weaving.tex

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\title{Mathematics of Weaving}
\author{Ken Levasseur\\
UMass Lowell
}
\date{August 20, 2016}
\begin{document}
\maketitle
\thispagestyle{empty}
\begin{abstract}
This article highlights some of the mathematics behind weaving. It is an outgrowth of two projects. The first was work with Shelley Rasmussen and students in the UMass Lowell Co-op Scholar program.  Students Mary Mersereau, Olivia Demers, and Matt D'Angelo developed materials that linked weaving with mathematics.  The second is \emph{Lowell Tex}, a collaboration between UMass Lowell faculty and local textile artists to develop STEM-related teaching materials.%
\end{abstract}
\typeout{************************************************}
\typeout{Section 1 Introduction}
\typeout{************************************************}
\section[Introduction]{Introduction}\label{section-introduction}
\leavevmode%
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{images/loom.png}
\caption{A loom
                \label{fig-loom}}
\end{figure}
Weaving that is considered here is what can be realized with a multi-harness loom such as the one shown in \hyperref[fig-loom]{Figure~\ref{fig-loom}}.
The main topics are %
\leavevmode%
\begin{itemize}[label=\textbullet]
\item{}an exposition of how the weaver's draft, consisting of a tie-up draft, treadlig matrix and lift plan can
		 be combined to visualize the final weave that will be produced, and%
\item{}an analysis of how harnesses can be tied in such a way to realize a given weave pattern.%
\end{itemize}
\typeout{************************************************}
\typeout{Subsection 1.1 Weaving Basics}
\typeout{************************************************}
\subsection[Weaving Basics]{Weaving Basics}\label{ss-weaving-def}
\index{Weaving Basics}A weave pattern is created by interleaving two sets of parallel lengths of fiber. Each of the fibers in one set, the \terminology{warp}, is connected to exactly one of \(n\) harnesses, \(n \geq 2\). Each \terminology{harness} will lift the threads that are connected to it.  A series of \terminology{treadles} are connected to the harnesses, with each treadle connected to between 1 and \(n-1\) of the harnesses. When a treadle is pressed, the warp fibers connected to that treadle through the harnesses are lifted, forming a \terminology{shed} between the warp fibers that are lifted and those that are not lifted. A \terminology{weft} thread can then be passed through the shed.  The weft runs perpendicular to the warp. Normally, the weft thread is passed through the shed with a \terminology{shuttle}. %
\par
By performing a sequence of pedal presses and shuttle passes that interleave the warp with the weft, a weaving pattern is realized.  Using different colors for the warp and weft makes the pattern visible. In this discussion we only consider "monochromatic weaving," where the warp are all of one color and the weft threads are all of a second color.%
\typeout{************************************************}
\typeout{Subsection 1.2 The Weaver's Draft}
\typeout{************************************************}
\subsection[The Weaver's Draft]{The Weaver's Draft}\label{ss-weaving-draft}
A weaver's draft is typically a array of four matrices, such as the small example in \hyperref[fig-draft]{Figure~\ref{fig-draft}}. The top left (A) matrix, called the harness draft, has as many columns as the number of warp threads, and there is a row for each harness.  Each column indicates to which harness the corresponding thread is tied. Each thread is tied to only one harness and so each column has one mark.  The top right (B) matrix, called the pedal plan, indicates which harnesses are tied to each treadle and determines which threads will make up the top of the shed when a pedal is pressed. The pedal has as many rows as there are harnesses and as many columns as there are pedals.  The bottom right (C) matrix is the lift plan, which indicates the sequence of pedal presses that are to performed for each successive weft thread. The lift plan has as many columns as there are pedals.  The number of rows is the number of weft threads, although the lift plan is frequently periodic and the rows on the draft would be repeated. The output from is sequence of presses and shuttle movements is indicated with the bottom left (D) matrix, the weave.  Some weavers use different  positioning schemes in their drafts, but the essential information is the same.  %
\leavevmode%
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{images/fig-draft.png}
\caption{A typical monochromatic weaving draft
                \label{fig-draft}}
\end{figure}
\typeout{************************************************}
\typeout{Subsection 1.3 Computing the weave pattern}
\typeout{************************************************}
\subsection[Computing the weave pattern]{Computing the weave pattern}\label{subsection-3}
Clearly the weaver has control over the harness matrix, pedal matrix and lift plan.   It is a simple matter of logic to determine whether any weft thread will lie above or below any warp thread.  In \hyperref[section-products]{Section~\ref{section-products}} we derive formulas to determine the whole weave based on matrix multiplication.%
\par
Not every draft will produce a weave that "hangs together."  A trivial example would be a draft for which one of the warp fibers is always on the upper part of the shed, which would leave that thread to float around.  Less obvious cases can arise though.  We will describe how this can be determined from the output matrix.%
\typeout{************************************************}
\typeout{Subsection 1.4 Realization of a Weave}
\typeout{************************************************}
\subsection[Realization of a Weave]{Realization of a Weave}\label{subsection-4}
An even more practical question is how to specify the harness matrix, pedal matrix and lift plan to attain a specific weave pattern, if at all possible for a given loom.  In \hyperref[section-realization]{Section~\ref{section-realization}}, we give a description of how to determine whether one can  realize a desired weave pattern and how the loom can be set to do so.%
\typeout{************************************************}
\typeout{Section 2 Weaving as a product of matrices}
\typeout{************************************************}
\section[Weaving as a product of matrices]{Weaving as a product of matrices}\label{section-products}
Recall that a weaving draft is a block array with the stucture


\(\begin{array}{|c|c|}
\hline
 H &P \\
\hline
 W &L \\
\hline
\end{array}\)

where \(H, T, W \textrm{ and } L\) are boolean matrices defined as follows, where True is represented as 1 and False as 0.  Expressions that derive the main formula in this section use the notation of Boolean arithmetic, were conjunction (``and'') is denoted by \(\cdot\) and disjunction (non-exclusive ``or'') is denoted by \(+\).%
\par
 \(H\)  is the harness draft it has as many rows as a loom has harnesses and as many columns as the number of warp threads. We define entries of this matrix by
\begin{equation*}H_{i j }=\textrm{ Is harness } i \textrm{ connected to  warp thread } j?\end{equation*}
%
\par
\(P\)  is the pedal matrix it has as many rows as a loom has harnesses and as many columns as the number of treadles in the loom. We define entries of this matrix by
\begin{equation*}P_{i j}= \textrm{Is harness } i \textrm{ lifted by treadle } j?\end{equation*}
%
\par
 \(L\)  is the lift plan matrix it has as many rows as the number of threads in the weft and as many columns as the number of treadles in the loom. We define entries of this matrix by
\begin{equation*}L_{i j}= \textrm{Is treadle } j \textrm{ pressed for weft tread } i?\end{equation*}
%
\par
 \(W\) is the matrix of the weave resulting  from the setting of  \(H\) ,  \(P\) , and  \(L\) . We derive  \(W\)  as a matrix
product in terms of these settings.
\begin{equation*}W_{i j}= \textrm{Is warp thead } j \textrm{ lifted above weft thread } i?\end{equation*}%
\begin{claim}[The main formula]\label{claim-1}
 \(W = L T^t H\) , where \(T^t\) is the transpose of \(T\).%
\end{claim}
\begin{proof}\hypertarget{proof-1}{}
To prove our claim we define the matrix  \(C\),  where \begin{equation*}C_{i j}= \textrm{Is warp thead } j \textrm{ lifted when treadle } i \textrm{ is pressed?}\end{equation*}%
\par
We derive a formula for the \(C\):
\begin{equation*}
\begin{split}
C_{i j} &= \textrm{Is warp thead } j \textrm{ lifted when treadle } i \textrm{ is pressed?}\\
	  	& = P_{1i}H_{1j }+P_{2i}H_{2j }+\cdots \\
		& = H^t{}_{j 1}P_{1 i}+H^t{}_{j 2}P_{2 i} +\cdots \\
		& = \left( H^t P \right)_{j i}\\
		& = \left( H^t P \right)^t_{i j}\\
		& = \left(T^tH\right)_{i j}
\end{split}
\end{equation*}
%
\par
Now we can compute \(W\).
\begin{equation*}
\begin{split}
W_{i j} &= \textrm{Is warp thead } j \textrm{ lifted above weft thread } i?\\
		&= L_{i 1}C_{1j }+ L_{i 2}C_{2j }+\cdots \\
		& = (L C)_{i j}\\
		&= \left(L T^tH\right){}_{i j}
\end{split}
\end{equation*}
%
\end{proof}
\typeout{************************************************}
\typeout{Section 3 Realization of a Weave Pattern}
\typeout{************************************************}
\section[Realization of a Weave Pattern]{Realization of a Weave Pattern}\label{section-realization}
\index{Minsets} 
Whether a pattern can be realized is determined by whether each of the distinct horizontal rows of the pattern can be realized. Let \(r_1,\dots , r_n\) be those rows. Each row \(r_i\) is realized be lifting a set \(R_i\) of warp fibers. Give two warp fibers \(f_1\) and \(f_2\), if \(f_1, f_2 \in R_i\) for some \(i\) yet \(f_1 \in R_j\) while \(f_2 \notin R_j\), we cannot connect \(f_1\) and \(f_2\) to the same harness.   If we presume that all warp fibers are connected to exactly one harness, with the set of warp fibers connected to harness \(i\) equal to \(H_i\), then the warp fibers lifted to realize a row must be a union of the \(H_i's\). These observations imply that \(\{H_i\}\) would be the minsets generated by \(\{R_j\}\).  For each \(i\),

\begin{equation*}H_i = \bigcap_{j=1}^{n}{\tilde{R}_j}\end{equation*} 
where \(\tilde{R}_j\) is either \(R_j\)  or \(R_j^c\).
%
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\appendix
%
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\typeout{Appendix A Notation}
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\section[Notation]{Notation}\label{appendix-1}
This is some notation introduced in the article.%
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\textbf{Symbol}&\textbf{Description}&\textbf{Page}\\[1em]
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\textbf{Symbol}&\textbf{Description}&\textbf{Page}\\[1em]
\endhead
\multicolumn{3}{r}{(Continued on next page)}\\
\endfoot
\endlastfoot
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\typeout{References  References}
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\section*{References}\label{references-1}
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\bibitem[1]{biblio-rasmussen-harness}\hypertarget{biblio-rasmussen-harness}{}Shelley Rasmussen, \textit{On 3-Harness Weaving: Cataloging Designs Generated by Fundamental Blocks Having Distinct Rows and Columns}.The Electronic Journal of Combinatorics, 2008, \textbf{15} no.\@\,R1, 57\textendash{}62.
                
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This article was authored in MathBook XML.%
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