deletions | additions       diff --git a/the dilated triple model.tex b/the dilated triple model.tex  index bd46f86..26f99dd 100644  --- a/the dilated triple model.tex  +++ b/the dilated triple model.tex 
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In the proposed model, every triple in an RDF graph is supplemented with other triples from the same RDF graph. The triple and its supplements form what is called a \textit{dilated triple}.\footnote{The Oxford English dictionary provides two definitions for the word "dilate": "to expand" and "to speak or write at length". It will become clear through the remainder of this article that both definitions suffice to succinctly summarize the presented model.  \textbf{Definition Definition  1. A Dilated Triple} Triple  Given a set of triples $R$ and a triple $\tau \in R$, a dilation of $\tau$ is a set of triples $T_\tau \subset R$ such that $\tau \in T_\tau$.  The dilated form of $\tau \in R$ is $T_\tau$. Informally, $T_\tau$ servers to elaborate the meaning of $\tau$. Formally, $T_\tau$ is a graph that at minimum contains only $\tau$ and at maximum contains all triples in $R$. The set of all non-$\tau$ triples in $T_\tau$ (i.e.~$T_\tau \setminus \tau$) are called \textit{supplementary triples} as they serve to contextualize, or supplement, the meaning of $\tau$. Finally, it is worth noting that every supplemental triple in $T_\tau$ has an associated dilated form, so that $T_\tau$ can be considered a set of nested sets.\footnote{The set of all dilated triples forms a \textit{dilated graph} denoted $\mca{T} = \bigcup_{\tau \in R} \{ T_\tau \}$.} An instance of $\tau$, its subject $s$, predicate $p$, object $o$, and its dilated form $T_\tau$, are diagrammed in Figure \ref{fig:dilated-triple}.