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Torus

physicsstring theorytopologygeometryfundamental domainsymmetriesperiodicitytikz

Three-dimensional visualization of a torus with major radius R and minor radius r. The torus is a fundamental geometric object in string theory, where strings can wrap around its two independent cycles. Its topology allows for periodic boundary conditions in two directions, which is relevant when studying compactification, modular invariance, and dualities in string theory.


Torus

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torus.tex (34 lines)

\documentclass{standalone}

\usepackage{pgfplots}
\pgfplotsset{compat=newest}

\begin{document}
\begin{tikzpicture}
  \begin{axis}[
      axis equal image,
      axis lines=middle,
      xmax=18,zmax=5,
      ticks=none,
      clip bounding box=upper bound,
      colormap/blackwhite
    ]

    \addplot3[domain=0:360,y domain=0:320, samples=50,surf,z buffer=sort]
    ({(12 + 3 * cos(x)) * cos(y)} ,
    {(12 + 3 * cos(x)) * sin(y)},
    {3 * sin(x)});
    % use axis coordinate system to draw the radii
    \draw [thick,blue] (axis cs: 0,0,0) -- (axis cs: 12,0,0) node [midway,above=-2] {$R$};
    \draw [thick,red] (axis cs: 12,-0.2,0) -- (axis cs: 12,3.7,0) node [midway,below right=-3] {$r$};

    % use axis coordinate system to draw fake x, y and z axes
    \draw [-latex] (axis cs: 0,0,0) -- node [pos=0.9, xshift=0.5em]{$z$}(axis cs: 0,0,10);
    \draw [-latex] (axis cs: 0,-15,0) --
    node [pos=0.9, xshift=-1em, yshift=0.5em]{$y$}(axis cs: 0,-20,0);
    \draw (axis cs: 0,0,0) -- (axis cs: 0,9,0);
    \draw (axis cs: 0,0,0) -- (axis cs: -9,0,0);
  \end{axis}
\end{tikzpicture}
\end{document}