TikZ-PGF – How to Create a Sierpinski Triangle in LaTeX

pstrickstikz-pgf

I have been trying to recreate the following image in LaTeX

and for me personally it has to be done with pdflatex. Although answers using pst-fractals
From Jake I got a good start which led to the following code

\documentclass{standalone}
\usepackage{tikz}
\def\level{5}
\usetikzlibrary{lindenmayersystems}

\begin{document}
\begin{tikzpicture}[l-system={step=5pt, order=\level, angle=120},rotate=180]
    \pgfdeclarelindenmayersystem{Sierpinski triangle}{
    \symbol{X}{\pgflsystemdrawforward}
    \symbol{Y}{\pgflsystemdrawforward}
    \rule{X -> X-Y+X+Y-X}
    \rule{Y -> YY}
    }
    \draw [black] (3,2) l-system
    [l-system={Sierpinski triangle, axiom=X, anchor=north east},fill=white];
\end{tikzpicture}

\end{document}

There are however two problems with the code.

I layer a black triangle behind the lindenmayersystem?
How do I keep the size of the triangle consistent with different number of recursion levels.

Best Answer

The width of the background triangles is 2^order*stepsize, so you can calculate the step size as a function of the desired overall width:

\documentclass[border=5mm]{standalone}
\usepackage{tikz}

\usetikzlibrary{lindenmayersystems}

\begin{document}%
\def\trianglewidth{2cm}%
\pgfdeclarelindenmayersystem{Sierpinski triangle}{
    \symbol{X}{\pgflsystemdrawforward}
    \symbol{Y}{\pgflsystemdrawforward}
    \rule{X -> X-Y+X+Y-X}
    \rule{Y -> YY}
}%
\foreach \level in {0,...,3}{%
\tikzset{
    l-system={step=\trianglewidth/(2^\level), order=\level, angle=-120}
}%
\begin{tikzpicture}
    \fill [black] (0,0) -- ++(0:\trianglewidth) -- ++(120:\trianglewidth) -- cycle;
    \draw [draw=none] (0,0) l-system
    [l-system={Sierpinski triangle, axiom=X},fill=white];
\end{tikzpicture}
}%
\end{document}

Related Solutions

[Tex/LaTex] TikZ: Drawing an arc from an intersection to an intersection

A solution which allows to draw intersection segments of any two intersections is available as tikz library fillbetween.

This library works as general purpose tikz library, but it is shipped with pgfplots and you need to load pgfplots in order to make it work:

\documentclass{standalone}

\usepackage{tikz}
\usepackage{pgfplots}
\usetikzlibrary{fillbetween}

\begin{document}

\begin{tikzpicture}

\draw [name path=red,red] (120:1.06) circle (1.9);
%\draw [name path=yellow,yellow] (0:1.06) circle (2.12);
\draw [name path=green,green!50!black] (0:0.77) circle (2.41);
\draw [name path=blue,blue] (0:0) circle (1.06);

% substitute this temp path by `\path` to make it invisible:
\draw[name path=temp1, intersection segments={of=red and blue,sequence=L1}];
\draw[red,-stealth,ultra thick, intersection segments={of=temp1 and green,sequence=L3}];

\end{tikzpicture}

\end{document}

enter image description here

The key intersection segments is described in all detail in the pgfplots reference manual section "5.6.6 Intersection Segment Recombination"; the key idea in this case is to

create a temporary path temp1 which is the first intersection segment of red and blue, more precisely, it is the first intersection segment in the Left argument in red and blue : red. This path is drawn as thin black path. Substitute its \draw statement by \path to make it invisible.
Compute the desired intersection segment by intersecting temp1 and green and use the correct intersection segment. By trial and error I figured that it is the third segment of path temp1 which is written as L3 (L = left argument in temp1 and green and 3 means third segment of that path).

The argument involves some trial and error because fillbetween is unaware of the fact that end and startpoint are connected -- and we as end users do not see start and end point.

Note that you can connect these path segments with other paths. If such an intersection segment should be the continuation of another path, use -- as before the first argument in sequence. This allows to fill paths segments:

\documentclass{standalone}

\usepackage{tikz}
\usepackage{pgfplots}
\usetikzlibrary{fillbetween}

\begin{document}

\begin{tikzpicture}

\draw [name path=red,red] (120:1.06) circle (1.9);
%\draw [name path=yellow,yellow] (0:1.06) circle (2.12);
\draw [name path=green,green!50!black] (0:0.77) circle (2.41);
\draw [name path=blue,blue] (0:0) circle (1.06);

% substitute this temp path by `\path` to make it invisible:
\draw[name path=temp1, intersection segments={of=red and blue,sequence=L1}];
\draw[red,fill=blue,-stealth,ultra thick, intersection segments={of=temp1 and green,sequence=L3}]  
    [intersection segments={of=temp1 and green, sequence={--R2}}]
;

\end{tikzpicture}

\end{document}

enter image description here

[Tex/LaTex] How to create a Sierpinski triangle in LaTeX with numbers in the triangles

You could use Mark Wibrow's solution, but it employs lualatex and if you don't already use it, it might not be worth it to switch to a different engine. Besides, that solution uses hexagons instead of triangles and it doesn't color them.

So here's an alternative version based on that answer but that has all the necessary changes. Plus it works with regular pdflatex, since we use pgfmath to do the proper calculations.

Output

enter image description here

Code

\documentclass[tikz,margin=10pt]{standalone}
\usetikzlibrary{shapes.geometric}

\tikzset{
    tris/.style={#1,
    font=\ttfamily,
    regular polygon, regular polygon sides=3, 
    minimum size=2cm, inner sep=0pt,
}
}

\begin{document}
\begin{tikzpicture}[x=2cm*sin 60, y=3cm*cos 60]

\foreach \n in {0,...,7}{
    \foreach \k in {0,...,\n}{ 
        \pgfmathtruncatemacro\fact{factorial(\n)/(factorial(\n-\k)*factorial(\k))}
    \ifodd\fact
        \node[tris={draw,fill, text=white}] at (-\n/2+\k, -\n) {\fact};
    \else
        \node[tris] at (-\n/2+\k, -\n) {\fact};
    \fi
    }
}
\end{tikzpicture}
\end{document}

Best Answer

Related Solutions

[Tex/LaTex] TikZ: Drawing an arc from an intersection to an intersection

[Tex/LaTex] How to create a Sierpinski triangle in LaTeX with numbers in the triangles

Output

Code

Related Question