How to make a 3D Lattice?

Question

How could I create a lattice like this:

10 by 10 lattice

In 3D, with the ability to change the x,y and z spacing independently of each other? Any solution using any LaTeX program would be great!

I have tried to do it with asymptote but it was a resounding failure. I can't make out the structure, see any grid, nor can I adjust the values independently.

3d lattice

Here is the code I used:

size(200);
import graph3;
import grid3;

real L=10;
triple s;
currentprojection=orthographic(0.25,0.25,0.75);
surface site = scale3(0.314)*unitsphere;

for(int i=1;i<=L;++i)
{
  for(int j=1;j<=L;++j)
  {
    for(int k=1;k<=L;++k)
    {
      s=(i,j,k);
      draw(shift(s)*site,red);
    }
  }
}

+1: Looks good to me. Just change the view angle and maybe increase the spacing of the spheres a bit. — Henri Menke, Dec 15 '13 at 09:50
Heh, kinda the problem... changing the view angle is wonky in asymptote and I can't find a good one. I also can't get a grid connecting the points to work correctly with adjustable spacing. — NictraSavios, Dec 15 '13 at 09:51
You might be interested in http://dominique.vrel.free.fr/latex2.htm — Chris Chudzicki, Dec 15 '13 at 16:29
See this question: http://tex.stackexchange.com/questions/97271/drawing-3d-crystal-lattice-with-tikz-pstricks — Alex, Dec 15 '13 at 22:30

Thomas · Answer 1 · 2013-12-15T16:02:14.837

Here is an example (probably inefficient, but working great !), with Tikz :

\documentclass{standalone}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}

\def \dx{2};
\def \dy{3};
\def \dz{2};
\def \nbx{4};
\def \nby{4};
\def \nbz{4};

\foreach \x in {1,...,\nbx} {
    \foreach \y in {1,...,\nby} {
        \foreach \z in {1,...,\nbz} {
            \node at (\x*\dx,\y*\dy,\z*\dz) [circle, fill=black] {};
        }
    }
}

% z lines
\foreach \x in {1,...,\nbx} {
    \foreach \z in {1,...,\nbz}{
        \draw (\x*\dx,\dy,\z*\dz) -- ( \x*\dx,\nby*\dy,\z*\dz);
    }
}

% x lines
\foreach \y in {1,...,\nbx} {
    \foreach \z in {1,...,\nbz}{
        \draw (\dx,\y*\dy,\z*\dz) -- ( \nbx*\dx,\y*\dy,\z*\dz);
    }
}

% y lines
\foreach \x in {1,...,\nbx} {
    \foreach \y in {1,...,\nbz}{
        \draw (\x*\dx,\y*\dy,\dz) -- ( \x*\dx,\y*\dy,\nbz*\dz);
    }
}

\end{tikzpicture}
\end{document}

enter image description here

EDIT : Here is the version with arrows, it's pretty ugly :)

\documentclass{standalone}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}
\tikzset{>=latex}

\def \dx{2};
\def \dy{3};
\def \dz{2};
\def \nbx{4};
\def \nby{4};
\def \nbz{4};

\foreach \x in {1,...,\nbx} {
    \foreach \y in {1,...,\nby} {
        \foreach \z in {1,...,\nbz} {
            \node at (\x*\dx,\y*\dy,\z*\dz) [circle, fill=black] {};
        }
    }
}

% z lines
\foreach \x in {1,...,\nbx} {
    \foreach \z in {1,...,\nbz}{
        \foreach \y in {2,...,\nby}{
            \draw [->, color = red, line width = 2pt](\x*\dx,\y*\dy - \dy,\z*\dz) -- ( \x*\dx , \y*\dy, \z*\dz);
        }
    }
}

% x lines
\foreach \y in {1,...,\nbx} {
    \foreach \z in {1,...,\nbz}{
        \foreach \x in {2,...,\nbx}{
            \draw[->, color = red, line width = 2pt](\x * \dx - \dx,\y*\dy,\z*\dz) -- ( \x * \dx,\y*\dy,\z*\dz);
        }
    }
}

% y lines
\foreach \x in {1,...,\nbx} {
    \foreach \y in {1,...,\nbz}{
        \foreach \z in {2,...,\nby}{
            \draw[->, color = red, line width = 2pt](\x*\dx,\y*\dy,\z*\dz - \dz) -- ( \x*\dx,\y*\dy,\z*\dz);
        }
    }
}

\end{tikzpicture}
\end{document}

enter image description here

Your solution works great, although... if it isn't too much trouble, could you modify it to draw the lines individually? That was they can be converted to arrow to show progression from one point to the next.
If its too much hassle though, don't worry! — NictraSavios, Dec 15 '13 at 11:21
I'm afraid my knowledge of Tikz has already reached its limit here (I'm just starting to learn it), it looks like you're expecting an use of the animate package. I hope someone will answer it, I suggest you to edit your question. — Thomas, Dec 15 '13 at 11:25
Not an animation, something like what I use here in the third graphic: http://tex.stackexchange.com/questions/148510/how-to-plot-a-lattice-of-points-on-the-surface-of-a-torus — NictraSavios, Dec 15 '13 at 11:36
Exactly!, Its related to the Metropolis Algorithm of the Ising Model. — NictraSavios, Dec 15 '13 at 13:52

score 7 · Accepted Answer · answered Dec 15 '13 at 21:05

Using @Tomas 's code, and adding in the arrows as per a comment, I ended up with this:

enter image description here

Code: (All credits to Thomas for the base!)

\documentclass{standalone}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}
\tikzset{>=latex}

\def \dx{3};
\def \dy{3};
\def \dz{3};
\def \nbx{4};
\def \nby{4};
\def \nbz{4};

\foreach \x in {1,...,\nbx} {
    \foreach \y in {1,...,\nby} {
        \foreach \z in {1,...,\nbz} {
            \node at (\x*\dx,\y*\dy,\z*\dz) [circle, fill=red] {};
        }
    }
}

% z lines
\foreach \x in {1,...,\nbx} {
    \foreach \z in {1,...,\nbz}{
        \foreach \y in {2,...,\nby}{
            \draw [->,shorten >=0.5\dz cm,line width = 2pt]( \x*\dx , \y*\dy, \z*\dz) -- (\x*\dx,\y*\dy - \dy,\z*\dz);
        }
    }
}

% x lines
\foreach \y in {1,...,\nbx} {
    \foreach \z in {1,...,\nbz}{
        \foreach \x in {2,...,\nbx}{
            \draw[->,shorten >=0.5\dx cm, line width = 2pt](\x * \dx - \dx,\y*\dy,\z*\dz) -- ( \x * \dx,\y*\dy,\z*\dz);
        }
    }
}

% y lines
\foreach \x in {1,...,\nbx} {
    \foreach \y in {1,...,\nbz}{
        \foreach \z in {2,...,\nby}{
            \draw[->,shorten >=0.5\dy cm, line width = 2pt] ( \x*\dx,\y*\dy,\z*\dz) -- (\x*\dx,\y*\dy,\z*\dz - \dz);
        }
    }
}

\end{tikzpicture}
\end{document}

score 4 · Answer 3 · answered Sep 30 '15 at 16:19

You were unable to find a good viewpoint because

Your spheres were far too large relative to the space between them, and
Your grid was too large (specifically, too deep).

Here is a modification of your Asymptote code that does the job.

\documentclass{standalone}
\usepackage{asypictureB}
\begin{document}
\begin{asypicture}{name=3dgrid}
settings.render=8;
settings.outformat='png';
size(200);
import graph3;

int L=4;
triple s;
currentprojection=perspective(27,19,11);
real r = 0.1;
surface site = scale3(r)*unitsphere;
real dist = 1.5;

for(int i=1;i<=L;++i)
{
  for(int j=1;j<=L;++j)
  {
    for(int k=1;k<=L;++k)
    {
      s=dist*(i,j,k);
      draw(shift(s)*site,red);
      if (i < L) draw(shift(s) * (O -- (dist-r)*X), arrow=Arrow3(TeXHead2, emissive(blue)),
              p=blue+0.7pt);
      if (j < L) draw(shift(s) * (O -- (dist-r)*Y), arrow=Arrow3(TeXHead2, emissive(gray)),
              p=gray+0.7pt);
      if (k < L) draw(shift(s) * (O -- (dist-r)*Z), arrow=Arrow3(TeXHead2, emissive(green)),
              p=green+0.7pt);
    }
  }
}
\end{asypicture}
\end{document}

score 2 · Answer 4 · edited Apr 13 '17 at 12:34

A somewhat more realistic solution can be achieved by using the package asymptote (as you mentioned):

\documentclass{standalone}
\usepackage{asymptote}

\begin{document}
\begin{asy}
import three;
settings.render=8;
settings.prc=false;
size(10cm);

//currentprojection=perspective((45,45,30));
currentprojection = orthographic((3,6,1));

material sphereCcolor = material(diffusepen=black, ambientpen=gray(0.1), specularpen=white);
material cylcolor = material(diffusepen=white, ambientpen=white);

real cylRadius = 0.1;
real sphereRadius = 0.25;

void drawRod(triple a, triple b) {
  surface rod = extrude(scale(cylRadius)*unitcircle, axis=length(b-a)*Z);
  triple orthovector = cross(Z, b-a);
  if (length(orthovector) > .01) {
    real angle = aCos(dot(Z, b-a) / length(b-a));
    rod = rotate(angle, orthovector) * rod;
  }
  draw(shift(a)*rod, surfacepen=cylcolor);
}

void drawCarbon(triple center) {
     draw(shift(center)*scale3(sphereRadius)*unitsphere, surfacepen=sphereCcolor);
}

triple P000 = (0,0,0);
triple P100 = 4X;
triple P010 = 4Y;
triple P001 = 4Z;
triple P011 = 4Y+4Z;
triple P101 = 4X+4Z;
triple P110 = 4X+4Y;
triple P111 = 4X+4Y+4Z;

drawRod(P000,P100);
drawRod(P000,P010);
drawRod(P000,P001);
drawRod(P010,P011);
drawRod(P001,P011);
drawRod(P100,P101);
drawRod(P001,P101);
drawRod(P100,P110);
drawRod(P010,P110);
drawRod(P011,P111);
drawRod(P101,P111);
drawRod(P110,P111);

drawCarbon(P000);
drawCarbon(P100);
drawCarbon(P010);
drawCarbon(P001);
drawCarbon(P011);
drawCarbon(P101);
drawCarbon(P110);
drawCarbon(P111);
\end{asy}
\end{document}

The lattice sites and connecting rods are in the right order with respect to each other.

For a similar example, check out the following question: Draw realistic 3D crystal structures (diamond)

Sorry for the bad rendering. I promise you the pdf version looks better. ;) — strpeter, Sep 27 '15 at 16:18

How to make a 3D Lattice?

4 Answers4

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