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This plot was originally made for my blog post on TikZ.de: http://tikz.de/periodisch/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

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}