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\begin{document}

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\begin{center}

\huge{Commentary on the ``Parallelogram Triple Boxes'' Concept}

\huge{Bill Ackerman}

\huge{7 June 2014}

\end{center}

Copyright \copyright\ 2004, 2005, 2012, 2014, William B. Ackerman.  Permission is granted to
make and redistribute non-commercial verbatim copies of this document.

\bigskip

This is an explanation of the deprecated ``Parallelogram Triple Boxes'' concept.  This
material previously resided in the ``Combining Offset Concepts and Phantom Concepts''
section of \url{http://www.challengedance.org/sd/book3.pdf}

The ``Offset Triple Boxes'' concept has replaced this concept.  See
\url{http://www.challengedance.org/sd/book2.pdf}.

\frontmatter
\mainmatter

\pagestyle{plain}

When the direction in which the phantoms are placed is parallel to the
offset direction, there are problems.  Consider Parallelogram Triple
Boxes as an example.  Once the offset is measured (50\%) and is taken
out, we have a virtual 2x4.  When we do Triple Boxes in a 2x4, we
place phantoms at each end to make a 2x6.  After doing the call, the
result of the Triple Box call is often a 2x6 from which phantoms can't
be stripped.  50\% of that is 3 people, so, when the offset is
re-imposed, the setup is a 2x9.  (But it often goes to a 2x7 or 2x5 because
phantoms can now be stripped.)  This formulation has sometimes been used.

\begin{displaydance}
\sxhBBBs{...,g1s,...,b2n,b1s,g4s} & \syhlines{b3n,g4s,b2n,g1s} \cr
\cbox{before Parallelogram\\Triple Boxes\\Circulate} & \cbox{remove 50\% offset} \\
\sxhBBBs{b1n,.+.,g1s,.+.,b4n,g2n} & \sxhBBFs{...,g2n,b1n,b2s,g1s} \cr
\cbox{do the Triple Box call} & \cbox{re-impose 50\% offset on a\\2x6.  Is this correct?} \cr
\end{displaydance}

While knowledgeable and widely respected people embrace this interpretation,
I believe there are problems with it.

\medskip

The Triple Box call doesn't necessarily end in a 2x6.  It could remain
a 2x4 after phantoms are stripped (Triple Boxes Zing, for example.)
In that case, if the phantoms are stripped before re-imposing the
offset (which seems logical, since they were placed after measuring
the offset), the result is a 2x6 rather than a 2x9, 2x7, or 2x5.  What's
wrong with that?  It means that the people in the center triple box
need to know what happened in the outer triple boxes in order to
figure out their own local offset.  Callers sometimes use the ``Triple
Formation'' concept when the inner and outer formations are different
and/or the calls are different.  People in the center formation
shouldn't need to know where the phantom spots in the outer formations
are located in order to figure out whether their result is a 2x2 box
or some kind of ``Z''.

\medskip

If the phantoms are not stripped before re-imposing the offset, there
is another problem, admittedly a theoretical one.  Everyone wants the
formulation of ``Parallelogram Triple Boxes'' to be a simple nesting
of the two concepts, just as ``Tandem Stable'' is a simple nesting of
two concepts.  ``Parallelogram Nothing'' is the same as
``Nothing''---it goes from a 2x6 Parallelogram to the same thing.
``Triple Boxes Nothing'' is also ``Nothing''---it goes from a 2x4 to
the same thing.  If the two concepts nest properly, ``Parallelogram
Triple Boxes Nothing'' on a 2x6 Parallelogram should be ``Nothing''.
But if the phantoms are not stripped before re-imposing the offset,
people would move.  The result is a 2x7.

\begin{displaydance}
\sxhBBBs{...,g1s,...,b2n,b1s,g4s} & \sxhBBVs{...,g1s,...,b2n,...,g4s,b1s} \cr
\cbox{before Parallelogram\\Triple Boxes Nothing} & \cbox{result?} \cr
\end{displaydance}

This is an unsettling result.

\medskip

There is a formulation that solves these problems, but it is not
particularly palatable or intuitive.  Do things in the following order:

\begin{quote}

Place the phantoms for the Triple Box concept or other phantom
concept.  In this example, the setup is now two 1x6 formations offset
by 2 people.

Measure the offset in terms of these ``positions'', even though they include
phantoms.  It is 33\% in this example.

Remove that offset, do the Triple Box or other phantom call in the
expanded setup, and re-impose the offset.

Strip phantoms from the actual result.

\end{quote}

\begin{displaydance}
\sxhBBBs{...,g1w,...,b1w,b2e,g4w} & \sxhBBBBs{...,.+.,...,g1w,.+.,b1w,b2e,g4w} \cr
\cbox{before Parallelogram\\Triple Boxes\\Circulate} & \cbox{place phantoms} \\
\sxhBBBs{.+.,.+.,b2e,g1w,g2e,b1w} & \sxhBBBs{b2e,.+.,g1e,.+.,b3e,g2w} \cr
\cbox{remove 33\% offset} & \cbox{do the Triple Box call} \\
\sxhBBBBs{...,.+.,...,.+.,b2e,g2w,g1e,b1w} & \syhlines{b4w,g3w,b1w,g2w} \cr
\cbox{re-impose 33\% offset} & \cbox{finished} \cr
\end{displaydance}

This method is logically consistent but not very intuitive.

\medskip

Because of these problems, Offset and Phantom concepts, in which the shear
direction is parallel to the phantom placement direction, are hardly
ever used.

\medskip

Here is another example:

\begin{displaydance}
\syhBBBBs{...,...,...,...,g3n,g4n,b3s,b4s} & \syhKKKKs{.+.,g3n,g4n,.+.,.+.,b3s,b4s,.+.} \cr
\cbox{before Offset Waves\\Split Phantom Waves\\Settle Back} & \cbox{place phantoms} \\
\syhBBBBs{.+.,b2n,.+.,g2s,.+.,g3n,.+.,b3s} & \syvBBBBs{.+.,b3w,.+.,g2e,.+.,g3w,.+.,b2e} \cr
\cbox{remove 50\% offset} & \cbox{do the Split Phantom Waves call} \\
\syvLLLLs{.+.,.+.,.+.,.+.,b4w,g1e,g4w,b1e} & \sxvlines{b4w,g1e,g4w,b1e} \cr
\cbox{re-impose 50\% offset} & \cbox{finished} \cr
\end{displaydance}

\end{document}
