Aperiodic tiling shapes draw in tikz

Question

I need a grid made up of pieces that look like a hat in tikz. Here is the piece with four hats. Could you help? Can we use a tikzlibrary for tiling of this?

\documentclass[border=3pt]{standalone}

\usepackage{tikz}

\begin{document}
    
    
\begin{tikzpicture}[rotate=-0]
\def\ra{1}
    \coordinate(O) at(0,0);
\coordinate(A1) at(0:\ra);
\coordinate(A2) at(60:\ra);
\coordinate(A3) at(120:\ra);
\path(A2)--(A1)--([turn]-60:\ra/2)coordinate(A01);
\draw[fill=yellow!6, thick](O)--(A01)--(A1)--(A2)--++(-\ra/2,0)--++(0,\ra)--([turn]60:\ra)--([turn]90:\ra/2)--([turn]60:\ra/2)--([turn]-90:\ra)--++(0,-\ra)coordinate(A3)--++(\ra/2,0)coordinate(A4)--([turn]-60:\ra/2)coordinate(A5)--cycle;

\draw[fill=gray!6, thick](O)--(A5)--(A4)--(A3)--++(-\ra/2,0)--([turn]60:\ra/2)coordinate(B1)--([turn]90:\ra)coordinate(B2) --++(0,-\ra)coordinate(B3)--++(\ra/2,0)coordinate(B4)--([turn]-60:\ra/2)coordinate(B5)--([turn]90:\ra)coordinate(B6)--++(0,\ra)coordinate(B7)--++(\ra/2,0)coordinate(B8)--(A01)--cycle;

\draw[fill=purple!6, thick](A01)--(B8)--(B7)--(B6)--(B5)--([turn]90:\ra/2)coordinate(D1)--++(\ra, 0)--([turn]60:\ra/2)--([turn]-90:\ra)--([turn]60:\ra)--([turn]90:\ra/2)--++(-\ra/2,0)--++(0,\ra)--(A01); 

\draw[fill=teal!6, thick](B5)--(B4)--(B3)--(B2)--(B1)--([turn]90:\ra/2)coordinate(C1)--([turn]60:\ra/2)--([turn]-90:\ra)--++(0,-\ra)--([turn]90:\ra/2)--([turn]-60:\ra/2)--([turn]90:\ra)--([turn]-60:\ra)--(D1)--cycle;

\end{tikzpicture}

\end{document}

Answer 1

There is a TikZ library for this, originally designed for Penrose tilings from https://tex.stackexchange.com/a/440412/86, and I am in the process of updating it for the new tile. Currently, I have the hat and turtle with a command for defining a Tile(a,b), and the meta and super cluster tiles. I'm now working on the substitution system for the super clusters, and wondering about the subclusters.

You can find the latest version on github, there's a release with all the necessary files in it, or for the bleeding edge version then the key file is penrose_code.dtx, download that and run tex penrose_code.dtx to generate the necessary files.

\documentclass{article}
%\url{https://tex.stackexchange.com/q/681708/86}
\usepackage{tikz}
\usetikzlibrary{
  tilings.polykite,
  calc,
}

\tikzset{
  hexagon/.pic={
    \path[pic actions] (1,0)
    foreach \k in {1,...,6} { -- (\k*60:1)}
    foreach \k in {1,...,6} {(0,0) -- (\k*60:1)}
    foreach \k in {1,...,6} {(0,0) -- (\k*60+30:{sqrt(3)/2})}
    ;
  },
  every tile/.style={draw}
}

\begin{document}

\begin{tikzpicture}[
  every aperiodical hat/.style={
    draw,
    ultra thick
  },
]
\pic[draw=none,fill=gray!50] {aperiodical hat};
\pic[draw,gray] at (-1,0) {hexagon};
\pic[draw,gray] at (60:1) {hexagon};
\pic[draw,gray] at (-60:1) {hexagon};
\pic[aperiodical hat,name=hat-a];
\pic[aperiodical hat,name=hat-b,align with=hat-a along 11 using 4];
\pic[aperiodical hat,name=hat-c,align with=hat-a along 26 using 7];
\pic[aperiodical hat,name=hat-d,align with=hat-a along 24 using 5];
\pic[aperiodical hat,name=hat-e,align with=hat-a along 13 using 2];
\pic[aperiodical hat,name=hat-f,align with=hat-a along 16 using 5];
\pic[red,flip tile,aperiodical hat,name=hat-g,align with=hat-a along 21 using 3];
\pic[aperiodical hat,name=hat-h,align with=hat-g back along 14 using 6];
\end{tikzpicture}
\end{document}

The above produces:

Answer 2

First, let's draw the tile more precisely and also choose a corner as coordinate zero:

\documentclass[border=10pt]{standalone}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}

\draw (0,0) coordinate (-i0)
    -- ++(60:1/2) coordinate (-i1)
    -- ++(120:1) coordinate (-i2)
    -- ++(180:1/2) coordinate (-i3)
    -- ++(90:{sin(60)}) coordinate (-i4)
    -- ++(150:{sin(60)}) coordinate (-i5)
    -- ++(240:1/2) coordinate (-i6)
    -- ++(300:1/2) coordinate (-i7)
    -- ++(210:{sin(60)}) coordinate (-i8)
    -- ++(270:{sin(60)}) coordinate (-i9) 
    -- ++(0:1/2) coordinate (-i10)
    -- ++(300:1/2) coordinate (-i11)
    -- ++(30:{sin(60)}) coordinate (-i12)
    -- cycle;

% adding the labels
\foreach \i in {0,...,12} {
    \pgfmathparse{int(\i == 3 || \i == 9 ? 1 : 0)}
    \draw[red] (-i\i) circle[radius=2pt];
    \ifnum\pgfmathresult=1\relax
        \node[below, red, font=\ttfamily\tiny] at (-i\i) {(-i\i)};
    \else
        \node[right, red, font=\ttfamily\tiny] at (-i\i) {(-i\i)};
    \fi
}

\end{tikzpicture}
\end{document}

Having this, we can create a pic from it. Since we would like to mirror and rotate this pic and also place it relative to others, we need to create it in a special way so that we first create the coordinates and only then draw the path having the possibility to shift and rotate the coordinates:

\documentclass[border=10pt]{standalone}
\usepackage{tikz}
\usetikzlibrary{calc}

\newif\ifaperiodicalhatmirrored
\tikzset{
    pics/aperiodical hat/.style={
        code={
            \tikzset{
                aperiodical hat/.cd, 
                #1,
                coordinate transformation/.style={
                    rotate around={\pgfkeysvalueof{/tikz/aperiodical hat/rotate around inner anchor}:(-inner anchor)},
                    shift={($(-outer anchor)-(-inner anchor)$)},
                }
            }
            \ifaperiodicalhatmirrored
                \path[xscale=-1]
            \else
                \path
            \fi
                (0,0) coordinate (-i0)
                -- ++(60:{\pgfkeysvalueof{/tikz/aperiodical hat/base unit}/2}) coordinate (-i1)
                -- ++(120:{\pgfkeysvalueof{/tikz/aperiodical hat/base unit}}) coordinate (-i2)
                -- ++(180:{\pgfkeysvalueof{/tikz/aperiodical hat/base unit}/2}) coordinate (-i3)
                -- ++(90:{\pgfkeysvalueof{/tikz/aperiodical hat/base unit}*sin(60)}) coordinate (-i4)
                -- ++(150:{\pgfkeysvalueof{/tikz/aperiodical hat/base unit}*sin(60)}) coordinate (-i5)
                -- ++(240:{\pgfkeysvalueof{/tikz/aperiodical hat/base unit}/2}) coordinate (-i6)
                -- ++(300:{\pgfkeysvalueof{/tikz/aperiodical hat/base unit}/2}) coordinate (-i7)
                -- ++(210:{\pgfkeysvalueof{/tikz/aperiodical hat/base unit}*sin(60)}) coordinate (-i8)
                -- ++(270:{\pgfkeysvalueof{/tikz/aperiodical hat/base unit}*sin(60)}) coordinate (-i9) 
                -- ++(0:{\pgfkeysvalueof{/tikz/aperiodical hat/base unit}/2}) coordinate (-i10)
                -- ++(300:{\pgfkeysvalueof{/tikz/aperiodical hat/base unit}/2}) coordinate (-i11)
                -- ++(30:{\pgfkeysvalueof{/tikz/aperiodical hat/base unit}*sin(60)}) coordinate (-i12)
                -- cycle;
            \coordinate (-inner anchor) at (\pgfkeysvalueof{/tikz/aperiodical hat/inner anchor});
            \coordinate (-outer anchor) at (\pgfkeysvalueof{/tikz/aperiodical hat/outer anchor});
            \path[pic actions]
                ([aperiodical hat/coordinate transformation]-i0) coordinate (-o0)
                    foreach \x in {1,...,12} { 
                        -- ([aperiodical hat/coordinate transformation]-i\x) coordinate (-o\x)
                    } -- cycle;
        }
    },
    aperiodical hat/base unit/.initial={1},
    aperiodical hat/inner anchor/.initial={0,0},
    aperiodical hat/outer anchor/.initial={0,0},
    aperiodical hat/rotate around inner anchor/.initial={0},
    aperiodical hat/mirrored/.is if={aperiodicalhatmirrored}
}

\begin{document}
\begin{tikzpicture}

\pic[draw, fill=yellow!6] (tile1) {aperiodical hat};

\pic[draw, fill=black!6] (tile2) {aperiodical hat={inner anchor={-i5}, rotate around inner anchor=-60, mirrored}};

\pic[draw, fill=red!6] (tile3) {aperiodical hat={inner anchor={-i9}, rotate around inner anchor=240}};

\pic[draw, fill=green!6] (tile4) {aperiodical hat={inner anchor={-i2}, outer anchor={tile2-o10}}};

\end{tikzpicture}
\end{document}

The pic aperiodical hat takes the following options:

• base unit: factor to scale the tile.
• inner anchor: anchor of the current tile.
• outer anchor: anchor of another tile (or some coordinate) which the anchor specified with inner anchor should sit at.
• rotate around inner anchor: degrees the tile should be rotated around the coordinate specified by inner anchor.
• mirrored: if used, the tile is mirrored.

There are two sets of anchors in each pic. The first set is -i0 to -i12 and represents the corners of the tile before it is rotated and shifted (but after it has been mirrored). The second set is -o0 to -o12 and represents the corners of the tile after all transformations are applied. For inner anchor the first set (-i0 to -i12) should typically be used, for outer anchor the second set (-o0 to -o12) should be used.

Having this functionality, I assume it should be possible to create also larger tilings ...

Answer 3

Another \pic version. This one uses a hexagonal grid with coordinates to place the tiles. As we have an 'aperiodic tiling' we need to place the tiles by hand but the grid helps with that.

For example, trying to replicate part of the first image here the code could be:

\documentclass[tikz]{standalone}
\usetikzlibrary{calc}

\tikzset
{%
  tile1/.style={thick,draw=black,fill=blue!40!cyan},
  tile2/.style={thick,draw=black,fill=cyan!50},
  tile3/.style={thick,draw=black,fill=gray!30},
  tile4/.style={thick,draw=black,fill=white},
  pics/tile/.style={
    code={%
    \path[pic actions] (0,0)            --++ (330:{cos(30)})  --++ (60:0.5)  --++ (120:1)   -|++
                       (-0.5,{cos(30)}) --++ (150:{cos(30)})  --++ (240:0.5) --++ (300:0.5) --++
                       (210:{cos(30)})  |-++ (0.5,{-cos(30)}) --++ (300:0.5) -- cycle;
    % to show the hexagons, remove it or comment it if you want
    \draw[very thin,gray] (30:{cos(30)}) -- (0,0) -- (150:{2*cos(30)})
                     (0,0) -- (90:{cos(30)}) --++ (180:0.5) --++ (120:0.5) --++ (30:{cos(30)})
                     (-1,0) --++ (60:1);
    }}
}

\begin{document}
\begin{tikzpicture}
% grid and clip
\foreach\i in {0,...,9} \foreach[count=\j]\jj in {A,...,H} 
  \coordinate (\jj\i) at ({\j+0.5*mod(\i,2)},{\i*cos(30)});
\clip[draw] ($(A1)+(-0.2,0.2)$) rectangle ($(G8)+(0.2,-0.2)$);
% tiles
\pic[tile2]                      at (B1) {tile};
\pic[tile2,rotate=240]           at (D2) {tile};
\pic[tile4,rotate=120]           at (G2) {tile};
\pic[tile1,rotate=300,xscale=-1] at (B3) {tile};
\pic[tile3,rotate=300]           at (E3) {tile};
\pic[tile2,rotate=60]            at (H3) {tile};
\pic[tile4,rotate=120]           at (A4) {tile};
\pic[tile2]                      at (D4) {tile};
\pic[tile4,rotate=60]            at (B5) {tile};
\pic[tile3]                      at (E5) {tile};
\pic[tile4,rotate=180]           at (G6) {tile};
\pic[tile3,rotate=240]           at (B7) {tile};
\pic[tile3,rotate=180]           at (A8) {tile};
\pic[tile2,rotate=120]           at (G8) {tile};
\pic[tile4,rotate=120]           at (E9) {tile};
% to show the grid
%\foreach\i in {0,...,9} \foreach[count=\j]\jj in {A,...,H} 
%  \fill (\jj\i) node[right] {\footnotesize$\jj\i$} circle (0.4mm);
\end{tikzpicture}
\end{document}

And produces:

Update: A 'map' to help with placing the tiles could be