How can I add stippling to elements of my tikz diagrams?

Question

Okay so I think everyone knows how to draw a sphere in Tikz

\tikz{
\fill [black] (-1,-1) rectangle (1,1);
\shade [ball color=white] (0,0) circle [radius=1cm];
}

But how can I add to it this effect that can be found in Penrose's drawings? (those dots around)

And in general how can one create that effect in Tikz for any shape? (Here are a couple of other images for your viewing pleasure)

Answer 1

This takes ages. Also some care is needed with the direction of the path and best effects are achieved by selectively applying the stippling to parts of the path. So, a real pain, basically. But (as with many things) it looks quite good from a distance.

\documentclass[tikz,border=5]{standalone}
\usetikzlibrary{decorations}
\pgfkeys{/pgf/decoration/.cd,
  stipple density/.store in=\pgfstippledensity,
  stipple density=.1,
  stipple scaling function/.store in=\pgfstipplescalingfunction,
  stipple scaling function=sin(\pgfstipplex*180)*0.875+0.125,
  stipple radius/.store in=\pgfstippleradius,
  stipple radius=0.25pt
}
\pgfdeclaredecoration{stipple}{draw}{
\state{draw}[width=\pgfdecorationsegmentlength]{%
  \pgfmathparse{\pgfdecoratedcompleteddistance/\pgfdecoratedpathlength}%
  \let\pgfstipplex=\pgfmathresult%
  \pgfmathparse{int(\pgfstippledensity*100)}%
  \let\pgfstipplen=\pgfmathresult%
  \pgfmathloop%
  \ifnum\pgfmathcounter<\pgfmathresult\relax%
    \pgfpathcircle{%
      \pgfpoint{(rnd)*\pgfdecorationsegmentlength}%
        {(\pgfstipplescalingfunction)*(rnd^4)*\pgfdecorationsegmentamplitude+\pgfstippleradius}}% 
    {\pgfstippleradius}%
  \repeatpgfmathloop%
}
}

\tikzset{stipple/.style={
  decoration={stipple, segment length=2pt, #1},
  decorate,
  fill
}}
\begin{document}
\begin{tikzpicture}
\draw [postaction={stipple={amplitude=0.125cm}}] 
  (0,0) [rotate=45]  circle [radius=1];
\path [postaction={stipple={amplitude=0.25cm, stipple density=.35}},
  postaction={stipple={amplitude=0.35cm, stipple density=.15}}]
  (135:1) arc (135:315:1);
\end{tikzpicture}

\end{document}

And this takes even longer:

\documentclass[tikz,border=5]{standalone}
\usetikzlibrary{decorations}
\pgfkeys{/pgf/decoration/.cd,
  stipple density/.store in=\pgfstippledensity,
  stipple density=(sin(\pgfstipplex*180)*0.25+0.1),
  stipple amplitude/.store in=\pgfstippleamplitude,
  stipple amplitude=(rnd^3)*\pgfstippley*(sin(\pgfstipplex*180)^2+0.05),
  stipple radius/.store in=\pgfstippleradius,
  stipple radius=0.25pt
}
\pgfdeclaredecoration{stipple}{draw}{
\state{draw}[width=\pgfdecorationsegmentlength]{%
  \pgfmathparse{\pgfdecoratedcompleteddistance/\pgfdecoratedpathlength}%
  \let\pgfstipplex=\pgfmathresult%
  \let\pgfstippley=\pgfdecorationsegmentamplitude%
  \pgfmathparse{int(abs((\pgfstippledensity)*100))}%
  \let\pgfstipplen=\pgfmathresult%
  \pgfmathloop%
  \ifnum\pgfmathcounter<\pgfmathresult\relax%
    \pgfpathcircle{%
      \pgfpoint{(rnd-0.5)*\pgfdecorationsegmentlength}%
        {(\pgfstippleamplitude)+\pgfstippleradius}}% 
    {\pgfstippleradius}%
  \repeatpgfmathloop%
}
}

\tikzset{stipple/.style={
  decoration={stipple, segment length=2pt, #1},
  decorate,
  fill
}}
\begin{document}
\begin{tikzpicture}[x=2em,y=2em]
\draw [postaction={stipple={amplitude=0.5cm}},
      postaction={stipple={amplitude=0.25cm}}] 
  (2,1) ..controls ++(135:1) and ++(90:1) .. 
  (0,0) .. controls ++(270:1) and ++(180:1.5) .. 
  (2,-2) .. controls ++(0:1.5) and ++(270:2) .. 
  (5,1) .. controls ++(90:2) and ++(75:1) .. 
  (2,1) .. controls ++(255:1/4) and ++(0:1/2) .. (3/2,0);
\draw [postaction={stipple={amplitude=0.125cm, stipple density=0.05}},
  postaction={stipple={amplitude=0.5cm, reverse path}},
  postaction={stipple={amplitude=0.25cm, reverse path}}] 
  (3,0) .. controls ++(135:1) and ++(90:1) .. (4,1);
\end{tikzpicture}
\end{document}

Answer 2

Looks like I’ll have to step up my game here. Mark Wibrow added a great answer, but this one has a few points in its favor, including grayscaling the dots so the ones on the inside are lighter, getting exactly predictable and replicable results on each run, and having control over all the parameters. (For example, you can change how much the thickness of the stippling varies by changing the exponent of \thisrowno{1} in both places.) It also runs pretty quick while working in engines other than LuaLaTeX.

\PassOptionsToPackage{svgnames}{xcolor}
\documentclass{standalone}
\usepackage{pgfplots}

\pgfplotsset{width=\textwidth,compat=1.12}

\begin{document}

\begin{tikzpicture}
\begin{axis}[trig format plots=rad, width=5cm,axis equal,
             xmin = -1, xmax = 1,
             axis x line = none, axis y line = none]
  \addplot [variable=\t,domain=0:2*pi,samples=30,smooth]({cos t},{sin t});
%% θ(u) = 2πku, r(t) = t^c
  \addplot+ [scatter,scatter src=0.6*(1-\thisrowno{0}^0.125)+0.2,
             only marks,mark=*,mark size=0.001cm,colormap/blackwhite]
            table[header=false,
                  x expr=\thisrowno{0}^0.125*cos(7*pi/3+sign(\thisrowno{2}-0.5)*pi*\thisrowno{1}^0.5),
                  y expr=\thisrowno{0}^0.125*sin(7*pi/3+sign(\thisrowno{2}-0.5)*pi*\thisrowno{1}^0.5),
                 ]
                 {randtuple.dat}; % A file of three columns of random numbers from [0,1).
\end{axis}
\end{tikzpicture}

\end{document}

For completeness, here’s the program that generated the random data, although it would be possible to generate it within pgfmath using rand:

#include <cmath>
#include <ctime>
#include <cstdint>
#include <iomanip>
#include <iostream>
#include <random>

using std::cout;

int main(void)
{
  static const ssize_t ncols = 1000;
  static const unsigned sigfigs = 17;
  static const unsigned width = 22;
  static const double interval = std::exp2(-64.0L);

  std::mt19937_64 rng( static_cast<std::mt19937_64::result_type>(
                         time(NULL)*CLOCKS_PER_SEC+clock() ) );

  cout.precision(sigfigs);

  for ( ssize_t i = 0; i < ncols; ++i ) {
    const double x = rng() * interval;
    const double y = rng() * interval;
    const double z = rng() * interval;

    cout << std::setw(width) << x << " "
         << std::setw(width) << y << " "
         << std::setw(width) << z << "\n";
  }

  return EXIT_SUCCESS;
}

We might also take columns of points (t,u) ∊ [0,1]×[0,1] drawn from a uniform random distribution, and map them to r(t) = t^c, θ(u) = 2πu.

And here’s what that looks like:

\PassOptionsToPackage{svgnames}{xcolor}
\documentclass{standalone}
\usepackage{pgfplots}

\pgfplotsset{width=\textwidth,compat=1.12}

\begin{document}

\begin{tikzpicture}
\begin{axis}[trig format plots=rad, width=5cm,axis equal,
             xmin = -1, xmax = 1,
             axis x line = none, axis y line = none]
  \addplot [variable=\t,domain=0:2*pi,samples=30,smooth]({cos t},{sin t});
%% θ(u) = 2πku, r(t) = t^c
  \addplot+ [scatter,scatter src=0.8*(1-\thisrowno{0}^0.125),
             only marks,mark=*,mark size=0.001cm,colormap/blackwhite]
            table[header=false,
                  x expr=\thisrowno{0}^0.125*cos(10*pi*\thisrowno{1}),
                  y expr=\thisrowno{0}^0.125*sin(10*pi*\thisrowno{1}),
                 ]
                 {randpairs.dat}; % A file of two columns of random numbers from [0,1).
\end{axis}
\end{tikzpicture}

\end{document}

And the old version

Maybe get a noisy distribution of points on a line, paramaterize those as a spiral that coils more tightly on the outside then the inside, e.g. for t ∊ [0,1], k>1, 0<c<1, θ(t) = 2πk√t, r(t) = t^c, and plot them?

\PassOptionsToPackage{svgnames}{xcolor}
\documentclass{standalone}
\usepackage{pgfplots}

\pgfplotsset{width=\textwidth,compat=1.12}

\begin{document}

\begin{tikzpicture}
\begin{axis}[trig format plots=rad, width=5cm,axis equal,
             xmin = -1, xmax = 1,
             axis x line = none, axis y line = none]
  \addplot[variable=\t,domain=0:2*pi,samples=30,smooth]({cos t},{sin t});
%% θ(t) = 2πk·√t, r(t) = t^c
%% x(t) = r(t) cos θ(t) = t^c cos 2πk√t
%% y(t) = r(t) sin θ(t) = t^c sin 2πk√t
%% Where t ∊ [0,1], k>1, 0<c<1
%%
%% To eliminate the first half-revolution, solve for 2πk√t = π.  A prime
%% number in the sample size is less likely to produce unattractive patterns.
  \addplot+[variable=\t,domain=0.01:1,samples=193,
            only marks,mark=*,mark size=0.01cm,
            mark options={draw=DimGray,fill=DimGray}]
           ({t^0.125*cos(10*pi*t^0.5)},{t^0.125*sin(10*pi*t^0.5});
\end{axis}
\end{tikzpicture}

\end{document}

For simplicity, there’s no noise in the distribution of the points, but it still looks fairly nice to me.

Answer 3

Ok, here goes, compile with lualatex.

\documentclass{article}

\usepackage{tikz}
\usepackage{luacode}



\begin{document}


\begin{tikzpicture}
\draw (0,0) circle(5cm);
\begin{luacode*}
math.randomseed(os.time())
for i=1,1000 do
r=math.random()*3.145926535*2
s=math.random()+3.9
tex.print("\\draw[fill] (" .. s*math.cos(r) .. "," .. s*math.sin(r) ..") circle(0.2mm);")
end
 \end{luacode*}
\end{tikzpicture}

\end{document}

And a somewhat more sophisticated example:

\documentclass{article}

\usepackage{tikz}
\usepackage{luacode}



\begin{document}


\begin{tikzpicture}
\draw (0,0) circle(5cm);
\begin{luacode*}
math.randomseed(os.time())
for i=1,1000 do
r=math.sqrt(math.random())*3.145926535*2
s=math.pow(math.random(),0.2)+3.99
tex.print("\\draw[fill] (" .. s*math.cos(r) .. "," .. s*math.sin(r) ..") circle(0.2mm);")
end
 \end{luacode*}
\end{tikzpicture}

\end{document}

Better yet:

\documentclass{article}

\usepackage{tikz}
\usepackage{luacode}



\begin{document}


\begin{tikzpicture}
\draw (0,0) circle(5cm);
\begin{luacode*}
math.randomseed(os.time())
for i=1,1000 do
r=math.sqrt(math.random())*3.145926535*2
s=math.pow(math.random(),0.1)*4.99
tex.print("\\draw[fill] (" .. s*math.cos(r) .. "," .. s*math.sin(r) ..") circle(0.2mm);")
end
 \end{luacode*}
\end{tikzpicture}

\end{document}

Best:

\documentclass{article}

\usepackage{tikz}
\usepackage{luacode}



\begin{document}


\begin{tikzpicture}
\draw (0,0) circle(5cm);
\begin{luacode*}
math.randomseed(os.time())
for i=1,1000 do
if math.random() > 0.5 then b=1 else b= -1 end
r=(b*math.sqrt(math.random())+1)*math.pi
s=math.pow(math.random(),0.1)*4.99
tex.print("\\draw[fill] (" .. s*math.cos(r) .. "," .. s*math.sin(r) ..") circle(0.3mm);")
end
 \end{luacode*}
\end{tikzpicture}

\end{document}

Answer 4

I guess this is not exactly what you want. But it looks like pictures from old books as well.

I just applied ordered dithering to the sphere here.

Theoretically, given any functional shading, it is always possible to apply a dithering by a post script. (unless the stack overflows.)

The resolution is hard-coded, but still it is possible to change the resolution. (But it is meaningless if the resolution is too high.)

\documentclass[tikz,border=9]{standalone}

\pgfdeclarefunctionalshading{ordered dithering sphere}{\pgfpoint{0bp}{0bp}}{\pgfpoint{50bp}{50bp}}{}{
    2 copy
    %
    50 div 128 mul floor 63.5 sub 64 div exch
    50 div 128 mul floor 63.5 sub 64 div exch
    2 copy
    dup mul exch dup mul add sqrt 3 1 roll
    %
    2 copy 
    dup mul exch
    dup mul add
    1.0 sub
    0.3 dup mul
    -0.5 dup mul add
    1.0 sub
    mul abs sqrt
    exch 0.3 mul add
    exch -0.5 mul add
    dup abs add 2.0 div 
    0.6 mul 0.4 add
    %
    exch .98 ge {pop 1} if
    3 1 roll
    %
    50 div exch 50 div 1 % y x 1
    %
    3 1 roll 8 mul dup floor sub
    2 1 roll 8 mul dup floor sub
    3 2 roll
    %
    3 1 roll 2 mul dup floor dup 3 1 roll sub 3 2 roll 2 mul dup floor dup 3 1 roll sub
    4 3 roll 3 2 roll 2 copy -4 mul mul exch 3 mul add exch 2 mul add 4 mul 4 3 roll add
    %
    3 1 roll 2 mul dup floor dup 3 1 roll sub 3 2 roll 2 mul dup floor dup 3 1 roll sub
    4 3 roll 3 2 roll 2 copy -4 mul mul exch 3 mul add exch 2 mul add 16 mul 4 3 roll add
    %
    3 1 roll 2 mul dup floor dup 3 1 roll sub 3 2 roll 2 mul dup floor dup 3 1 roll sub
    4 3 roll 3 2 roll 2 copy -4 mul mul exch 3 mul add exch 2 mul add 64 mul 4 3 roll add
    %
    3 1 roll 2 mul dup floor dup 3 1 roll sub 3 2 roll 2 mul dup floor dup 3 1 roll sub
    4 3 roll 3 2 roll 2 copy -4 mul mul exch 3 mul add exch 2 mul add 256 mul 4 3 roll add
    %
    1025 div
    3 index
    le
    {1}{0}ifelse
    dup dup
}

\begin{document}
    \tikz\shade[shading=ordered dithering sphere](0,0)circle[radius=5cm];
\end{document}