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\documentclass[tikz,11pt]{standalone}
\usepackage{tikz}
\pgfmathsetmacro{\gridLineWidth}{0.6pt}
\pgfmathsetmacro{\polytopeLineWidth}{1.2pt}
\pgfmathsetmacro{\dotSize}{0.5mm}
\definecolor{fillColor}{rgb}{0.8,0.8,0.8}
\pgfmathsetmacro{\opacity}{0.7}
\begin{document}
\begin{tikzpicture}
% (1, 1, 1)
\begin{scope}[yshift=5.2cm, scale=0.7]
\node at (-5, 0) {$\mathbf{q} = (1, 1, 1)$};
\begin{scope}
\node at (0, -3.3) {$\nabla = \Delta^{\!*}$};
\clip (0,0) circle (2.7);
\path [draw, fill opacity=\opacity, fill=fillColor, line width=\polytopeLineWidth] (1, 0) --(0, 1) --(-1, -1) --cycle;
% \draw[step=1, dotted, line width=\gridLineWidth] (-10, -10) grid (10, 10);
\path [draw, line width=\gridLineWidth] (-10, 0) --(10, 0) (0, -10) --(0, 10);
\foreach \i in {-2,...,2}
\foreach \j in {-2,...,2}{
\node at (\i, \j) [circle, fill, inner sep=\dotSize] {};
};
\end{scope}
\begin{scope}[xshift=6.5cm]
\node at (0, -3.3) {$\nabla^* = \Delta$};
\clip (0,0) circle (2.7);
\begin{scope}
\path [draw, fill opacity=\opacity, fill=fillColor, line width=\polytopeLineWidth] (-1, 2) --(2, -1) --(-1, -1) --cycle;
% \draw[step=1, dotted, line width=\gridLineWidth] (-10, -10) grid (10, 10);
\path [draw, line width=\gridLineWidth] (-10, 0) --(10, 0) (0, -10) --(0, 10);
\foreach \i in {-2,...,2}
\foreach \j in {-2,...,2}{
\node at (\i, \j) [circle, fill, inner sep=\dotSize] {};
};
\end{scope}
\end{scope}
\end{scope}
% (1, 1, 2)
\begin{scope}[yshift=0cm, scale=0.7]
\node at (-5, 0) {$\mathbf{q} = (1, 1, 2)$};
\begin{scope}
\node at (0, -3.3) {$\nabla = \Delta^{\!*}$};
\clip (0,0) circle (2.7);
\begin{scope}[scale=1.4142] % sqrt(2)
\path [draw, fill opacity=\opacity, fill=fillColor, line width=\polytopeLineWidth] (1, 0) --(0, 1) --(-1/2, -1/2) --cycle;
% \draw[step=1, dotted, line width=\gridLineWidth] (-10, -10) grid (10, 10);
\path [draw, line width=\gridLineWidth] (-10, 0) --(10, 0) (0, -10) --(0, 10);
\foreach \i in {-2,...,2}
\foreach \j in {-2,...,2}{
\node at (\i / 2 - \j / 2, \i / 2 + \j / 2) [circle, fill, inner sep=\dotSize] {};
};
\end{scope}
\end{scope}
\begin{scope}[xshift=6.5cm]
\node at (0, -3.3) {$\nabla^* = \Delta$};
\clip (0,0) circle (2.7);
\begin{scope}[scale=0.707]
\path [draw, fill opacity=\opacity, fill=fillColor, line width=\polytopeLineWidth] (-1, 3) --(3, -1) --(-1, -1) --cycle;
% \draw[step=1, dotted, line width=\gridLineWidth] (-10, -10) grid (10, 10);
\path [draw, line width=\gridLineWidth] (-10, 0) --(10, 0) (0, -10) --(0, 10);
\foreach \i in {-2,...,2}
\foreach \j in {-2,...,2}{
\node at (\i - \j, \i + \j) [circle, fill, inner sep=\dotSize] {};
};
\end{scope}
\end{scope}
\end{scope}
% (1, 2, 3)
\begin{scope}[yshift=-5.2cm, scale=0.7]
\node at (-5, 0) {$\mathbf{q} = (1, 2, 3)$};
\begin{scope}
\node at (0, -3.3) {$\nabla = \Delta^{\!*}$};
\clip (0,0) circle (2.7);
\begin{scope}[scale=1.732] % sqrt(3)
\path [draw, fill opacity=\opacity, fill=fillColor, line width=\polytopeLineWidth] (1, 0) --(0, 1) --(-1/3, -2/3) --cycle;
% \draw[step=1, dotted, line width=\gridLineWidth] (-10, -10) grid (10, 10);
\path [draw, line width=\gridLineWidth] (-10, 0) --(10, 0) (0, -10) --(0, 10);
\foreach \i in {-2,...,2}
\foreach \j in {-4,...,4}{
\node at (\i / 3 - \j / 3, \i * 2 / 3 + \j / 3) [circle, fill, inner sep=\dotSize] {};
};
\end{scope}
\end{scope}
\begin{scope}[xshift=6.5cm]
\node at (0, -3.3) {$\nabla^* = \Delta$};
\clip (0,0) circle (2.7);
\begin{scope}[scale=0.577]
\begin{scope}[xshift=-2cm]
\path [draw, fill opacity=\opacity, fill=fillColor, line width=\polytopeLineWidth] (5, -1) --(-1, 2) --(-1, -1) --cycle;
% \draw[step=1, dotted, line width=\gridLineWidth] (-10, -10) grid (10, 10);
\path [draw, line width=\gridLineWidth] (-10, 0) --(10, 0) (0, -10) --(0, 10);
\foreach \i in {-1,...,3}
\foreach \j in {-4,...,4}{
\node at (3 * \i + \j, \j) [circle, fill, inner sep=\dotSize] {};
};
\end{scope}
\end{scope}
\end{scope}
\end{scope}
\end{tikzpicture}
\end{document}
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