7

enter image description here

I was trying to draw these two lattices in LaTeX for hours, but making no progress. Could someone please remind me the key code of doing so? Thanks a lot! Happy New Year!

percusse
  • 157,807
Gaoran Yu
  • 71
  • The second one is easier since it is easy to see the pattern to be translated. The first one also has a pattern, but it is more complicated. – Sigur Jan 01 '15 at 22:13
  • 2
    Since you tried for hours, you got already something? Please show that, so we can help to improve it. – Stefan Kottwitz Jan 01 '15 at 22:16
  • Is there a rule between those numbers and the shape? – percusse Jan 01 '15 at 22:16
  • @percusse, the number of sides of the polygons. – Sigur Jan 01 '15 at 22:17
  • @Sigur How do we understand which one is which? – percusse Jan 01 '15 at 22:18
  • @percusse, on each vertex you have 3 polygons and those numbers say what they are. – Sigur Jan 01 '15 at 22:21
  • @StefanKottwitz I was working out the coordinates for every vertices and use \foreach command to make the graph periodic. But for the (4,6,12) lattices, it's really painful to do things in this way. I was trying to use commands like \node [draw,minimum size = 0.5cm, regular polygon, regular polygon sides=12] at (0,0) {} in geometric shape library, but it does not look like a regular polygon at all. Sorry I was new to tikz package and this forum so I do not know much of them. – Gaoran Yu Jan 01 '15 at 22:34
  • @GaoranYu, you draw it using \draw (-15:2cm) \foreach \x in {15,45,...,360} {-- (\x:2cm) };. Now the problem is to draw the square. You have to compute its side length. – Sigur Jan 01 '15 at 22:52
  • 1
    @Sigur Thank you so much for your help. I can work out (4,8^2) now. It's just the (4,6,12) is quite annoying :( – Gaoran Yu Jan 01 '15 at 23:12

5 Answers5

16

Here's a quick attempt at (4,6,12) in Metapost that might provide a starting point for you in TikZ. It's based on repeating a unit shape, shown highlighted in pink below.

For more details on Metapost, you can start with the answer to this question.

enter image description here

prologues := 3;
outputtemplate := "%j%c.eps";

beginfig(1);

u = 20;
dx = u*(1+cosd(15)/sind(15));
picture unit; unit = image(
   path s; s = unitsquare shifted -(1/2,1/2) scaled u shifted (dx/2,0);
   for t=0 upto 2: 
     draw s rotated 120t; 
     for tt=1 upto 3:
       draw subpath(3,4) of s rotated (120t+30tt);
     endfor
   endfor
);

draw unit withpen pencircle scaled 4 withcolor .8[red,white];

for x=-5dx step dx until 5dx:
  for y=-5dx step dx until 5dx:
    draw unit shifted (x,0) shifted ((y,0) rotated 60);
  endfor
endfor

path box; box = unitsquare shifted -(1/2,1/2) scaled 12u;
clip currentpicture to box; draw box dashed evenly;

endfig;
end.
Community
  • 1
Thruston
  • 42,268
7

Here is one TikZ solution for (4,6,12) :

\documentclass[border=50]{standalone}
\usepackage{tikz}
\usetikzlibrary{math}
\usetikzlibrary{calc}

\tikzset{square/.style= { to path={ let \p1=(\tikztostart), \p2=(\tikztotarget),
  \p3=($(\p2)!1!90:(\p1)$), \p4=($(\p1)!1!-90:(\p2)$) in
  (\p4) -- (\p1) -- (\p2) -- (\p3) (\p2)}}}

\tikzmath{
  function drawone(\x,\y,\size) {
    {
      \draw[shift={(\x,\y)}] foreach \a in {-120,0,120} {
        [rotate=\a] (-15:\size) to[square] (15:\size) -- (45:\size) -- (75:\size) -- (105:\size) };
    };
  };
}

\begin{document}
  \begin{tikzpicture}
    \tikzmath{
      \size = 2; \nx = 3; \ny = 4;
      \xstep = (cos(15)+sin(15))*\size;
      \ystep = 3*\xstep/tan(60);
      for \i in {1,...,\nx}{
        for \j in {1,...,\ny}{
          drawone((2*\i+mod(\j,2))*\xstep,\j*\ystep,\size);
        };
      };
    }
    \begin{scope}[red,thick]
      \tikzmath{drawone(2,3,\size);}
    \end{scope}
  \end{tikzpicture}
\end{document}

enter image description here

UPDATE: Here is the (4,82) case which is simpler : 

\documentclass[border=50]{standalone}
\usepackage{tikz}
\usetikzlibrary{math}

\tikzmath{
  function drawone(\x,\y,\size) {
    {
      \draw[shift={(\x,\y)},rotate=-22.5] (0:\size) foreach \a in {0,45,...,360}{ -- (\a:\size)};
    };
  };
}

\begin{document}
  \begin{tikzpicture}
    \tikzmath{
      \size = .7; \nx = 8; \ny = 5;
      \step = 2*\size*cos(22.5);
      for \i in {1,...,\nx}{
        for \j in {1,...,\ny}{
          drawone(\i*\step,\j*\step,\size);
        };
      };
    }
  \end{tikzpicture}
\end{document}

enter image description here

Kpym
  • 23,002
5

A layman's approach (with no computations):

\documentclass[tikz,border=5]{standalone}
\usetikzlibrary{shapes.geometric}
\tikzset{
    myshape/.pic = {
    \node[name=s,regular polygon, regular polygon sides=12, minimum width=1cm, draw,
        outer sep=0pt] at (0,0){};
    \draw (s.corner 3) -- ([shift={(120:2mm)}]s.corner 3) --
           coordinate (b)([shift={(120:2mm)}]s.corner 2)
        -- (s.corner 2);
    \draw (s.corner 7) -- ([shift={(240:2mm)}]s.corner 7) -- ([shift={(240:2mm)}]s.corner 6)
        -- (s.corner 6);
    \draw (s.corner 11) -- ([shift={(0:2mm)}]s.corner 11) --
            coordinate[midway] (a)([shift={(0:2mm)}]s.corner 10)
        -- (s.corner 10);
}
}
\begin{document}
  \begin{tikzpicture}
    \pic (p) at (0,0) {myshape};
    \pic[anchor=west] (q) at (pa) {myshape};
    \pic[anchor=west] (r) at (qa) {myshape};
    \pic[anchor=west] (s) at (ra) {myshape};

    \pic[anchor=south east] (pp) at (pb) {myshape};
    \pic[anchor=south east] (qq) at (qb) {myshape};
    \pic[anchor=south east] (rr) at (rb) {myshape};
    \pic[anchor=south east] (ss) at (sb) {myshape};

    \pic[anchor=south east] (ppp) at (ppb) {myshape};
    \pic[anchor=south east] (qqq) at (qqb) {myshape};
    \pic[anchor=south east] (rrr) at (rrb) {myshape};
    \pic[anchor=south east] (sss) at (ssb) {myshape};
    \pic[anchor=west]  at (sssa) {myshape};

  \end{tikzpicture}
\end{document}

or

\documentclass[tikz,border=5]{standalone}
\usetikzlibrary{shapes.geometric}
\tikzset{
    myshape/.pic = {
    \node[name=s,regular polygon, regular polygon sides=12, minimum width=1cm+2.5\pgflinewidth, draw,
        outer sep=0pt] at (0,0){};
    \draw (s.corner 3) -- ([shift={(120:2mm)}]s.corner 3) --
           coordinate (b)([shift={(120:2mm)}]s.corner 2)
        -- (s.corner 2);
    \draw (s.corner 7) -- ([shift={(240:2mm)}]s.corner 7) -- ([shift={(240:2mm)}]s.corner 6)
        -- (s.corner 6);
    \draw (s.corner 11) -- ([shift={(0:2mm)}]s.corner 11) --
            coordinate[midway] (a)([shift={(0:2mm)}]s.corner 10)
        -- (s.corner 10);
}
}
\begin{document}
  \begin{tikzpicture}
    \foreach \x in {0,1.2,2.4,3.6,4.8}{
     \pic (p) at (\x,0) {myshape};
    }
    \foreach \x in {0.6,1.8,3,4.2,5.4}{
     \pic (p) at (\x,1.05cm-\pgflinewidth) {myshape};
    }
    \foreach \x in {0,1.2,2.4,3.6,4.8}{
     \pic (p) at (\x,2.1cm-2\pgflinewidth) {myshape};
    }
    \foreach \x in {0.6,1.8,3,4.2,5.4}{
     \pic (p) at (\x,3.15cm-3\pgflinewidth) {myshape};
    }
  \end{tikzpicture}
\end{document}

and

\documentclass[tikz,border=5]{standalone}
\usetikzlibrary{shapes.geometric}
\tikzset{
    myshape/.pic = {
    \node[name=s,regular polygon, regular polygon sides=8, minimum width=1.1cm-\pgflinewidth, draw,
        outer sep=0pt] at (0,0){};
}
}
\begin{document}
  \begin{tikzpicture}
    \foreach \x in {0,1,2,3,4}{
     \foreach \y in {0,1,2,3,4}{
     \pic (p) at (\x,\y) {myshape};
    }
    }
  \end{tikzpicture}
\end{document}
user43963
  • 1,570
  • (+1) for the (4,8^2) case. I think that using the idea of this case you can make the first one realy in a simpler way : create a pic using one node with regular polygon sides=12 and 3 square nodes with \foreach \i in {-120,0,120} \path[rotate=\i] (0:some distance) node[transform shape,...]. Then adapt your two step loop to draw two lines of shapes, one half xshifted. – Kpym Jan 02 '15 at 15:38
2

Here is a solution for the (4,6,12) case using to path.

\documentclass[tikz,border=5]{standalone}
\usetikzlibrary{calc}
\tikzset{
  xy/.style args={(#1,#2)}{ x={(#1,#2)}, y={({-1*(#2)},#1)} },
  sq/.style= { to path={ let \p1=(\tikztostart), \p2=(\tikztotarget), \p3=($(\p1)!.5!(\p2)$) in
      {[shift={(\p3)},xy={(\x2-\x1,\y2-\y1)},scale={.5/(1+cot(15))}]
        foreach \i in {1,-1}{ [scale=\i] (1,-1) -- (1,1) -- +(60:2) (1,1) -- (-1,1)} (\p2)}}}}
\begin{document}
  \begin{tikzpicture}
    % demonstrate to[sq]
    \draw[green] (0,0) to ++(0:1) (0,0) to ++(120:1);
    \draw[ultra thick,red] (0,0) to[sq] ++(1,0) (0,0) to[sq] ++(120:1);

    % draw the patern
    \foreach \x in {-2,...,2}{
      \foreach \y in {-2,...,2}{
        \draw ({\x+.5*\y},{.5*\y*tan(60)}) to[sq] ++(0:1) to[sq] ++(120:1) to[sq] ++(-120:1);
      }
    }
  \end{tikzpicture}
\end{document}

enter image description here

Kpym
  • 23,002
  • Thanks a lot for your answers! I can not be more grateful for your help. And I'm wondering if your code requires the latest version of tikz package. Cause there are several commands that my tex does not recognize. – Gaoran Yu Jan 04 '15 at 04:03
  • @GaoranYu Yes you need the last version of TikZ : 3.0.0. The solution that use markings, works almost under 2.10, but you have to comment \scoped (or replace it by scope environment). I think that it is a good think to switch to the last version of TikZ. – Kpym Jan 04 '15 at 10:48
2

One more TikZ solution for (4,6,12). This time using slanted grid decorated with regular polygone shapes.

\documentclass[tikz,border=5]{standalone}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{shapes.geometric}
\tikzset{
  polygon/.style 2 args = { draw, regular polygon, regular polygon sides=#1,
    minimum size=#2*1cm, inner sep=0pt, outer sep=0pt
  },
  p12/.style= {draw=red, opacity=.2,
    preaction={decorate, decoration={ markings, mark=between positions 0 and .999 step 1cm
      with { \node[polygon={12}{1/(sin(15)+cos(15))}]{}; }}}
  },
  p4/.style= { draw=blue, opacity=.2,
    preaction={decorate, decoration={ markings, mark=between positions .5cm and .999 step 1cm
    with { \node[transform shape, polygon={4}{sqrt(2)*sin(15)/(1+tan(15))}]{}; }}}
  }
}
\begin{document}
  \begin{tikzpicture}
    \draw[clip] (2,1) rectangle +(3,3);

    \begin{scope}[yscale=sqrt(3)/2,xslant=.5]
      % demonstrate p4 and p12 decorations
      \scoped[red,ultra thick,opacity=.5] \path[p12] (2,2) -- ++(1,0);
      \scoped[green,ultra thick,opacity=.5] \path[p4] (2,2) -- ++(1,0);

      % decorate slanted grids
      \draw[p4,p12,ystep=10] (0,0) grid +(5,5);
      \draw[xslant=-1,p4] (2,0) grid +(5,5);
    \end{scope}
  \end{tikzpicture}
\end{document}

enter image description here

Kpym
  • 23,002
  • I am sure now you are a mathematician ;-) –  Jan 03 '15 at 10:54
  • 1
    @jfbu Is that so obvious ... just because I put three solutions to one problem? ;) – Kpym Jan 03 '15 at 11:01
  • @jfbu I do not put the most "mathematical" solution that use the Coxeter group of symmetries to generate the figure starting from 1 edge. The problem is that I can't (easily) generate only the reduced words, and it draws multiple times the same edge :( – Kpym Jan 03 '15 at 11:10
  • what are generators and relations? –  Jan 03 '15 at 11:14
  • about drawing things multiple times, do you know whether when you define the styles the math within is evaluated once and for all by TikZ ? –  Jan 03 '15 at 11:17
  • 1
    @jfbu for generator we can take the three symmetries (s_1,s_2,s_3, elements of order 2) relating the centers of the three regular polygons. And, as the composition of two of them is a rotation of pi/2,pi/3 or pi/6, we have that they are of order 4,6 and 12. So we have the relations {s_1^2, s_2^2, s_3^2, (s_1 s_2)^4, (s_2 s_3)^6, (s_2 s_3)^12}. But please, don't do this ! ;) – Kpym Jan 03 '15 at 11:52
  • @jfbu the math in the style is evaluated at each style call. Here : once for any path of the grid. But the things are not drawn multiple times in the clipped area ... outside yes in some places :( – Kpym Jan 03 '15 at 11:57
  • thanks for the math explanations and also for the ones on the inner workings of TikZ... –  Jan 03 '15 at 23:16