TeXblog

Dynamic systems, bifurcations, procedural worlds

I’m using pgfplots a lot, so I will share some examples today.

Here, I benefit from these features of pgfplots, going beyond base TikZ:

• Simple plotting with 3d coordinates and axonometric projection
• Presentation of required coordinate axes
• Using color gradients
• Reading in files with externally calculated data

In any case we can use Lua for calculating data. Lua generates the TeX commands for printing, which will be processed in a pgfplots axis environment.

Here are the samples, just click on it to get to the corresponding thread on TeXwelt with full source code.

Lorenz attractor (dynamic system)

While I posted a Python calculated version on TeXwelt.de, Henri added one, which bases on LuaTeX. Let’s see his picture at first:

Of pgfplots I used the transparency feature besides the standard 3d plot, so I got an impression of the density:

Once we calculated the data, the code is simple:

\documentclass[border=10pt]{standalone}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
  \begin{axis}[
      xmin = -25, xmax = 25,
      ymin = -25, ymax = 25,
      zmin =   0, zmax = 50,
      hide axis,
    ]
    \addplot3[mark=none, mesh, shader=interp, color=black, opacity=0.2]
      file { lorenz.dat }; 
  \end{axis}
\end{tikzpicture}
\end{document}

Fractal landscapes – the base for producing procedural worlds

Between adjacent points, new points will be calculated, with random but limited variation. Finally we will get a mountainous landscaoe. The calculated points get color according to their height: blue for sea level and below, green for mountains and white above the snowline.

Next step: specify nice starting values, for beginning with a certain base structure, such as an island in the water.

Feigenbaum diagram (bifurkations)

This is a classic of the chaos theory und closely related to the Mandelbrot set. Also here, we use transparency for an impression of the point density.

I often started such topics on TeXwelt.de. LaTeX support for thesis writers is not the only talking point there. It became established, that TeX connoisseurs post their ideas in shape of a question, often themselves posting the first answer, opening a discussion. The final goal is a knowledge database, built on top of questions and answers.

08. July 2014 by stefan
Categories: Graphics, Mathematics, pgf/TikZ | Leave a comment

Iterated fractals

I recently dealt with  Iterated function systems , in short IFS. Here we got repeated transformations: the space will be mapped onto itself. There can be different map specifications. We do this an infinite number of times 🙂 and take a look at the set in space, which stays invariant. This can be a fractal object.

Enough of theory, there are great books on it, and Wikipedia provides a nice starting point. How do we generate such a fractal image? The simplest approach is the so called chaos game: wie nehmen einen Punkt her, and apply one of the transformations, randomly chosen. Because we got point sets, which are invariant under those transformations, the mapped point will be in the set again. We take the new point and repeat it, thousands of times, until a clear shape appears.

Let’s do this with the famous Barnsley fern!

But how? We need loops and the possibility of calculating affine transformations. It can be done with pgfmath, but I think it’s hardly readable. So I rather take Lua, integrating a programming language in the classical sense into the macro expansion language TeX. It’s easily written in Lua. I put the transformation parameters and probabilities into a matrix, so it can easily be changed for experiments. Let’s start the chaos game!

For compiling, we need LuaTeX and patience. For testing and playing with parameters and probabilities, it’s recommendable to choose a low number of iterations.

\documentclass[tikz,border=10pt]{standalone}
\usepackage{luacode}
\begin{luacode*}
  function barnsley(iterations,options)
    local x = math.random()
    local y = math.random()
    local m = {
        0.0,   0.0,   0.0, 0.16, 0.0,  0.0, 0.01,
       0.85,  0.04, -0.04, 0.85, 0.0,  1.6, 0.85,
        0.2, -0.26,  0.23, 0.22, 0.0,  1.6, 0.07,
      -0.15,  0.28,  0.26, 0.24, 0.0, 0.44, 0.07
    }
    local pm = { m[7], m[7] + m[14], m[7] + m[14] + m[21] }
    if options ~= [[]] then
      tex.sprint("\\draw[" .. options .. "] ")
    else
      tex.sprint("\\addplot coordinates{")
    end
    for i=1, iterations do
      p = math.random()
      if     p < pm[1] then
        case = 0
      elseif p < pm[2] then
        case = 1
      elseif p < pm[3] then
        case = 2
      else
        case = 3
      end
      newx = (m[7*case+1] * x) + (m[7*case+2] * y) + m[7*case+5]
         y = (m[7*case+3] * x) + (m[7*case+4] * y) + m[7*case+6]
         x = newx
      tex.sprint("("..x..","..y..") circle (0.05pt)")
    end
    tex.sprint(";")
  end
\end{luacode*}
\begin{document}
\begin{tikzpicture}
  \directlua{barnsley(100000, [[color=green!50!black,fill]])}
\end{tikzpicture}
\end{document}

On TeXwelt.de I produced variations, printed using pgfplots.

Also the famous Sierpinski triangle can be generated using the chaos game instead of the L-System approach, the similar source code is on TeXwelt.de, like linked above:

Now in three dimensions? 🙂 No joke – in analogy to the triangle there’s the squarish Sierpinski carpet, wich becomes the so called Menger sponge in three dimensions.

07. July 2014 by stefan
Categories: Graphics, Mathematics, pgf/TikZ | Leave a comment

Commutative diagrams with crossing edges

When one edge of a graph passes over or under another edge there may be a small gap needed in the drawing of one of those edges. An easy trick is to draw the upper edge two times: first with the background color and more thickness, afterwards normally, like
<node1> edge [-,line width=6pt,draw=white] <node2> edge <node2>
Here’s an example diagram used in the definition of the pull-back of a Banach bundle:

The source code:

\begin{tikzpicture}
  \matrix (m) [matrix of math nodes, row sep=3em,
    column sep=3em]{
    & f^\ast E_V& & \vphantom{f^\ast}E_V \\
    f^\ast E & & \vphantom{f^\ast}E & \\
    & U & & V \\
    M & & N & \\};
  \path[-stealth]
    (m-1-2) edge (m-1-4) edge (m-2-1)
            edge [densely dotted] (m-3-2)
    (m-1-4) edge (m-3-4) edge (m-2-3)
    (m-2-1) edge [-,line width=6pt,draw=white] (m-2-3)
            edge (m-2-3) edge (m-4-1)
    (m-3-2) edge [densely dotted] (m-3-4)
            edge [densely dotted] (m-4-1)
    (m-4-1) edge (m-4-3)
    (m-3-4) edge (m-4-3)
    (m-2-3) edge [-,line width=6pt,draw=white] (m-4-3)
            edge (m-4-3);
\end{tikzpicture}

This topic was discussed on Matheplanet.com.

If you want to read more about commutative diagrams with TikZ have a look here: Chains with labeled edges.

04. January 2014 by stefan
Categories: Graphics, Mathematics, pgf/TikZ | Leave a comment