Visualize a multi dimensional mathematical ring with TikZ

Question

I'm trying to visualize a multi dimensional ring with a torus. I draw a single ring like this:

\tikzset{
  dot node/.style={
    shape=circle,
    fill=white,
    draw,
    inner sep=+0pt,
    minimum size=+5mm
  },
  arc style/.style={
    o->|,
    shorten >=+-.5\pgflinewidth,
    shorten <=+-.5\pgflinewidth,
  }
}

\begin{tikzpicture}[
  thick,
  every pin edge/.style={<-},
  >=latex,
  declare function/.list={
    outerR=3.0;,
    innerR=2.4;,
    angleofNode(\a)=\a/12*360;}
  ]
  \node [draw,circle through=(0:outerR)] (c) {};

  \foreach \iAngle in {0,...,11}
    \node[dot node, label=center:\iAngle] at (c.{angleofNode(\iAngle)}) {};

  \foreach \sAngle/\eAngle/\tLabel in {-1/3/+4}
    \draw[arc style] ({angleofNode(\sAngle)}:innerR) arc[radius=innerR, start angle=angleofNode(\sAngle), end angle=angleofNode(\eAngle)]
         node at ({angleofNode(\sAngle+\eAngle)/2}:innerR-.5) {${\tLabel}$} ;
\end{tikzpicture}

How would you visualize a few of them ordered around a tours like this?

I've tried to combine my TikZ drawing with a torus example but I didn't come far. Does somebody have an idea how to do this and maybe being able to draw arrows?

Answer 1

You can use tikz-3dplot and the 3d library. To very first approximation one could do

\documentclass[tikz,border=3mm]{standalone}
\usepackage{tikz-3dplot}
\usetikzlibrary{3d}
\tikzset{
  dot node/.style={
    shape=circle,
    fill=white,
    draw,
    inner sep=+0pt,
    minimum size=+5mm
  },
  arc style/.style={
    o->|,
    shorten >=+-.5\pgflinewidth,
    shorten <=+-.5\pgflinewidth,
  }
}

\begin{document}

\tdplotsetmaincoords{60}{0}
\begin{tikzpicture}[tdplot_main_coords,
    declare function={outerR=3.0;
    innerR=2.4;
    angleofNode(\a)=\a/12*360;},
    pics/ring/.style={code={%
    \draw circle[radius=outerR];
    \foreach \iAngle in {0,...,11}
    {\node[dot node, label=center:\iAngle] at ({angleofNode(\iAngle)}:outerR) {};}
    }}]
   \begin{scope}[canvas is xy plane at z=0]
    \draw circle[radius=8];
   \end{scope}  
   \foreach \XX in {0,...,7}    
   {\tdplotsetrotatedcoords{0}{0}{22.5+\XX*45}
    \begin{scope}[tdplot_rotated_coords,canvas is xz plane at y=0]
    \pgfmathtruncatemacro{\itest}{sign(cos(22.5+\XX*45))}
    \path (8,0) node[dot node]{$x^\XX$} pic[transform shape,xscale=\itest]{ring};
    \ifnum\XX=1
     \foreach \sAngle/\eAngle/\tLabel in {-1/3/+4}
    \draw[arc style] (8,0) +({angleofNode(\sAngle)}:innerR) arc[radius=innerR, start angle=angleofNode(\sAngle), end angle=angleofNode(\eAngle)]
         node[transform shape] at ($(8,0)+({angleofNode(\sAngle+\eAngle)/2}:innerR-.5)$) {${\tLabel}$} ;
    \fi
    \end{scope}
    }
   \begin{scope}[canvas is xy plane at z=3.6]
    \pgflowlevelsynccm
    \draw[arc style] (3*45+22.5:8) arc[start angle=3*45+22.5,end angle=1*45+22.5,radius=8];
   \end{scope}  
\end{tikzpicture}
\end{document}