Plane to Torus

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\documentclass[tikz]{standalone}
\usetikzlibrary{decorations.markings,arrows}
\begin{document}
\begin{tikzpicture}
  % lattice
  \draw[->,ultra thick] (-2,0) -- (2,0);
  \draw[->,ultra thick] (0,-2) -- (0,2);
  \draw (-2,-2) grid (2,2);
  \node [draw,ultra thick,rectangle,minimum width=1cm,minimum height=1cm,label=below:$2 \pi$,label=right:$2 \pi$] at (0.5,0.5) {};
  % square
  \begin{scope}[xshift=4cm,yshift=1cm]
    \draw[postaction={decorate},decoration={
          markings,
          mark=at position .145 with {\arrow{latex}},
          mark=at position .375 with {\arrow{latex}},
          mark=at position .635 with {\arrowreversed{latex}},
          mark=at position .875 with {\arrowreversed{latex}},
        }
    ]
    (0,-1) -- +(2,0) -- +(2,2) -- +(0,2) -- cycle;
  \end{scope}
  % bracelet
  \begin{scope}[xshift=9.5cm,yshift=1cm]
    \draw[postaction={decorate},decoration={
          markings,
          mark=at position .5 with {\arrow{latex}}
        }
    ]
    (0,.25) -- ++(2,0);
    \draw[postaction={decorate},decoration={
          markings,
          mark=at position .5 with {\arrow{latex}}
        }
    ]
    (0,-.25) -- ++(2,0);
    \draw[postaction={decorate},decoration={
 
 
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