I need help to build a tikzpicture of the covering of S^1

Question

I want to make a picture the looks like the picture in this video, if you pause it at 42 seconds.

https://www.youtube.com/watch?v=vP7NAeeKjrw

If anyone could help, I would really appreciate it!

Thanks,

Answer 1

Without too much bling bling

\documentclass{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}
\usetikzlibrary{backgrounds}
\begin{document}
\begin{tikzpicture}
\begin{axis}[axis lines=none,unit vector ratio={1 1 1.2},width=10cm,
axis background/.style={
shade,top color=white,bottom color=black!80!gray,middle color=gray},view={10}{40},xmax=1.5,xmin=-1.5,ymax=1.5,ymin=-1.5]
\addplot3+[gray,domain=0:6*pi,samples y=0,samples=200,no marks, smooth](
  {cos(deg(x))},
  {sin(deg(x))},
  {1+x/5}
) node[circle,inner sep=1pt,ball color=gray,pos=0.40]{};
\addplot3+[white,domain=0:2*pi,samples y=0,samples=100,no marks, smooth](
  {cos(deg(x))},
  {sin(deg(x))},
  {0}
) node[circle,inner sep=1pt,ball color=gray,pos=0.2]{};
\end{axis}

\end{tikzpicture}
\end{document}