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fractional-emulate.tex
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fractional-emulate.tex
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\documentclass[11pt]{article}
\usepackage{fullpage}
\usepackage{parskip}
\usepackage{tex-squares}
%\usepackage{doublespace}
%\usepackage{graphicx}
%\usepackage{moreverb}
\title{The Fractional Emulate Concept (DRAFT)}
\author{Rachel Chasin and Laura Dean}
\date{\today}
%\markright{foo}
\begin{document}
\thispagestyle{empty}
\maketitle
%\tableofcontents
In this paper we describe the fractional emulate concept. This concept
is analogous to fractional stable: do the given call until you've
turned the given fraction, and then continue with the rest of the
call, but emulate it -- that is, do any subsequent turning motions
but remain in place.
As with the fractional stable concept, different people may start
working emulate at different times, although unlike with fractional
stable, this may result in an illegal call (see
Section~\ref{sec:nonanalogy}).
The motivation for this concept came about when Laura expressed
frustration that Zip the Top had no apparent relation to the other
``the top'' calls -- in particular, nobody was casting off 3/4.
``Well, actually, it's Fan the Top, but emulate after turning 1/4,''
Janet helpfully noted. And so it is!
Fractional emulate feels appropriate for describing calls
involving a U-Turn Back -- for example, Zip the Top and Shazam.
Likewise, Follow Your Neighbor involves substantial turning in place for the leaders,
and can be described as: leaders work 1/4 emulate,
trailers work 3/4 emulate, Follow Your Leader.
%The effect of the trailers working 3/4 emulate despite having turned
%the total amount of the call is to stop them from moving ahead to be 1 and 2 in a column.
Some more fun equivalences:
\begin{tabular}{|l|l|}
\hline
1/4 emulate Fan the Top & Zip the Top \\
1/4 emulate Cast off 3/4 (from mini-wave) & Shazam \\
1/4 emulate Box Transfer & Follow Thru and U-Turn Back \\
& (tandem twosome Shazam) \\
1/4 emulate Follow Your Neighbor & Same as previous\\
1/4 emulate Trade & Single Wheel \\
1/4 emulate Scoot Back & Single Ferris Wheel \\
& (tandem twosome Single Wheel)\\
1/4 emulate Couples Trade & Stack the Wheel \\
1/4 emulate Box Counter Rotate 1/2 & Couple Up \\
1/4 emulate Flip the Line ($n$/4) & Patch the centers \\
(but not 1/4 emulate 2/3 Recycle) & \\
\hline
\end{tabular}
\section{An interesting non-analogy}
\label{sec:nonanalogy}
The fractional emulate concept is analogous to fractional stable,
but (at least) one property is different.
With fractional stable,
if a given call X is legal from a given starting setup,
then fractional stable X is also possible.
(This is also true for just plain stable, or just plain emulate.)
With fractional emulate, this is not the case!
Consider, for example: 1/2 emulate, Box Circulate twice.
In this call, people try to end up on the same spot,
with the same facing direction
(so they can't even think about taking right hands).
Other applications might be legal, but awkward:
1/4 emulate Box Circulate, for example,
would leave the original leaders in single quarter tag spots,
and the original trailers in box spots.
It's not clear what a caller would want to do from there!
(TODO: should the previous para go here, or with Limitations?
Do they all go together? Maybe we want the worked example,
and then that stuff. Hm.)
\section{A worked example}
An interesting case with fractional emulate using fractions larger
than 1/4 arises with box transfer. 1/4 emulate Box Transfer
follows plainly from the definition,
but 1/2 emulate and 3/4 emulate
Box Transfer require breathing. For the 1/2 emulate case, after the leaders
have turned 1/2 and stop changing position, they have box circulated
one spot from their original position, and are therefore still in
adjacent planes. However, the original trailers extend and arm turn 1/2
before they stop changing positions, putting them on the center line
of the box and forcing everyone to breathe to create three planes.
This makes the ending formation a Z.
A similar Z is formed by
3/4 emulate Box Transfer. This marks another difference from
fractional stable: if a call involves turning $n$/4, the $n$/4 stable
version will be equivalent to the original call while the $n$/4
emulate version may not. The concepts differ when the call involves
changing position but not direction after the last turn (for example,
extending in the case of box transfer).
\displaytwo{
\dancer{1}{n} & \dancer{2}{s} \\
\dancer{3}{n} & \dancer{4}{s} \\
}{starting formation for $n$/4 emulate box transfer}{
\dancer{1}{w} \\
\dancer{3}{w} \\
\dancer{2}{e} \\
\dancer{4}{e} \\
}{1/4 emulate box transfer}
\displaytwo{
\idancer & \dancer{1}{w} \\
\dancer{2}{e} & \dancer{3}{w} \\
\dancer{4}{e} & \idancer
}{1/2 emulate box transfer}{
\dancer{4}{e} & \dancer{2}{e} & \idancer \\
\idancer & \dancer{3}{w} & \dancer{1}{w} \\
}{3/4 emulate box transfer}
Likewise, Curl Thru can be defined
as a 1/2 emulate Crossfire.
\section{Relations to Existing Concepts}
Many fractional emulate calls end up being ``and roll'' or ``and roll and
roll'' endings of existing calls,
or (relatedly) fractional twosome versions of calls.
One counterexample is 1/2 emulate Swing Thru
(or other calls in the $n$/4 Thru family, with $n$/4 emulate).
On this call, the new centers U-Turn back toward each other.
Calls of this ilk may or may not end up feeling good to dance!
Another counter-example, of course, is seen above:
calls (such as Box Transfer) that end with walking ahead
(or sliding) rather than turning.
\section{Limitations}
Some calls won't work well, because the position isn't well-defined
when you've turned a certain number of quarters. Consider, for
example, zoom: the leader's position is well-defined after 1/2, but
not after 1/4 or 3/4.
\section{Thanks!}
Thanks to Janet Chuang for discovering this concept,
by noticing that Zip the Top actually does have something to do
with other ``The Top'' calls -- contrary to Laura's complaints!
(These complaints are now rescinded.)
Thanks also to Andy Latto for the excellent example
of an awkward ending position (1/4 emulate Box Circulate),
and for a sanity check of some other ideas.
\end{document}