Draw the polyhedra tessellation in the 3-dimensional-space

Question

Consider a polyhedron composed by

• 24 vertices (4×6)
• 14 faces (contain 6 squares and 8 hexagons)
• 36 edges (4×6+6×82=36).

This polyhedron can pack and tessellate the 3-dimensional-space like this.

Question: What are the available methods to draw this in LaTex? (My attempt was that I showed such packing is possible and prepared this figure [not using LaTex..].)

Answer 1

This draws such polyhedra and illustrates the point that they provide a tessellation of 3d space.

1. Symbol 1 correctly identified the polyhedron as truncated octahedron.
2. The data can of (such) polyhedra can be obtained from Mathematica via N[PolyhedronData["TruncatedOctahedron", "GraphicsComplex"]].
3. The main purpose of this answer is to provide one way to draw and order polyhedron with TikZ.
4. This answer requires the experimental 3dtools library.

In Mathematica's conventions, these polyhedra sit at lattice points with lattice vectors (0,2,-\sqrt{2}), (2,0,-\sqrt{2}), (0,2,\sqrt{2}). 3d ordering can be obtained by projecting the loci of the polyhedra on the normal of the screen, and sort according to the projections. (To speed up the compilation a bit the polyhedra are stored in \saveboxes, which is implicitly suggested here.)

\documentclass[tikz,border=3mm]{standalone}
\usepackage{tikz-3dplot}
\usetikzlibrary{backgrounds,3dtools}
\newsavebox\TruncatedOctahedronBlue
\newsavebox\TruncatedOctahedronOrange
\tdplotsetmaincoords{80}{105}
\newcommand{\TruncatedOctahedron}{%
\begin{tikzpicture}[tdplot_main_coords,line cap=round,line join=round]
\path foreach \Coord [count=\X] in 
{(-1.5,-0.5,0.), (-1.5,0.5,0.), (-1.,-1.,-0.707107), (-1.,-1.,0.707107), 
(-1.,1.,-0.707107), (-1.,1.,0.707107), (-0.5,-1.5,0.), (-0.5, -0.5,-1.41421),
(-0.5,-0.5,1.41421), (-0.5,0.5,-1.41421), (-0.5,0.5, 1.41421), (-0.5,1.5,0.),
(0.5,-1.5,0.), (0.5,-0.5,-1.41421), (0.5,-0.5, 1.41421), (0.5,0.5,-1.41421),
(0.5,0.5,1.41421), (0.5,1.5,0.), (1.,-1., -0.707107), (1.,-1.,0.707107),
(1.,1.,-0.707107), (1.,1.,0.707107), (1.5, -0.5,0.), (1.5,0.5,0.)}
  {\Coord coordinate (p\X) \pgfextra{\xdef\NumVertices{\X}}};
 %\message{number of vertices is \NumVertices^^J} 
 % normal of screen 
 \path[overlay] ({sin(\tdplotmaintheta)*sin(\tdplotmainphi)},
    {-1*sin(\tdplotmaintheta)*cos(\tdplotmainphi)},
    {cos(\tdplotmaintheta)}) coordinate (n)
    (0.5,0.5,{0.5*sqrt(2)})  coordinate (L); 
 \edef\lstPast{0}
 \foreach \poly in 
 {{17,11,9,15}, {14,8,10,16}, {22,24,21,18}, {12,5,2,6}, {13,19,23,20},
    {4,1,3,7}, {19,13,7,3, 8,14}, {15,9,4,7,13,20}, {16,10,5,12,18,21}, 
    {22,18,12,6,11,17}, {20,23,24,22,17,15}, {14,16,21,24,23, 19}, {9,11,6,2,1,4},  
    {3,1,2,5,10,8}}
 {
  \pgfmathtruncatemacro{\ione}{{\poly}[0]}
  \pgfmathtruncatemacro{\itwo}{{\poly}[1]}
  \pgfmathtruncatemacro{\ithree}{{\poly}[2]}
  \path[overlay,3d coordinate={(dA)=(p\itwo)-(p\ione)},
   3d coordinate={(dB)=(p\itwo)-(p\ithree)},
   3d coordinate={(nA)=(dA)x(dB)}] ;
  \pgfmathtruncatemacro{\jtest}{sign(TD("(nA)o(p\ione)"))}
  % make sure that the normal points outwards
  \ifnum\jtest<0
   \path[overlay,3d coordinate={(nA)=(dB)x(dA)}];
  \fi
  % compute projection the normal of the polygon on the normal of screen    
  \pgfmathsetmacro\myproj{TD("(nA)o(n)")}
  \pgfmathsetmacro\lproj{TD("(nA)o(L)")}
  \pgfmathtruncatemacro{\itest}{sign(\myproj)}
  \pgfmathtruncatemacro{\cf}{70+20*\lproj}% color fraction between 50 and 90
  \ifnum\itest>-1 
   \draw[ultra thin] [fill=mypolyhedroncolor!\cf]
   plot[samples at=\poly,variable=\x](p\x) -- cycle; 
  \else
   \begin{scope}[on background layer] 
    \draw[gray!20,ultra thin] [fill=mypolyhedroncolor!\cf!black]
    plot[samples at=\poly,variable=\x](p\x) -- cycle;  
   \end{scope}
  \fi
  }  
\end{tikzpicture}}
\colorlet{mypolyhedroncolor}{blue}
\sbox\TruncatedOctahedronBlue{\TruncatedOctahedron}
\colorlet{mypolyhedroncolor}{orange}
\sbox\TruncatedOctahedronOrange{\TruncatedOctahedron}
\begin{document}
\begin{tikzpicture}[tdplot_main_coords,line cap=round,line join=round]
\path foreach \Y in {0,1,2} {foreach \X in {0,1,2} 
  {({2*\Y}, {2*\X}, {-sqrt(2)*\X-sqrt(2)*\Y})
  node{\pgfmathtruncatemacro{\Z}{\X+\Y}
  \ifodd\Z
  \usebox{\TruncatedOctahedronOrange}
  \else
  \usebox{\TruncatedOctahedronBlue}
  \fi} }};
\end{tikzpicture}
\end{document}

To illustrate the point that this is a tessellation, one may want to draw them one by one.

\documentclass[tikz,border=3mm]{standalone}
\usepackage{tikz-3dplot}
\usetikzlibrary{backgrounds,3dtools}
\newsavebox\TruncatedOctahedronBlue
\newsavebox\TruncatedOctahedronOrange
\tdplotsetmaincoords{80}{105}
\newcommand{\TruncatedOctahedron}{%
\begin{tikzpicture}[tdplot_main_coords,line cap=round,line join=round]
\path foreach \Coord [count=\X] in 
{(-1.5,-0.5,0.), (-1.5,0.5,0.), (-1.,-1.,-0.707107), (-1.,-1.,0.707107), 
(-1.,1.,-0.707107), (-1.,1.,0.707107), (-0.5,-1.5,0.), (-0.5, -0.5,-1.41421),
(-0.5,-0.5,1.41421), (-0.5,0.5,-1.41421), (-0.5,0.5, 1.41421), (-0.5,1.5,0.),
(0.5,-1.5,0.), (0.5,-0.5,-1.41421), (0.5,-0.5, 1.41421), (0.5,0.5,-1.41421),
(0.5,0.5,1.41421), (0.5,1.5,0.), (1.,-1., -0.707107), (1.,-1.,0.707107),
(1.,1.,-0.707107), (1.,1.,0.707107), (1.5, -0.5,0.), (1.5,0.5,0.)}
  {\Coord coordinate (p\X) \pgfextra{\xdef\NumVertices{\X}}};
 %\message{number of vertices is \NumVertices^^J} 
 % normal of screen 
 \path[overlay] ({sin(\tdplotmaintheta)*sin(\tdplotmainphi)},
    {-1*sin(\tdplotmaintheta)*cos(\tdplotmainphi)},
    {cos(\tdplotmaintheta)}) coordinate (n)
    (0.5,0.5,{0.5*sqrt(2)})  coordinate (L); 
 \edef\lstPast{0}
 \foreach \poly in 
 {{17,11,9,15}, {14,8,10,16}, {22,24,21,18}, {12,5,2,6}, {13,19,23,20},
    {4,1,3,7}, {19,13,7,3, 8,14}, {15,9,4,7,13,20}, {16,10,5,12,18,21}, 
    {22,18,12,6,11,17}, {20,23,24,22,17,15}, {14,16,21,24,23, 19}, {9,11,6,2,1,4},  
    {3,1,2,5,10,8}}
 {
  \pgfmathtruncatemacro{\ione}{{\poly}[0]}
  \pgfmathtruncatemacro{\itwo}{{\poly}[1]}
  \pgfmathtruncatemacro{\ithree}{{\poly}[2]}
  \path[overlay,3d coordinate={(dA)=(p\itwo)-(p\ione)},
   3d coordinate={(dB)=(p\itwo)-(p\ithree)},
   3d coordinate={(nA)=(dA)x(dB)}] ;
  \pgfmathtruncatemacro{\jtest}{sign(TD("(nA)o(p\ione)"))}
  % make sure that the normal points outwards
  \ifnum\jtest<0
   \path[overlay,3d coordinate={(nA)=(dB)x(dA)}];
  \fi
  % compute projection the normal of the polygon on the normal of screen    
  \pgfmathsetmacro\myproj{TD("(nA)o(n)")}
  \pgfmathsetmacro\lproj{TD("(nA)o(L)")}
  \pgfmathtruncatemacro{\itest}{sign(\myproj)}
  \pgfmathtruncatemacro{\cf}{70+20*\lproj}% color fraction between 50 and 90
  \ifnum\itest>-1 
   \draw[ultra thin] [fill=mypolyhedroncolor!\cf]
   plot[samples at=\poly,variable=\x](p\x) -- cycle; 
  \else
   \begin{scope}[on background layer] 
    \draw[gray,ultra thin] [fill=mypolyhedroncolor!\cf!black]
    plot[samples at=\poly,variable=\x](p\x) -- cycle;  
   \end{scope}
  \fi
  }  
\end{tikzpicture}}
\colorlet{mypolyhedroncolor}{blue}
\sbox\TruncatedOctahedronBlue{\TruncatedOctahedron}
\colorlet{mypolyhedroncolor}{orange}
\sbox\TruncatedOctahedronOrange{\TruncatedOctahedron}
\begin{document}
\foreach \Ani in {1,...,27}
{\begin{tikzpicture}[tdplot_main_coords,line cap=round,line join=round]
\path[tdplot_screen_coords] (-3,-8.2) rectangle (10,4.5);
\path foreach \Y in {0,1,2} {foreach \Z in {0,1,2} 
    {foreach \X in {0,1,2} 
  {({2*\Y}, {2*\X+2*\Z},
  {-sqrt(2)*\X-sqrt(2)*\Y+sqrt(2)*\Z})
  node{\pgfmathtruncatemacro{\QQ}{\X+3*\Z+9*\Y}
  \ifnum\Ani>\QQ
  \ifodd\QQ
  \usebox{\TruncatedOctahedronOrange}
  \else
  \usebox{\TruncatedOctahedronBlue}
  \fi
  \fi} }}};
\end{tikzpicture}}
\end{document}