The sine function is a well known periodic function. Choose an argument factor, which doesn’t divide the period well, and add another sine function with a different factor, and you can see a complicated path around the origin.

This plot was originally made for my blog post on TikZ.de: http://tikz.de/periodisch/

See also the other polar sine plot examples there.

Polar plot of sine based function

Edit and compile if you like:

\documentclass[border=10pt]{standalone}
\usepackage{pgfplots}
\usepgfplotslibrary{polar}
\begin{document}
\begin{tikzpicture}
  \begin{polaraxis}[
      domain = -3600:3600,
      samples = 4000
    ]
    \addplot[blue!50!black] {1 - sin(50*x/49) - sin(8*x)};
  \end{polaraxis}
\end{tikzpicture}
\end{document}

Click to download: sine-polar.tex • sine-polar.pdf

See also: Original Source by Stefan Kottwitz

Note: The copyright belongs to the blog author and the blog. For the license, please see the linked original source blog.