Jtdylktuy

During the current review of the tensors I have arrived at a page of Wikipedia where you can see the symbol of Levi-Civita in a beautiful three-dimensional matrix.

I hope that nobody will be angry with me if I do not produce any MWE but for me it would be nice to see the construction of a matrix so made and can be made available to other users.

More or less:

documentclass[tikz,border=2mm]{standalone} 

usetikzlibrary{positioning, matrix}

usepackage{amsmath}



newcommand{arrayfilling}[2]{

fill[#2!30, opacity=.5] ([shift={(1mm,1mm)}]#1.north west) coordinate(#1auxnw)--([shift={(1mm,1mm)}]#1.north east)coordinate(#1auxne) to[out=-75, in=75] ([shift={(1mm,-1mm)}]#1.south east)coordinate(#1auxse)--([shift={(1mm,-1mm)}]#1.south west)coordinate(#1auxsw) to[out=105, in=-105] cycle;

fill[#2!80!black, opacity=1] (#1auxne) to[out=-75, in=75] (#1auxse) to[out=78, in=-78] cycle;

fill[#2!80!black, opacity=1] (#1auxnw) to[out=-105, in=105] (#1auxsw) to[out=102, in=-102] cycle;

}





begin{document}

begin{tikzpicture}[font=ttfamily,

    mymatrix/.style={

        matrix of math nodes, inner sep=0pt, color=#1,

        column sep=-pgflinewidth, row sep=-pgflinewidth, anchor=south west,   

        nodes={anchor=center, minimum width=5mm, 

            minimum height=3mm, outer sep=0pt, inner sep=0pt, 

            text width=5mm, align=right,

            draw=none, font=small},

}

]



matrix (C) [mymatrix=green] at (6mm,5mm) 

{0 & 1 & 0 \ -1 & 0 & 0\ 0 & 0 & 0\};

arrayfilling{C}{green}



matrix (B) [mymatrix=red] at (3mm,2.5mm)

{0 & 0 & -1 \ 0 & 0 & 0\ 1 & 0 & 0\};

arrayfilling{B}{red}



matrix (A) [mymatrix=blue] at (0,0)

{0 & 0 & 0 \ 0 & 0 & 1\ 0 & -1 & 0\};

arrayfilling{A}{blue}



foreach i in {auxnw, auxne, auxse, auxsw}

draw[brown, ultra thin] (Ai)--(Ci);



node[below left=-1mm and 5mm of B.west] {$epsilon_{ijk} =$};

end{tikzpicture}

end{document}

Something like that?

documentclass[tikz,border=3.14mm]{standalone}

usepackage{mathtools}

usetikzlibrary{matrix,backgrounds,3d}

usepackage{tikz-3dplot}

%definecolor{mygreen}{RGB}{12,252,12}

begin{document}

tdplotsetmaincoords{75}{20}

begin{tikzpicture}[tdplot_main_coords]

  begin{scope}[canvas is xz plane at y=1,transform shape]

    node[inner sep=0pt,text=green!70!black,opacity=0.8] (mat1)

    {$displaystylebegin{pmatrix*}[r]

        0 & 1 & 0 \

        -1 & 0 & 0 \

        0 & 0 & 0 \

    end{pmatrix*}$};

    begin{scope}[on background layer]

     fill[green!70!black,opacity=0.2] ([xshift=8.5pt]mat1.south west) 

     coordinate (blb) to[out=140,in=-140,looseness=0.7]

     ([xshift=8.5pt]mat1.north west) coordinate (tlb) -- 

     ([xshift=-8.5pt]mat1.north east) coordinate (trb)

      to[out=-40,in=40,looseness=0.7] ([xshift=-8.5pt]mat1.south east)

      coordinate (brb)

       -- cycle;

    end{scope} 

  end{scope}

  %

  begin{scope}[canvas is xz plane at y=0,transform shape]

    node[inner sep=0pt,text=red,opacity=0.8] (mat2) {$displaystyle

    begin{pmatrix*}[r]

        0 & 0 & -1 \

        0 & 0 & 0 \

        1 & 0 & 0 \

    end{pmatrix*}$};

    begin{scope}[on background layer]

     fill[red,opacity=0.2] ([xshift=8.5pt]mat2.south west) to[out=140,in=-140,looseness=0.7]

     ([xshift=8.5pt]mat2.north west) -- ([xshift=-8.5pt]mat2.north east) 

      to[out=-40,in=40,looseness=0.7] ([xshift=-8.5pt]mat2.south east) -- cycle;

    end{scope} 

  end{scope}

  %

  begin{scope}[canvas is xz plane at y=-1,transform shape]

    node[inner sep=0pt,text=blue,opacity=0.8] (mat3) {$displaystyle

    begin{pmatrix*}[r]

        0 & 0 & 0 \

        0 & 0 & 1 \

        0 & -1 & 0 \

    end{pmatrix*}$};

    begin{scope}[on background layer]

     fill[blue,opacity=0.2]

      ([xshift=8.5pt]mat3.south west) coordinate (blf)

      to[out=140,in=-140,looseness=0.7]

     ([xshift=8.5pt]mat3.north west) coordinate (tlf) 

     -- ([xshift=-8.5pt]mat3.north east) coordinate (trf)

      to[out=-40,in=40,looseness=0.7] ([xshift=-8.5pt]mat3.south east) 

      coordinate (brf) -- cycle;

    end{scope} 

  end{scope}

  foreach X in {tl,tr,br}

  {draw[thin,orange] (X f) -- (X b);}

  begin{scope}[on background layer]

   draw[thin,orange] (blf) -- (blb);

  end{scope}

node[left] at (mat3.west) {$varepsilon_{ijk}=$};

end{tikzpicture}

end{document}

EDIT: Aligned the entries right, big thanks to Barbara Beeton. (I just wonder why no one complained that the Levi-Civita tensor is not a tensor, but a tensor density. ;-)

2nd EDIT: Response to Anush's comment (well-taken! ;-).

documentclass[tikz,border=3.14mm]{standalone}

usepackage{mathtools}

usetikzlibrary{matrix,backgrounds,3d}

usepackage{tikz-3dplot}

begin{document}

tdplotsetmaincoords{75}{20}

begin{tikzpicture}[tdplot_main_coords]

  begin{scope}[canvas is xz plane at y=1,transform shape]

    node[inner sep=0pt,text=green!70!black,opacity=0.8] (mat1)

    {$displaystylebegin{pmatrix*}[r]

        0 & hphantom{-}1 & hphantom{-}0 \

        -1 & 0 & 0 \

        0 & 0 & 0 \

    end{pmatrix*}$};

    begin{scope}[on background layer]

     fill[green!70!black,opacity=0.2] ([xshift=8.5pt]mat1.south west) 

     coordinate (blb) to[out=140,in=-140,looseness=0.7]

     ([xshift=8.5pt]mat1.north west) coordinate (tlb) -- 

     ([xshift=-8.5pt]mat1.north east) coordinate (trb)

      to[out=-40,in=40,looseness=0.7] ([xshift=-8.5pt]mat1.south east)

      coordinate (brb)

       -- cycle;

    end{scope} 

  end{scope}

  %

  begin{scope}[canvas is xz plane at y=0,transform shape]

    node[inner sep=0pt,text=red,opacity=0.8] (mat2) {$displaystyle

    begin{pmatrix*}[r]

        hphantom{-}0 & hphantom{-}0 & -1 \

        0 & 0 & 0 \

        1 & 0 & 0 \

    end{pmatrix*}$};

    begin{scope}[on background layer]

     fill[red,opacity=0.2] ([xshift=8.5pt]mat2.south west) to[out=140,in=-140,looseness=0.7]

     ([xshift=8.5pt]mat2.north west) -- ([xshift=-8.5pt]mat2.north east) 

      to[out=-40,in=40,looseness=0.7] ([xshift=-8.5pt]mat2.south east) -- cycle;

    end{scope} 

  end{scope}

  %

  begin{scope}[canvas is xz plane at y=-1,transform shape]

    node[inner sep=0pt,text=blue,opacity=0.8] (mat3) {$displaystyle

    begin{pmatrix*}[r]

        hphantom{-}0 & 0 & hphantom{-}0 \

        0 & 0 & 1 \

        0 & -1 & 0 \

    end{pmatrix*}$};

    begin{scope}[on background layer]

     fill[blue,opacity=0.2]

      ([xshift=8.5pt]mat3.south west) coordinate (blf)

      to[out=140,in=-140,looseness=0.7]

     ([xshift=8.5pt]mat3.north west) coordinate (tlf) 

     -- ([xshift=-8.5pt]mat3.north east) coordinate (trf)

      to[out=-40,in=40,looseness=0.7] ([xshift=-8.5pt]mat3.south east) 

      coordinate (brf) -- cycle;

    end{scope} 

  end{scope}

  foreach X in {tl,tr,br}

  {draw[thin,orange] (X f) -- (X b);}

  begin{scope}[on background layer]

   draw[thin,orange] (blf) -- (blb);

  end{scope}

  begin{scope}[canvas is xz plane at y=0,transform shape]

   node[left] at (mat2.west -| mat3.west) {$varepsilon_{ijk}=$};

  end{scope}

end{tikzpicture}

end{document}