\documentclass{minimal}
\usepackage{tikz}


\begin{document}

Pavage Hexagonal

Le motif de base

\bigskip
\begin{tikzpicture}
	\draw[very thick,red] (0:0) -- ++ (60:1cm)-- ++ (0:1cm)--++(-60:1cm)--++(0:1cm);
	\draw[very thick, blue] (0:0)--  ++ (-60:1cm) ++ (0:1cm) --++(60:1cm);
\end{tikzpicture}

\bigskip
Le pavage construit à partir de ce motif

\bigskip
	\begin{tikzpicture}
		\foreach \i in {0,3,...,6}{
 			\foreach \j in {0,...,3}{
				\begin{scope}[xshift=\i cm,yshift={\j*1.73052 cm}]
					\draw[very thick,red] (0:0) -- ++ (60:1cm)-- ++ (0:1cm)--++(-60:1cm)--++(0:1cm);
					\draw[very thick,blue] (0:0)--  ++ (-60:1cm) ++ (0:1cm) --++(60:1cm);
				\end{scope}
				}
			}
	\end{tikzpicture}


\newpage

Les racines cubiques de l'unit\'e et leurs images par les translations $t_1$ et $t_2$.

	\bigskip
	\begin{tikzpicture}[scale=0.8]
		\foreach \i in {-1,...,2}{
 			\begin{scope}[yshift={\i*1.73052 cm}]
	 			\foreach \j in {-2,...,3}{
					\begin{scope}[xshift={\j*1.5 cm},yshift={\j*0.86603 cm}]		
						\draw[->,thick] (0:0)--(0:0.95cm);
						\draw[->,thick] (0:0)--(120:0.95cm);
						\draw[->,thick] (0:0)--(240:0.95cm);
					\end{scope}
					}
			\end{scope}
			}
			\draw[->,very thick,red] (0:0)--(0:0.95cm);
			\draw[->,very thick,red] (0:0)--(120:0.95cm);
			\draw[->,very thick,red] (0:0)--(240:0.95cm);
			\draw[->,very thick,blue] (0,0)--(0,1.73)node[right,midway]{$\vec{t_1}$};
			\draw[->,very thick,blue] (0,0)--(1.5,0.866)node[above,midway]{$\vec{t_2}$};
	\end{tikzpicture}

\bigskip
Rep\'erage des points sur le pavage

\bigskip
\newcommand{\pointhexaA}[3]{
 	\fill[fill=red!20,draw=red,thick] ({#1*1.5},{#1*0.86603+#2*1.73052}) circle (1mm)node[above right]{$#3$};
	}
\newcommand{\pointhexaB}[3]{
 	\fill[fill=blue!20,draw=blue,thick] ({#1*1.5+1},{#1*0.86603+#2*1.73052}) circle (0.8mm)node[right]{$#3$};
	}

\bigskip
	\begin{tikzpicture}
		\clip (-5.5,-4) rectangle (5,4);
		\foreach \i in {-3,...,4}{
 			\begin{scope}[yshift={\i*1.73052 cm}]
	 			\foreach \j in {-4,...,3}{
					\begin{scope}[xshift={\j*1.5 cm},yshift={\j*0.86603 cm}]		
						\draw[thick] (0:0)--(0:1cm);
						\draw[thick] (0:0)--(120:1cm);
						\draw[thick] (0:0)--(240:1cm);
					\end{scope}
					}
			\end{scope}
			}
			\pointhexaA{0}{0}{A'}
			\pointhexaA{1}{0}{B'}
			\pointhexaA{1}{1}{C'}
			\pointhexaA{0}{1}{D'}
			\pointhexaA{-1}{1}{E'}
			\pointhexaA{-1}{0}{F'}
			\pointhexaA{-1}{-1}{G'}
			\pointhexaA{0}{-1}{H'}
			\pointhexaA{1}{-1}{I'}
			%
			\pointhexaB{0}{0}{A}
			\pointhexaB{1}{0}{B}
			\pointhexaB{1}{1}{C}
			\pointhexaB{0}{1}{D}
			\pointhexaB{-1}{1}{E}
			\pointhexaB{-1}{0}{F}
			\pointhexaB{-1}{-1}{G}
			\pointhexaB{0}{-1}{H}
			\pointhexaB{1}{-1}{I}
	\end{tikzpicture}
\end{document}

