How to draw a “halftone” spiral made of circles in LaTeX?

Question

I would like to use LaTeX to draw a spiral like the one below, but it's way out of my league. Any help would be much appreciated.

NEW EDIT: (now with a better 3d look!) after playing a bit with Jasper Habicht's solution, this was the best I could get up to now. I'm pretty happy. Thank you!

My customized version of Jasper Habicht's solution! (Give him a “up” vote!)

% Credits to Jasper Habicht (See below)
\documentclass[tikz]{standalone}
\usetikzlibrary{calc}

\begin{document}
  
\definecolor{mygreencolor}{RGB}{133,196,96}  

\begin{tikzpicture}

\pgfmathsetmacro\maxradius{.3}
\pgfmathsetmacro\inner{10}
\pgfmathsetmacro\outer{40}
\pgfmathsetmacro\range{\outer-\inner}

\foreach \stepy in {\inner, ..., \outer}
    \pgfmathsetmacro\stepstart{60/\stepy}
    \pgfmathsetmacro\steplast{360-\stepstart}
    \pgfmathsetmacro\stepcount{floor(360/\stepstart)} 
    \foreach \stepx in {0, \stepstart, ..., \steplast}
        \pgfmathsetmacro\stepsingle{floor(\stepx/\stepstart)}
        \pgfmathsetmacro\stepradius{(\maxradius/2)*sin(deg(\stepsingle*pi/(\stepcount/2) - pi - (\stepy/5))) + (\maxradius/2)+0.13}
        \pgfmathsetmacro\mybase{90*(\stepy-\inner)/\range)}
        \pgfmathsetmacro\mycoef{((1-cos(\mybase)+sin(\mybase))/2}
        \pgfmathsetmacro\mystepy{\inner+\range*\mycoef}
        \fill[mygreencolor] ({-(\stepx+.25*\stepcount)}:-\mystepy mm) circle (\stepradius mm);

\end{tikzpicture}

\end{document}

Answer 1

A different approach that also needs a lot of fine tuning.

What does it do? It draws several rings consisting of a varying number of small circles and increases the number of small circles as the radii of the rings grow in such a way that the distribution of the small circles is eventually relatively even. The radii of the small circles are calculated using a sine-wave function that commutes between a radius of zero and 0.5 mm. The sine wave also gets a shift for each of the rings in order to produce the spiral effect.

\documentclass[tikz]{standalone}
\usetikzlibrary{calc}

\begin{document}

\begin{tikzpicture}

\pgfmathsetmacro\maxradius{.5}

\foreach \stepy in {10, ..., 30}
    \pgfmathsetmacro\stepstart{60/\stepy}
    \pgfmathsetmacro\steplast{360-\stepstart}
    \pgfmathsetmacro\stepcount{floor(360/\stepstart)}
    \foreach \stepx in {0, \stepstart, ..., \steplast}
        \pgfmathsetmacro\stepsingle{floor(\stepx/\stepstart)}
        \pgfmathsetmacro\stepradius{(\maxradius/2)*cos(deg(\stepsingle*pi/(\stepcount/2) - pi - (\stepy/5))) + (\maxradius/2)}
        \fill[green] (\stepx:\stepy mm) circle (\stepradius mm);

\end{tikzpicture}

\end{document}

---

Edited to give it a more of a 3D effect:

\documentclass[tikz]{standalone}
\usetikzlibrary{calc}

\begin{document}

\begin{tikzpicture}

\pgfmathsetmacro\maxradius{.5}

\foreach \stepy in {10, ..., 30}
    \pgfmathsetmacro\stepstart{60/\stepy}
    \pgfmathsetmacro\steplast{360-\stepstart}
    \pgfmathsetmacro\stepcount{floor(360/\stepstart)}
    \foreach \stepx in {0, \stepstart, ..., \steplast}
        \pgfmathsetmacro\stepsingle{floor(\stepx/\stepstart)}
        \pgfmathsetmacro\stepradius{(\maxradius/2)*cos(deg(\stepsingle*pi/(\stepcount/2) - pi - (\stepy/5))) + (\maxradius/2)}
        \fill[green] ({\stepx+.25*\stepcount}:\stepy mm) circle (\stepradius mm);

\end{tikzpicture}

\end{document}

In the original picture, the inner circles are smaller and closer together. I guess, it is not too difficult to achieve this effect.

Answer 2

Here is a brief example that will probably require several adjustments. I add a commented spiral that I use as a reference.

\documentclass[tikz, border=2mm]{standalone}

\begin{document}
\begin{tikzpicture}[line cap=round]
  \foreach\a in {0,2,...,359}       % angle to put each cricle (polar coordinates)
  {
    \pgfmathsetmacro\s{2.5+\a/90}   % theoretical point of the spiral (radius, polar coordinates)
    \foreach\r in {2.5,2.6,...,6.5} % radius to put each circle (polar coordinates)
    {
      \pgfmathsetmacro\w{0.1+0.5/(1+abs(\s-\r))} % width of the circles
      \fill[green] (\a:\r) circle (\w mm);
    }
    % Reference spiral:
    %\fill (\a:\s) circle (0.05);
  }
\end{tikzpicture}
\end{document}