What package(s) and commands do I need to create shapes as thought of by Jacob Barnett in LaTeX?

Question

I watched both the "60 minutes" talk and the "numberphile" youtube video discussing how Jacob Barnett sees numbers as shapes. In both of them they mention that Barnett sees numbers as fractal-like shapes consisting of polygons that overlap/encase the valence/outer vertices of subsequent polygons with the same shape. For example, the number 9 is thought of as a triangle with triangles that encase a vertex at each of the triangle's vertices. For 27, reproduce the shape for 9 but add triangles that encase each of the valence vertices of the shape for 9.

Links to Videos: (numberphile) https://www.youtube.com/watch?v=hP-DZMmQBng

(60 minutes talk/requires access to "CBS All Access") https://www.cbsnews.com/news/jake-math-prodigy-proud-of-his-autism/

As far I have gotten:

\documentclass[border=5mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric}

\tikzset{
    buffer/.style={
        draw,
        shape border rotate=180,
        regular polygon,
        regular polygon sides=3,
        fill=white,
        node distance=2cm,
        minimum height=4em
    }
}

\begin{document}
\begin{tikzpicture}
\node[buffer]{};
\end{tikzpicture}


\begin{tikzpicture}
\node[buffer]{};
\end{tikzpicture}
\end{document}

Are there packages and commands available to create such shapes?

Answer 1

I'm sure some tikz expert could do it better, but this could be a starting point.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric}
\usepackage{caption}

\tikzset{
    buffer/.style={
        draw,
        regular polygon,
        regular polygon sides=3,
        minimum size=20em
    },
    pics/nine/.style 2 args= {code={% 
        \node[buffer, fill=#1] (A) {};
        \foreach \mycorner in {1,2,3}
        \node[buffer, fill=#2, scale=.33, transform canvas] at (A.corner \mycorner) {};
    }},
}

\begin{document}
    \begin{figure}
        \centering
        \begin{tikzpicture}
            \node[buffer, fill=red]{};
        \end{tikzpicture}
        \caption{Number 3}
    \end{figure}
    \begin{figure}
    \centering
        \begin{tikzpicture}
            \node[buffer, fill=red]{};
            \node[buffer, yscale=-1, fill=orange]{};
        \end{tikzpicture}
        \caption{Number 6}
    \end{figure}
    \begin{figure}
    \centering
        \begin{tikzpicture}
            \pic {nine={red}{orange}}; 
        \end{tikzpicture}       
        \caption{Number 9}
    \end{figure}
    \begin{figure}
    \centering
        \begin{tikzpicture}
            \node[buffer, fill=red] (B) {};
            \foreach \mycorner in {1,2,3}
                \pic[scale=.33, transform shape] at (B.corner \mycorner) {nine={orange}{red}};
        \end{tikzpicture}
        \caption{Number 27}
    \end{figure}
    \begin{figure}
        \centering
        \begin{tikzpicture}
            \path (-4.4,-5.1) rectangle (4.4,5.1);
            \node[buffer, fill=red] (B) {};
            \foreach \mycorner in {1,2,3}
            \pic[scale=.33, transform shape] at (B.corner \mycorner)
                {nine={orange}{red}};
            \node[buffer, fill=orange, yscale=-1] (C) {};
            \begin{scope}[yscale=-1]
            \foreach \mycorner in {1,2,3}
            \pic[scale=.33, transform shape] at (C.corner \mycorner)
            {nine={red}{orange}};           
            \end{scope}
        \end{tikzpicture}
        \caption{Number 54}
    \end{figure}
\end{document}