The sine function is a well known periodic function. Choose an argument factor, which doesn’t divide the period well, 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This plot was originally made for my blog post on TikZ.de: http://tikz.de/periodisch/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,colormaps} \begin{document} \begin{tikzpicture} \begin{polaraxis}[ domain = -14400:14400, samples = 3000, colormap/cool, hide axis ] \addplot[no markers, mesh, opacity=0.5] {1-sin(40*x/39}; \end{polaraxis} \end{tikzpicture} \end{document}