How to draw a Klein Bottle

Question

My attempt:

\begin{tikzpicture}
  \node (n1) at (0,0)  {};
  \node (n2) at (0,1)  {};
  \node (n3) at (1,0)  {};
  \node (n4) at (1,1)  {};
  \foreach \from/\to in {}
    \draw (\from) -- (\to);
  \foreach \from/\to in {n1/n3,n4/n2}
    \draw (\from) edge [->] (\to);
     \foreach \from/\to in {n2/n3,n4/n1}
    \draw (\from) edge [->>] (\to);
     \foreach \from/\to in {n1/n4}
    \draw (\from) edge [->>>] (\to);
  \foreach \from/\to in {}
    \draw (\from) edge [bend right] (\to);
\end{tikzpicture}

doesn't work very well - this is what it outputs:

How can I get the arrow to not be at the tip, but rather in the middle of the edge?

The interwebz presents a much different view on what a klein bottle is... — Werner, Oct 16 '15 at 16:45
Sorry, should have been more specific: here, I'm looking at the delta-complex. If you're interested: https://www.youtube.com/watch?v=D6r66kd9XVs — man_in_green_shirt, Oct 16 '15 at 16:48

score 5 · Accepted Answer · edited Apr 13 '17 at 12:35

5

Two options:

With TikZ, using a style with a decoration:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{decorations.markings}

\def\Singlearrow{{\arrow[scale=1.5,xshift={0.5pt+2.25\pgflinewidth}]{>}}}
\def\Doublearrow{{\arrow[scale=1.5,xshift=1.35pt +2.47\pgflinewidth]{>>}}}
\def\Triplearrow{{\arrow[scale=1.5,xshift=1.75pt +2.47\pgflinewidth]{>>>}}}

\begin{document}

\begin{tikzpicture}[
line cap=round,
marr/.style={
  decoration={
    markings,
    mark=at position 0.5 with {#1}
    },
  postaction={decorate}
}
] 
\path
  coordinate (n1) at (0,0)
  coordinate (n2) at (0,2)
  coordinate (n3) at (2,0)
  coordinate (n4) at (2,2);
\foreach \from/\to in {n1/n3,n4/n2}
    \draw[marr=\Singlearrow] (\from) -- (\to);
\foreach \from/\to in {n1/n2,n3/n4}
    \draw[marr=\Doublearrow] (\from) -- (\to);
\draw[marr=\Triplearrow] (n1) -- (n4);
\draw[->]
  (0.35,1.3) arc (220:-40:8pt);
\draw[->]
  (1.7,0.35) arc (-40:220:8pt);
\end{tikzpicture}

\end{document}

Some remarks

I used \coordinates instead of the original \nodes, so the lines will meet.
Changing the line cap to round gives better result at the verticae.
By default a decoration places the arrow tip, not its center, at the specified position. This produces ugly results (specially noticeable for composed arrows). Addressing this issue, there are some posts such as Position arrow decoration by center, not by tip, but the automatic answers there are, unfortunately, outdated. I used instead a manual approach similar to the answer by Kpym.

Using tikz-cd:

\documentclass{article}
\usepackage{tikz-cd}
\usetikzlibrary{decorations.markings,arrows.meta}
\usetikzlibrary{backgrounds}

\begin{document}

\begin{tikzpicture}
\end{tikzpicture}

\begin{tikzcd}[
sep=2cm,
cells={shape=coordinate},
]
  \ar[draw=none,r, "{\tikz\node[rotate=90,inner sep=0pt] {\tikz\draw[->](0,0) ;};  }" description]
  \ar[dash,r] 
& 
{} 
\\
  \ar[draw=none,r, "{\tikz\node[rotate=270,inner sep=0pt] {\tikz\draw[->](0,0) ;};  }" description] 
  \ar[draw=none,u, "{\tikz\node[inner sep=0pt] {\tikz\draw[->>](0,0) ;};  }" description] 
  \ar[draw=none,ur, "{\tikz\node[rotate=-45,inner sep=0pt] {\tikz\draw[->>>](0,0) ;};  }" description] 
  \ar[dash,r] 
  \ar[dash,u] 
  \ar[dash,ur] 
& 
{}
  \ar[draw=none,u, "{\tikz\node[inner sep=0pt] {\tikz\draw[->>](0,0) ;};  }" description]
    \ar[dash,u]  
\end{tikzcd}

\end{document}

edited Apr 13 '17 at 12:35

Community

1

answered Oct 16 '15 at 16:20

Gonzalo Medina

505,128

Thanks! Also, is there a way to 'shrink' the nodes so that the edges meet at the common vertices? – man_in_green_shirt Oct 16 '15 at 16:22
1

@man_in_green_shirt Sure; see my updated answer. – Gonzalo Medina Oct 16 '15 at 16:24
1

Is there an easy way to center the arrow heads, so that the middle of the arrow heads are at the middle of the line and not the tip? – Heiko Oberdiek Oct 16 '15 at 16:28
1

@HeikoOberdiek This is possible, yes, but not as easy as I initially thought. I have a solution and will post it in some minutes. – Gonzalo Medina Oct 16 '15 at 16:30
Also, the arrows are a bit small like this. How can I make them bigger? – man_in_green_shirt Oct 16 '15 at 16:37
1

@man_in_green_shirt Please, make all requirements from the beginning. It's not nice to be editing answers every time you came up with a new request. – Gonzalo Medina Oct 16 '15 at 16:39
1

@man_in_green_shirt Please see my updated answer. Change the scale value according to your needs. – Gonzalo Medina Oct 16 '15 at 16:42
You're right, sorry - will do so in the future. Thank again! – man_in_green_shirt Oct 16 '15 at 16:43
@HeikoOberdiek I finally decided to use a "manual" solution that I think is enough for this case. Perhaps later I'll try to implement an automatic one, but I can't promise anything :) – Gonzalo Medina Oct 16 '15 at 16:49
1

@man_in_green_shirt You're welcome! I added now the missing arrow (I had only drawn initially the quotient space) and the arcs as in the linked video. – Gonzalo Medina Oct 16 '15 at 17:09
@man_in_green_shirt Answer updated again; this time I present an option using tikz-cd. – Gonzalo Medina Oct 16 '15 at 17:36

How to draw a Klein Bottle

1 Answers1

Some remarks

Linked