Creating an Altered Form of Sierpinski Gasket in Tikz

Question

I'd like to adapt the example here to create a triangle similar to the Sierpinski gasket: How to create a Sierpinski triangle in LaTeX?

Essentially I'd like to create the altered form of the gasket pictured below. Even just the first iteration of the fractal (with 3 triangles removed from the bigger triangle) would be great! However, I haven't been able to work out how to adapt the techniques from the link above. Any help would be greatly appreciated!

Answer 1

Welcome! This is a rather close reproduction of your screen shot with the lindenmayersystems library.

\documentclass[tikz,border=3mm]{standalone}
\usetikzlibrary{lindenmayersystems}
\begin{document}%
\def\trianglewidth{6cm}%

\pgfdeclarelindenmayersystem{Sierpinski triangle 3}{%
    \symbol{X}{\pgflsystemdrawforward}%
    \rule{X -> X-X+X+X-X-XX+X+X-X+X-X}%
}%
\pgfmathtruncatemacro{\level}{4}%
\tikzset{%
    l-system={step={\trianglewidth/pow(3,\level)}, order=\level, angle=-120}
}%
\begin{tikzpicture}
\fill [black] (0,0) -- ++(0:\trianglewidth) -- ++(120:\trianglewidth) -- cycle;
\clip (0,0) -- ++(0:\trianglewidth) -- ++(120:\trianglewidth) -- cycle;
    \draw [draw=none] (0,0) l-system
    [l-system={Sierpinski triangle 3,
    axiom=X},fill=white,draw,line width=1pt/\level,line join=round];
\end{tikzpicture}
\end{document}

Let's now discuss how one may get there. First of all,I think the post you link to can be simplified. As far as I can see, there is no need for the Y rule. This observation also allows one to generalize the construction.

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

\usetikzlibrary{lindenmayersystems,backgrounds}

\begin{document}%
\def\trianglewidth{2cm}%
\pgfdeclarelindenmayersystem{Sierpinski triangle}{
    \symbol{X}{\pgflsystemdrawforward}
    \rule{X -> X-X+X+X-X}
}%
\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}
}%

\pgfdeclarelindenmayersystem{Sierpinski triangle 3}{
    \symbol{X}{\pgflsystemdrawforward}
    \rule{X -> X-X+X+X-X-XX+X+X-X+X-X}
}%
\foreach \level in {1,...,4}{%
\tikzset{
    l-system={step={\trianglewidth/pow(3,\level)}, order=\level, angle=-120}
}%
\begin{tikzpicture}[background rectangle/.style={fill=blue},
show background rectangle]
\fill [black] (0,0) -- ++(0:\trianglewidth) -- ++(120:\trianglewidth) -- cycle;
    \draw [draw=none] (0,0) l-system
    [l-system={Sierpinski triangle 3, axiom=X},fill=white];
\end{tikzpicture}}
\end{document}

The regular Sierpinski triangles get reproduced:

The new ones are drawn here with a background to show that there is no cheating going on, i.e. no unnecessary stuff gets drawn in white.

And here is a "new only" background free version.

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

\usetikzlibrary{lindenmayersystems}

\begin{document}%
\def\trianglewidth{3cm}%

\pgfdeclarelindenmayersystem{Sierpinski triangle 3}{
    \symbol{X}{\pgflsystemdrawforward}
    \rule{X -> X-X+X+X-X-XX+X+X-X+X-X}
}%
\foreach \level in {1,...,3}{%
\tikzset{
    l-system={step={\trianglewidth/pow(3,\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 3, axiom=X},fill=white];
\end{tikzpicture}}
\end{document}

To get something that resembles your screen shot more, we can add a line width.

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

\usetikzlibrary{lindenmayersystems}

\begin{document}%
\def\trianglewidth{3cm}%

\pgfdeclarelindenmayersystem{Sierpinski triangle 3}{
    \symbol{X}{\pgflsystemdrawforward}
    \rule{X -> X-X+X+X-X-XX+X+X-X+X-X}
}%
\foreach \level in {1,...,3}{%
\tikzset{
    l-system={step={\trianglewidth/pow(3,\level)}, order=\level, angle=-120}
}%
\ifnum\level>1
~
\fi
\begin{tikzpicture}
\fill [black] (0,0) -- ++(0:\trianglewidth) -- ++(120:\trianglewidth) -- cycle;
\clip (0,0) -- ++(0:\trianglewidth) -- ++(120:\trianglewidth) -- cycle;
    \draw [draw=none] (0,0) l-system
    [l-system={Sierpinski triangle 3,
    axiom=X},fill=white,draw,line width=1pt/\level,line join=round];
\end{tikzpicture}}
\end{document}

If you are curious what's going on here: just imagine you are a turtle and crawl along some path. Whenever there is an X, this means you crawl some distance, if there is a +, you turn by -120 (why -120? because Jake said so and it is not a good idea to argue with an owl;-) and if there is a - you turn by 120 degrees. There is another library that allows us to illustrate this (and one can just search and replace the X, + and - by the corresponding turtle keys).

\documentclass[tikz,border=3mm]{standalone}
\usetikzlibrary{turtle}
\newcounter{iturt}
\begin{document}
\begin{tikzpicture}[scale=2,
pics/arrow/.style={code={\draw[-latex] (-0.5,0) -- (0,0);}}]
\draw [turtle={how/.style={to path={-- (\tikztotarget)
node[pos=0.7,auto]{\stepcounter{iturt}\number\value{iturt}}
pic[pos=1,sloped,allow upside down]{arrow}}},
home,right=90,forward,right=-120,forward,left=-120,forward,left=-120,forward,right=-120,forward,right=-120,forward,forward,left=-120,forward,left=-120,forward,right=-120,forward,left=-120,forward,right=-120,forward}];
\end{tikzpicture}
\end{document}

One can also use the parser module, maybe this is even simpler, but the bottom-line is that TikZ has tools that allow one to visualize what's going on here.