w }
{\ctablebox{\allemandedX \\ after \call{trail off}}}

\call{bend the line, slide thru, eight chain 3, allemande left}

\section{Diamonds}
\label{sec:diamonds}

Diamond setups are unusual in that the usual analysis of calls as \x\
or \o\ doesn't work because the dancers are T-boned.  We can
nevertheless handle diamonds thru a special trick, but you need to be
aware that some seemingly straightforward things that work for other
four-person setups can lead to the wrong result.  We will mention these
shortly.

The formulation that we make is:

\highlight{For the purposes of \x s and \o s, a diamond is equivalent
to the wave or 2-faced line that would result from \call{flip the
diamond} or \call{cut the diamond}.}

It follows from this that \call{flip the diamond} and \call{cut the
diamond} are \o\ calls by definition.

We can handle both normal and facing diamonds.  As usual, you are
responsible for formation management.  You must be aware of which kind
of diamond you have, and that \call{flip the diamond} turns normal
diamonds into waves and facing diamonds into 2-faced lines, while
\call{cut the diamond} does the opposite.

Here are some simple four-person calls from diamonds:

\begin{tabular}{ll}
\call{diamond circulate} & \x \\
\call{flip the diamond} & \o \\
\call{cut the diamond} & \o \\
\call{centers hinge} (from diamond) & \x \\
\call{centers hinge} (from 2-faced line) & \o \\
\call{switch to a diamond} & \x
\end{tabular}

How about \call{6x2 acey deucey} from the A1 program?  Sorry, the
system just can't handle it.  A few modules will be described in
Chapter~\ref{chap:something}.

Here is what you need to be careful about.  You are by now
sufficiently familiar with \x\ and \o\ calls to know that any call
that has just one sex turn around is an \x.  For example, from waves,
\call{boys/girls/centers/ends u-turn back} is an \x.  This is {\em
not\/} true for the centers of diamonds.  Observe that \call{flip the
diamond} and \call{centers u-turn back, flip the diamond} are
equivalent to each other in terms of facing direction, which is all
that matters for \x/\o\ analysis.

\section{\call{tag the line}}
\label{sec:tagtheline}

In their simplest form, certain \call{tag the line} calls are very
simple four-person calls that can be handled very easily:

\highlight{From a two-faced line, \call{tag the line right/left} is an
\o, and \call{1/2 tag} is an \x.}

There are a number of tricks we can use to enhance our ability to use
this call and to make it easier to remember the \x/\o\ nature of these
calls.  We will note a few things at the outset.  The system just
can't handle \call{tag the line 1/4} or \call{tag the line 3/4}.  The
centers would be all the same sex.  Also, \call{tag the line}, but
don't \call{face left/right}, would be legal, but the resulting setup
would be essentially useless, since it would not be a legal CDPT
setup.  So you should not try anything other than \call{1/2 tag} and
\call{tag the line right/left/zig-zag/zag-zig}.

Here is the trick that helps one to remember whether a call is an \x\
or an \o.  This works for \call{tag the line} and \call{flip the
line}, with a variety of facing directions that we will discuss in
detail later.

\highlight{If the call preserves wave-ness or couple-ness, it is an
\o.  If it converts waves (or miniwaves) to couples or vice-versa, it
is an \x.}

It should become clear in the following paragraphs what we mean by
``wave-ness'' and ``couple-ness.''

Here are some applications:

From a 2-faced line, \call{tag the line right/left} preserves couples,
and so it is an \o.  \call{face in/out} is not legal with mental
image, as the resulting four-person setup is not symmetric.  (But we will
presently discuss a context in which it can be handled, and it will
turn out that the result is considered couples.)

\call{tag the line, zig zag/zag zig} (A2) turns couples into
miniwaves, and so it is an \x.

We can do these calls starting in waves.  Smoothness dictates that they
be left-handed waves, unless we say \call{left tag the line}.  From
waves, \call{tag the line right/left} turns miniwaves into couples,
and so it is an \x.  \call{tag the line, zig zag/zag zig} preserves
miniwaves so it is an \o.

From either a 2-faced line (right-handed for smoothness) or a wave
(left-handed for smoothness) we can call \call{1/2 tag}.  This leaves
a box, which we consider to be wave-like.  So it is an \x\ from a
2-faced line and an \o\ from a wave.

This trick applies to \call{flip the line} (C1) also.  Of course, the
starting setup must be a wave.  \call{flip the line, zig zag/zag zig}
starts and ends in miniwaves, so it is an \o.  \call{flip the line,
face right/left} starts in waves but ends in couples, so it is an \x.
\call{flip the line 1/2} (or \call{2/3 recycle}) is an \o.  \call{flip
back} is also an \o, because \call{scoot back} is an \o.  \call{flip
your neighbor} is an \x\ because \call{follow your neighbor} is an \x.
From a 2-faced line, \call{tag back to a wave} is an \x, and
\call{tag your neighbor} is an \o.

\subsection{\call{tag the line} from Lines Back-to-Back}

So far we have only considered \call{tag the line} when each symmetric
setup is a general line, such as a 2-faced line or a wave, and we have
used it as a four-person call.

What if the symmetric setups are boxes?  That is, we have what the
dancers would consider to be lines back-to-back.  In that case, the
setups interact with each other, so it is an eight-person call.  We can
still handle it, as long as the final direction that we give is
\call{in} or \call{out}.

\highlight{\call{tag the line in/out} from lines facing or lines
back-to-back:\\
\begin{itemize}
\item Move the designee to his local (in his box of four) opposite.  Face
him in or out, depending on what you said to the dancers.
\item The final setup is lines facing or lines back-to-back.
\item The spot does not change or move.
\end{itemize}
}

You can do this from lines facing either in or out, but many people
consider it distasteful to call \call{tag the line} from
lines facing in.

\call{heads square thru 2, everyone square thru 2}

\displayone
{ \bdancer n & \gdancer n & \bdancer n & \gdancer n \\
  \gdancer s & \bdancer s & \gdancer s & \dbdancer s }
{\allemandebO}

\call{tag the line in}

\displayone
{ \bdancer s & \gdancer s & \dbdancer s & \gdancer s \\
  \gdancer n & \bdancer n & \gdancer n & \bdancer n }
{\allemandebO}

\call{star thru, trade by, allemande left}