# Need to plot a lattice and the topological building of a torus!

as told in the title I need to plot a lattice on the plane (not the standard one with basis (1,0), (0,1) ) and I'd like to show up how to build a torus starting from its topological square... If anyone could help it would be really appreciated!

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The lattice is easy enough, but what exactly do you want to show for the quotient to the torus? Which stages do you want? – Loop Space Apr 11 '11 at 18:06

## 1 Answer

As a topologist, I can attest to the necessity of drawing tori frequently, and also the steps in their construction. So I think that this will be a useful resource. Having said that, the actual tori in the following are a little "hackish" and have been merely adjusted to look okay rather than being perfect renderings.

Here's the code. As I commented, the lattice is easy enough. The first stages of the torus are also fairly straightforward. The only tricky bit is getting the partial torus and the torus to look reasonable. They could probably be improved. The code could also be condensed a little by using styles for the arrowed lines.

(I've just spotted that in the second stage, the right-hand edge is drawn the wrong way around so the arrows point in the wrong direction. This is easily corrected.)

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{decorations.markings,arrows}
\begin{document}
\begin{tikzpicture}
\draw[->,ultra thick,shorten >=-8pt] (-3,0) -- (3,0);
\draw[->,ultra thick,shorten >=-8pt] (0,-3) -- (0,3);
\begin{scope}
\clip (-3,-3) rectangle (3,3);
\begin{scope}[xslant=1]
\draw (-6,-6) grid (6,6);
\draw[ultra thick] (0,0) rectangle (1,1);
\end{scope}
\end{scope}
\end{tikzpicture}

\begin{tikzpicture}
\draw[postaction={decorate},decoration={
markings,
mark=at position .145 with {\arrow{latex}},
mark=at position .375 with {\arrow{latex}},
mark=at position .395 with {\arrow{latex}},
mark=at position .635 with {\arrowreversed{latex}},
mark=at position .875 with {\arrowreversed{latex}},
mark=at position .895 with {\arrowreversed{latex}}
}
]
(0,-1) -- +(2,0) -- +(2,2) -- +(0,2) -- cycle;

\begin{scope}[xshift=3cm]
\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={
markings,
mark=at position .5 with {\arrow{latex}},
mark=at position .55 with {\arrow{latex}}
}
] (0,-.25) to[out=-120,in=0] (-.35,-.75) to[out=180,in=180] (-.35,.75) to[out=0,in=120] (0,.25);
\draw (2,.25) to[out=120,in=0] (1.65,.75) -- (-.35,.75) (-.35,-.75) --  (1.65,-.75) to[out=0,in=-120] (2,-.25);
\begin{scope}
\clip (0,.25) rectangle (2,-.25);
\draw[postaction={decorate},decoration={
markings,
mark=at position .5 with {\arrow{latex}},
mark=at position .58 with {\arrow{latex}}
}
]  (1.65,.75) to[out=180,in=180] (1.65,-.75);
\end{scope}
\begin{scope}[xshift=3cm]
\draw[postaction={decorate},decoration={
markings,
mark=at position .5 with {\arrow{latex}},
}
]  (0,0) -- (2,0);
\draw[postaction={decorate},decoration={
markings,
mark=at position .5 with {\arrow{latex}},
mark=at position .55 with {\arrow{latex}},
}
] (0,0) arc[start angle=0,delta angle=-360,x radius=.35,y radius=.75];
\draw (2,0) arc[start angle=0,delta angle=-90,x radius=.35,y radius=.75] -- ++(-2,0);
\draw (2,0) arc[start angle=0,delta angle=90,x radius=.35,y radius=.75] -- ++(-2,0);
\end{scope}
\end{scope}
\begin{scope}[yshift=-3cm]
\draw[postaction={decorate},decoration={
markings,
mark=at position .5 with {\arrow{latex}},
}
]  (0,0) -- (3,0);
\draw[postaction={decorate},decoration={
markings,
mark=at position .5 with {\arrow{latex}},
mark=at position .6 with {\arrow{latex}},
}
] (0,0) arc[start angle=0,delta angle=-360,x radius=.15,y radius=.35];
\draw (3,0) arc[start angle=0,delta angle=-90,x radius=.15,y radius=.35] -- ++(-3,0);
\draw (3,0) arc[start angle=0,delta angle=90,x radius=.15,y radius=.35] -- ++(-3,0);
\begin{scope}[xshift=4cm]
\draw[postaction={decorate},decoration={
markings,
mark=at position .5 with {\arrow{latex}},
mark=at position .6 with {\arrow{latex}},
}
] (1,0) arc[start angle=0,delta angle=-360,x radius=.15,y radius=.35];
\draw (1,0) ++(-.15,-.35) .. controls +(170:1) and +(-90:.5) .. ++(-1.5,1) .. controls +(90:.5) and +(180:1) .. ++(2,1) .. controls +(0:1) and +(90:.5) .. ++(2,-1) .. controls +(-90:.5) and +(10:1) .. ++(-1.5,-1) coordinate (a);
\draw[postaction={decorate},decoration={
markings,
mark=at position .5 with {\arrow{latex}},
mark=at position .6 with {\arrow{latex}},
}
] (a) ++(-.15,.35) arc[start angle=0,delta angle=-360,x radius=-.15,y radius=.35];
\draw (1,0) ++(-.15,.35) .. controls +(170:.5) and +(-60:.25) .. ++(-.9,.5) coordinate (b);
\draw (a) ++(0,.7) .. controls +(10:.5) and +(240:.25) .. ++(.9,.5) coordinate (c);
\begin{scope}
\clip (1,0) ++(-.15,.35) .. controls +(170:.5) and +(-60:.25) .. ++(-.9,.5) -- ++(0,2) -| (c)  .. controls +(240:.25) and +(10:.5) .. ++(-.9,-.5);
\draw (1,0) ++(-.15,-.35) ++(0,-.7) .. controls +(170:1) and +(-90:.5) .. ++(-1.5,.8) .. controls +(90:.5) and +(180:1) .. ++(2,1.2) .. controls +(0:1) and +(90:.5) .. ++(2,-1.2) .. controls +(-90:.5) and +(10:1) .. ++(-1.5,-.8);
\draw[postaction={decorate},decoration={
markings,
mark=at position .5 with {\arrow{latex}},
}
] (1,0) ++(-.15,-.35) ++(0,-.8) .. controls +(170:1) and +(-90:.5) .. ++(-1.5,.8) .. controls +(90:.5) and +(180:1.2) .. ++(2,1.5) .. controls +(0:1.2) and +(90:.5) .. ++(2,-1.5) .. controls +(-90:.5) and +(10:1) .. ++(-1.5,-.8);
\end{scope}
\begin{scope}
\clip  (a) ++(-.15,.35) arc[start angle=0,delta angle=-360,x radius=-.15,y radius=.35];
\draw (a) ++(-.15,.35) .. controls +(10:1) and +(-90:.5) .. ++(1.5,.8);
\end{scope}
\begin{scope}
\clip (1,0) arc[start angle=0,delta angle=-360,x radius=.15,y radius=.35];
\draw (1,0) .. controls +(170:1) and +(-90:.5) .. ++(-1.5,.8);
\end{scope}
\begin{scope}[xshift=5cm]
\draw[postaction={decorate},decoration={
markings,
mark=at position .5 with {\arrow{latex}},
mark=at position .6 with {\arrow{latex}},
}
] (1.5,.35) arc[start angle=90,end angle=-90,y radius=.35,x radius=.1];
\draw (1.5,-.35) .. controls +(180:1) and +(-90:.65) .. ++(-2,1) .. controls +(90:.65) and +(180:1) .. ++(2,1) .. controls +(0:1) and +(90:.65) .. ++(2,-1) .. controls +(-90:.65) and +(0:1) .. ++(-2,-1); \draw (1.5,.35) .. controls +(180:.5) and +(-50:.25) .. ++(-1.3,.35) coordinate (b);
\draw (1.5,.35) .. controls +(0:.5) and +(230:.25) .. ++(1.3,.35) coordinate (c);
\begin{scope}
\clip (1.5,.35) .. controls +(180:.5) and +(-50:.25) .. ++(-1.3,.35) -- ++(0,2) -| (c)  .. controls +(230:.25) and +(0:.5) .. ++(-1.3,-.35);
\draw (1.5,-.35) ++(0,-.7) .. controls +(180:1) and +(-90:.65) .. ++(-1.5,1) .. controls +(90:.65) and +(180:1) .. ++(1.5,1) .. controls +(0:1) and +(90:.65) .. ++(1.5,-1) .. controls +(-90:.65) and +(0:1) .. ++(-1.5,-1);
\draw[postaction={decorate},decoration={
markings,
mark=at position .5 with {\arrow{latex}},
}
] (1.5,-.35) ++(0,-.6) .. controls +(180:1) and +(-90:.65) .. ++(-1.5,1) .. controls +(90:.65) and +(180:1) .. ++(1.5,1) .. controls +(0:1) and +(90:.65) .. ++(1.5,-1) .. controls +(-90:.65) and +(0:1) .. ++(-1.5,-1);
\end{scope}
\end{scope}
\end{scope}
\end{scope}
\end{tikzpicture}
\end{document}

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Very nice! I may be misunderstanding the arrows, but shouldn't the double arrow in the very last image (bottom right) be pointing up, like in the penultimate image (bottom middle)? – Jake May 14 '11 at 21:17
Great! One more small thing: the arrows on the left and right side of the square are not at the same height. – Caramdir May 14 '11 at 21:23
@Jake: Yup. Guilty as charged. – Loop Space May 15 '11 at 20:37
@Caramdir: The arrow placement is actually very tricky. I would like (for the single headed arrows) to place the centre of the arrow at the centre of the line. But I would also want the point of the arrow to lie on the line. So the arrow needs to be anchored at (desired position + arrow length). But the first is a dimension and the second a length. Presumably it's possible to come up with a more robust solution than "it looks vaguely right" but I wasn't going to bother until I'd seen if anyone was interested in this answer. With 7 votes so far, I'm tempted to fix it. – Loop Space May 15 '11 at 20:40
@Andrew: If you should come up with a more robust solution to arrow placement, please add an answer to tex.stackexchange.com/questions/4986/…. – Caramdir May 16 '11 at 20:13