How might I typeset the Fano plane in LaTeX?

Question

I'm sorry if this is a duplicate in any way or is otherwise a lousy question.

How might I typeset the Fano plane in LaTeX?

In particular, I need a vector graphic of it and I would prefer to use the tikz package. I don't want to copy & paste someone's answer either: I'd like to be able to customise it.

Here's a picture of the Fano plane for convenience:

I'm afraid I haven't got a clue. I can use tikz-cd for the average commutative diagram but that's about it.

Please help :)

Answer 1

Using some basic geometry and the calc library, you can leave all the calculations to TikZ:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc}

\newcommand\FanoPlane[1][1cm]{%
\begin{tikzpicture}[
mydot/.style={
  draw,
  circle,
  fill=black,
  inner sep=1.5pt}
]
\draw
  (0,0) coordinate (A) --
  (#1,0) coordinate (B) --
  ($ (A)!.5!(B) ! {sin(60)*2} ! 90:(B) $) coordinate (C) -- cycle;
\coordinate (O) at
  (barycentric cs:A=1,B=1,C=1);
\draw (O) circle [radius=#1*1.717/6];
\draw (C) -- ($ (A)!.5!(B) $) coordinate (LC); 
\draw (A) -- ($ (B)!.5!(C) $) coordinate (LA); 
\draw (B) -- ($ (C)!.5!(A) $) coordinate (LB); 
\foreach \Nodo in {A,B,C,O,LC,LA,LB}
  \node[mydot] at (\Nodo) {};    
\end{tikzpicture}%
}

\begin{document}

\FanoPlane\quad\FanoPlane[2cm]\quad\FanoPlane[3cm]

\end{document}

Answer 2

Not that much different from the others really...

\documentclass[tikz,border=5]{standalone}
\begin{document}
\tikz[every node/.style={circle, fill}]
  \draw circle [radius=1] (90:2) -- (210:2) -- (330:2) -- cycle (0,0) node {}
    \foreach \i in {90,210,330}{ (\i:2) node {} -- (\i+180:1) node {} };
\end{document}

Answer 3

Here are two short solutions:

\documentclass[tikz]{standalone}
\begin{document}
\begin{tikzpicture}
  \draw (30:1)  -- (210:2)
        (150:1) -- (330:2)
        (270:1) -- (90:2)
        (90:2)  -- (210:2) -- (330:2) -- cycle
        (0:0)   circle (1);
  \fill (0:0)   circle(3pt)
        (30:1)  circle(3pt)
        (90:2)  circle(3pt)
        (150:1) circle(3pt)
        (210:2) circle(3pt)
        (270:1) circle(3pt)
        (330:2) circle(3pt);
\end{tikzpicture}
\end{document}

\documentclass[tikz]{standalone}
\begin{document}
\begin{tikzpicture}
  \draw \foreach \a in {30,150,270}{(\a:1)  -- (180+\a:2)}
        (90:2)  -- (210:2) -- (330:2) -- cycle
        (0:0) circle (1);
  \fill \foreach \p in {(0:0),(30:1),(90:2),(150:1),(210:2),(270:1),(330:2)}
        {\p circle(3pt)};
\end{tikzpicture}
\end{document}

Answer 4

I read this answer on the site and tried to draw what you are questioning.

%pdflatex
\documentclass[margin=0.5mm]{standalone}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}

   \draw (0,0) circle (0.5);
   \draw (90:1) -- (-30:1)--(210:1)--cycle;

   \draw (90:1)--(0,0);
   \draw (210:1)--(0,0);
   \draw (-30:1)--(0,0);

   \draw (30:0.5)--(0,0);
   \draw (150:0.5)--(0,0);
   \draw (270:0.5)--(0,0);

   \fill (-0.866,-0.5) circle (1.5pt);
   \fill (0.866,-0.5) circle (1.5pt);
   \fill (0,-0.5) circle (1.5pt);
   \fill (0,1) circle (1.5pt);
   \fill (0,0) circle (1.5pt);
   \fill (0.433,0.25) circle (1.5pt);
   \fill (-0.433,0.25) circle (1.5pt);

\end{tikzpicture}
\end{document}

Answer 5

Another solution based on pstricks, with a short code thands to the pst-poly package. The pdf option allows for pdfLaTeX compilation, provided you set the --shell-escape switch for TeX Live or MacTeX, --enable-write18 for MiKTeX:

\documentclass[12pt, pdf, x11names]{standalone}

\usepackage{pst-poly}

\begin{document}
\psset{unit = 2cm,dimen = m}
\begin{pspicture*}(-1.25,-1)(1.25,1.1)
\providecommand{\PstPolygonNode}{%
\psdots[dotstyle = o, dotsize=4pt,fillstyle = solid, fillcolor=cyan](1;\INode)}
\psset{ linewidth = 0.6pt}
\rput{-30}(0,0){\PstTriangle[PolyName =A]}
\pscircle(A0){0.5}
 \rput{150}(0,0){\PstTriangle[unit=0.5, PolyOffset=2,PolyName=I, linestyle=none]}
 \multido{\i = 1 + 1}{3}{\ncline[nodesep = 2pt]{A\i}{I\i}}
 \psdot[dotstyle = o, dotsize=4pt,fillstyle = solid, fillcolor=cyan](A0)
\end{pspicture*}

\end{document} 

Answer 6

A PSTricks solution:

\documentclass{article}

\usepackage{pstricks-add}
\usepackage{xfp}

\begin{document}

\def\radius{3}
\begin{pspicture}(-0.1,-0.1)(\fpeval{2*sqrt(3)*\radius+0.1},\fpeval{3*\radius+0.1})
 \psset{
   dimen = m,
   linewidth = 1pt
 }
  \pnodes{P}%
    (\fpeval{sqrt(3)*\radius},\radius)%
    (0,0)%
    (\fpeval{0.5*sqrt(3)*\radius},\fpeval{1.5*\radius})%
    (\fpeval{sqrt(3)*\radius},\fpeval{3*\radius})%
    (\fpeval{1.5*sqrt(3)*\radius},\fpeval{1.5*\radius})%
    (\fpeval{2*sqrt(3)*\radius},0)%
    (\fpeval{sqrt(3)*\radius},0)
  \pspolygon(P1)(P3)(P5)
  \multido{\iA = 1+1, \iB = 4+1}{3}{\psline(P\iA)(P\iB)}
  \multido{\i = 0+1}{7}{\psdot[dotsize = 2pt 5](P\i)}
  \pscircle(P0){\radius}
\end{pspicture}

\end{document}

All you have to do is choose the value of \radius and the drawing will be adjusted accoringly.

Notes:

• Either pst-poly (see Bernard's answer) or pst-eucl can also do the job but I chose to actually calculate the coordinates of all the relevant points.

• The 'extra' 0.1s at the boundry are because of the dots.

Answer 7

Here's another variation using the through library for the circle:

\documentclass[tikz]{standalone}
\usetikzlibrary{through,shapes.geometric,calc}

\begin{document}

  \begin{tikzpicture}
    \node (triag) [draw, regular polygon, regular polygon sides=3, minimum width=20mm] {};
    \path [fill] circle (.5mm);
    \foreach \i/\j/\k in {1/2/3,2/3/1,3/1/2}
      \coordinate  (triag side \i) at ($(triag.corner \j)!1/2!(triag.corner \k)$);
    \foreach \i in {1,2,3}
      {
        \path [fill] (triag.corner \i) circle (.5mm) (triag side \i) circle (.5mm);
        \draw (triag.corner \i) -- (triag side \i);
      }
    \node [draw] [circle through={(triag side 1)}] {};
  \end{tikzpicture}

\end{document}

Some fancier pics:

\documentclass[tikz]{standalone}
\usetikzlibrary{through,shapes.geometric,calc}

\begin{document}

  \tikzset{
    pics/fano/.style n args={2}{%
      code={
        \node (triag) [draw, regular polygon, regular polygon sides=3, minimum width=#1, pic actions] {};
        \path [fill, pic actions] circle (#2);
        \foreach \i/\j/\k in {1/2/3,2/3/1,3/1/2}
          \coordinate  (triag side \i) at ($(triag.corner \j)!1/2!(triag.corner \k)$);
        \foreach \i in {1,2,3}
          {
            \path [fill, pic actions] (triag.corner \i) circle (#2) (triag side \i) circle (#2);
            \draw [pic actions] (triag.corner \i) -- (triag side \i);
          }
        \node [draw, pic actions] [circle through={(triag side 1)}] {};
      },
    },
  }

  \begin{tikzpicture}
    \draw pic {fano={25mm}{.75mm}};
    \draw [xshift=25mm, blue] pic {fano={15mm}{.5mm}};
    \draw [xshift=40mm, green] pic {fano={10mm}{.25mm}};
  \end{tikzpicture}

\end{document}

Answer 8

We had better have a Metapost solution as well. I've added the conventional numbering to the points.

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

beginfig(1);

z1 = (0,4cm);
z2 = z1 rotated 120; 
z4 = z2 rotated 120;
z3 = .5[z1,z2];
z5 = .5[z1,z4];
z6 = .5[z2,z4];
z7 = origin;

draw z1 -- z2 -- z4 -- cycle;
draw z3 .. z5 .. z6 .. cycle;

draw z1 -- z6;
draw z2 -- z5;
draw z3 -- z4;

for i=1 upto 7: 
  fill fullcircle scaled 12 shifted z[i]; 
  label(decimal i infont "phvr8r", z[i]) withcolor white;
endfor

endfig;
end.

Note that drawing through three points with .. approximates a circular arc in plain MP.

Answer 9

Here's another variation using only nodes with any computation

\documentclass{scrartcl}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric}

\begin{document}
\begin{tikzpicture}

\node[blue,circle, draw,inner sep=0,minimum size=3cm,outer sep=0] (c) at(0,0){};
\node[regular polygon, regular polygon sides=3, draw,
inner sep=0,minimum size=6cm,red] (t) at (0.0,0) {};

\foreach \nn in {c.30,c.150,c.270,t.90,t.210,t.330}{
\node[fill=black,circle] at(\nn){};
}
\draw[green] (c.30) -- (t.210) (c.150) -- (t.330)  (c.270) -- (t.90);
\end{tikzpicture}

\end{document}

Answer 10

The octonions in quaternion pairs,

\begin{center}
\newcommand*\circled[1]{
%\tikz[baseline=(char.base)]{
%            \node[shape=circle,draw=white,inner sep=0pt,fill=white] (char) 
%{
\small{\(#1\)}
%}
%;}
}
\begin{tikzpicture}
\begin{scope}[very thick,decoration={
    markings,
    mark=at position 0.5 with {\arrow{>}}}
    ] 
\draw[draw=white, double=black, thick,->] (0,0) circle (1);
\draw[thin,->] (1,0) -- (1,0.0001);
\draw[thin,->] (-1,0) -- (-1,-0.0001);

\draw[thin,->] ({-1/sqrt(2)},{-1/sqrt(2)}) -- ({-1/sqrt(2)+0.001},{-1/sqrt(2)-0.001});
\draw[thin,->] ({1/sqrt(2)},{-1/sqrt(2)}) -- ({1/sqrt(2)+0.001},{-1/sqrt(2)+0.001});

\node (0) at (0,0) {\circled{(0,1)}};

\foreach \i/\j/\k in {0/{(0,i)}/{(k,0)},1/{(0,j)}/{(i,0)},2/{(0,k)}/{(j,0)}}{%
    \node[fill=white] (\i;0) at ({2*cos(-30+120*\i)},{2*sin(-30+120*\i)}) {\(\j\)};
    \node[fill=white] (\i;1) at ({cos(30+120*\i)},{sin(30+120*\i)}) {\(\k\)};
}
\foreach \i/\j/\k/\l in {0/0/0/1,0/1/1/0,1/0/1/1,1/1/2/0,2/0/2/1,2/1/0/0}{%
    \draw[thin,postaction={decorate}] (\i;\j) -- (\k;\l);
}
\draw[thin,postaction={decorate}] (0) -- (0;1);
\draw[thin,postaction={decorate}] (0) -- (1;1);
\draw[thin,postaction={decorate}] (2;1) -- (0);
\draw[thin,postaction={decorate}] (0) -- (0;0);
\draw[thin,postaction={decorate}] (0) -- (1;0);
\draw[thin,postaction={decorate}] (0) -- (2;0);
%\draw[thin,postaction={decorate}] (1) -- (7);
%\draw[thin,postaction={decorate}] (2) -- (4);
%\draw[thin,postaction={decorate}] (2) -- (5);
%\draw[thin,postaction={decorate}] (3) -- (4);
%\draw[thin,postaction={decorate}] (3) -- (6);
%\draw[thin,postaction={decorate}] (4) -- (5);
%\draw[thin,postaction={decorate}] (4) -- (6);
%\draw[thin,postaction={decorate}] (4) -- (7);
%\draw[thin,postaction={decorate}] (5) -- (3);
%\draw[thin,postaction={decorate}] (6) -- (1);
%\draw[thin,postaction={decorate}] (7) -- (2);
\end{scope}
\end{tikzpicture}
\end{center}
Rotating the diagram by \(\sfrac{1}{3}\) of a revolution gives an automorphism of \(\Oct{}\).
Rotating by \(\sfrac{1}{6}\) doesn't work: the inner arrows don't go the right way, but if we change \(e_4 \mapsto -e_4\), we fix that.
Similarly, if we flip the diagram by a reflection preserving the vertical axis, and then change the signs of \(e_6\) and \(e_2\) we find another automorphism.

We can also draw this diagram as a cube:
\begin{center}
\newcommand{\Depth}{1}
\newcommand{\Height}{1}
\newcommand{\Width}{1}
\tdplotsetmaincoords{70}{110}
\begin{tikzpicture}[tdplot_main_coords]
\coordinate (0) at (0,0,0);
\coordinate (1) at (\Depth,0,\Height);
\coordinate (2) at (\Depth,\Width,0);
\coordinate (3) at (0,\Width,\Height);
\coordinate (4) at (\Depth,\Width,\Height);
\coordinate (5) at (0,\Width,0);
\coordinate (6) at (0,0,\Height);
\coordinate (7) at (\Depth,0,0);

\draw[gray!10,fill=gray!10] (0) -- (6) -- (1) -- (7) -- cycle;% Left Face
\draw[gray!10,fill=gray!30] (0) -- (7) -- (2) -- (5) -- cycle;% Bottom Face
\draw[gray!10,fill=gray!40] (0) -- (6) -- (3) -- (5) -- cycle;% Back Face
\draw[gray!10,fill=gray!20,opacity=0.6] (1) -- (4) -- (2) -- (7) -- cycle;% Front Face
\draw[gray!10,fill=gray!20,opacity=0.8] (1) -- (4) -- (3) -- (6) -- cycle;% Top Face
\draw[gray!10,fill=gray!20,opacity=0.8] (4) -- (3) -- (5) -- (2) -- cycle;% Right Face

\node at (1) {\small\(1\)};
\node at (2) {\small\(2\)};
\node at (3) {\small\(3\)};
\node at (4) {\small\(4\)};
\node at (5) {\small\(5\)};
\node at (6) {\small\(6\)};
\node at (7) {\small\(7\)};

%% Following is for debugging purposes so you can see where the points are
%% These are last so that they show up on top
%\foreach \xy in {O, A, B, C, D, E, F, G}{
%    \node at (\xy) {\xy};
%}
\end{tikzpicture}
\end{center}
with one vertex not marked.
If we let \(k=\Z{}/2\Z{}\), then the cube is the set of points in \(k^3\), or more precisely, since the unlabelled point is the origin and is deleted, the cube is the projective plane \(\Proj{2}_k\) over the finite field \(k\) of order 2.
The lines in this projective plane, suitably ordered, are
\begin{center}
\newcommand{\Depth}{1}
\newcommand{\Height}{1}
\newcommand{\Width}{1}
\tdplotsetmaincoords{70}{110}
\begin{tikzpicture}[tdplot_main_coords]%Highlight the bottom
\coordinate (0) at (0,0,0);
\coordinate (1) at (\Depth,0,\Height);
\coordinate (2) at (\Depth,\Width,0);
\coordinate (3) at (0,\Width,\Height);
\coordinate (4) at (\Depth,\Width,\Height);
\coordinate (5) at (0,\Width,0);
\coordinate (6) at (0,0,\Height);
\coordinate (7) at (\Depth,0,0);
%

\draw[gray!10,fill=gray!10] (0) -- (6) -- (1) -- (7) -- cycle;% Left Face
\draw[gray!10,fill=gray!110] (0) -- (7) -- (2) -- (5) -- cycle;% Bottom Face
\draw[gray!10,fill=gray!40] (0) -- (6) -- (3) -- (5) -- cycle;% Back Face
\draw[gray!10,fill=gray!20,opacity=0.6] (1) -- (4) -- (2) -- (7) -- cycle;% Front Face
\draw[gray!10,fill=gray!20,opacity=0.8] (1) -- (4) -- (3) -- (6) -- cycle;% Top Face
\draw[gray!10,fill=gray!20,opacity=0.8] (4) -- (3) -- (5) -- (2) -- cycle;% Right Face

\node at (1) {\small\(1\)};
\node at (2) {\small\(2\)};
\node at (3) {\small\(3\)};
\node at (4) {\small\(4\)};
\node at (5) {\small\(5\)};
\node at (6) {\small\(6\)};
\node at (7) {\small\(7\)};

%% Following is for debugging purposes so you can see where the points are
%% These are last so that they show up on top
%\foreach \xy in {O, A, B, C, D, E, F, G}{
%    \node at (\xy) {\xy};
%}
\end{tikzpicture}
\begin{tikzpicture}[tdplot_main_coords]%Highlight the left
\coordinate (0) at (0,0,0);
\coordinate (1) at (\Depth,0,\Height);
\coordinate (2) at (\Depth,\Width,0);
\coordinate (3) at (0,\Width,\Height);
\coordinate (4) at (\Depth,\Width,\Height);
\coordinate (5) at (0,\Width,0);
\coordinate (6) at (0,0,\Height);
\coordinate (7) at (\Depth,0,0);
%

\draw[gray!10,fill=gray!90] (0) -- (6) -- (1) -- (7) -- cycle;% Left Face
\draw[gray!10,fill=gray!30] (0) -- (7) -- (2) -- (5) -- cycle;% Bottom Face
\draw[gray!10,fill=gray!40] (0) -- (6) -- (3) -- (5) -- cycle;% Back Face
\draw[gray!10,fill=gray!20,opacity=0.6] (1) -- (4) -- (2) -- (7) -- cycle;% Front Face
\draw[gray!10,fill=gray!20,opacity=0.8] (1) -- (4) -- (3) -- (6) -- cycle;% Top Face
\draw[gray!10,fill=gray!20,opacity=0.8] (4) -- (3) -- (5) -- (2) -- cycle;% Right Face

\node at (1) {\small\(1\)};
\node at (2) {\small\(2\)};
\node at (3) {\small\(3\)};
\node at (4) {\small\(4\)};
\node at (5) {\small\(5\)};
\node at (6) {\small\(6\)};
\node at (7) {\small\(7\)};

%% Following is for debugging purposes so you can see where the points are
%% These are last so that they show up on top
%\foreach \xy in {O, A, B, C, D, E, F, G}{
%    \node at (\xy) {\xy};
%}
\end{tikzpicture}
\begin{tikzpicture}[tdplot_main_coords]%Highlight back face
\coordinate (0) at (0,0,0);
\coordinate (1) at (\Depth,0,\Height);
\coordinate (2) at (\Depth,\Width,0);
\coordinate (3) at (0,\Width,\Height);
\coordinate (4) at (\Depth,\Width,\Height);
\coordinate (5) at (0,\Width,0);
\coordinate (6) at (0,0,\Height);
\coordinate (7) at (\Depth,0,0);
%

\draw[gray!10,fill=gray!10] (0) -- (6) -- (1) -- (7) -- cycle;% Left Face
\draw[gray!10,fill=gray!30] (0) -- (7) -- (2) -- (5) -- cycle;% Bottom Face
\draw[gray!10,fill=gray!120] (0) -- (6) -- (3) -- (5) -- cycle;% Back Face
\draw[gray!10,fill=gray!20,opacity=0.6] (1) -- (4) -- (2) -- (7) -- cycle;% Front Face
\draw[gray!10,fill=gray!20,opacity=0.8] (1) -- (4) -- (3) -- (6) -- cycle;% Top Face
\draw[gray!10,fill=gray!20,opacity=0.8] (4) -- (3) -- (5) -- (2) -- cycle;% Right Face

\node at (1) {\small\(1\)};
\node at (2) {\small\(2\)};
\node at (3) {\small\(3\)};
\node at (4) {\small\(4\)};
\node at (5) {\small\(5\)};
\node at (6) {\small\(6\)};
\node at (7) {\small\(7\)};

%% Following is for debugging purposes so you can see where the points are
%% These are last so that they show up on top
%\foreach \xy in {O, A, B, C, D, E, F, G}{
%    \node at (\xy) {\xy};
%}
\end{tikzpicture}
\begin{tikzpicture}[tdplot_main_coords]
\coordinate (0) at (0,0,0);
\coordinate (1) at (\Depth,0,\Height);
\coordinate (2) at (\Depth,\Width,0);
\coordinate (3) at (0,\Width,\Height);
\coordinate (4) at (\Depth,\Width,\Height);
\coordinate (5) at (0,\Width,0);
\coordinate (6) at (0,0,\Height);
\coordinate (7) at (\Depth,0,0);
%

\draw[gray!10,fill=gray!10] (0) -- (6) -- (1) -- (7) -- cycle;% Left Face
\draw[gray!10,fill=gray!30] (0) -- (7) -- (2) -- (5) -- cycle;% Bottom Face
\draw[gray!10,fill=gray!40] (0) -- (6) -- (3) -- (5) -- cycle;% Back Face
\draw[gray!10,fill=gray!20,opacity=0.6] (1) -- (4) -- (2) -- (7) -- cycle;% Front Face
\draw[gray!10,fill=gray!20,opacity=0.8] (1) -- (4) -- (3) -- (6) -- cycle;% Top Face
\draw[gray!10,fill=gray!20,opacity=0.8] (4) -- (3) -- (5) -- (2) -- cycle;% Right Face

\draw[gray!10,fill=gray!80,opacity=0.8] (0) -- (6) -- (4) -- (2) -- cycle;

\node at (1) {\small\(1\)};
\node at (2) {\small\(2\)};
\node at (3) {\small\(3\)};
\node at (4) {\small\(4\)};
\node at (5) {\small\(5\)};
\node at (6) {\small\(6\)};
\node at (7) {\small\(7\)};

%% Following is for debugging purposes so you can see where the points are
%% These are last so that they show up on top
%\foreach \xy in {O, A, B, C, D, E, F, G}{
%    \node at (\xy) {\xy};
%}
\end{tikzpicture}
\begin{tikzpicture}[tdplot_main_coords]
\coordinate (0) at (0,0,0);
\coordinate (1) at (\Depth,0,\Height);
\coordinate (2) at (\Depth,\Width,0);
\coordinate (3) at (0,\Width,\Height);
\coordinate (4) at (\Depth,\Width,\Height);
\coordinate (5) at (0,\Width,0);
\coordinate (6) at (0,0,\Height);
\coordinate (7) at (\Depth,0,0);
%

\draw[gray!10,fill=gray!10] (0) -- (6) -- (1) -- (7) -- cycle;% Left Face
\draw[gray!10,fill=gray!30] (0) -- (7) -- (2) -- (5) -- cycle;% Bottom Face
\draw[gray!10,fill=gray!40] (0) -- (6) -- (3) -- (5) -- cycle;% Back Face
\draw[gray!10,fill=gray!20,opacity=0.6] (1) -- (4) -- (2) -- (7) -- cycle;% Front Face
\draw[gray!10,fill=gray!20,opacity=0.8] (1) -- (4) -- (3) -- (6) -- cycle;% Top Face
\draw[gray!10,fill=gray!20,opacity=0.8] (4) -- (3) -- (5) -- (2) -- cycle;% Right Face

\draw[gray!10,fill=gray!80,opacity=0.8] (0) -- (1) -- (4) -- (5) -- cycle;

\node at (1) {\small\(1\)};
\node at (2) {\small\(2\)};
\node at (3) {\small\(3\)};
\node at (4) {\small\(4\)};
\node at (5) {\small\(5\)};
\node at (6) {\small\(6\)};
\node at (7) {\small\(7\)};

%% Following is for debugging purposes so you can see where the points are
%% These are last so that they show up on top
%\foreach \xy in {O, A, B, C, D, E, F, G}{
%    \node at (\xy) {\xy};
%}
\end{tikzpicture}
{
\tdplotsetmaincoords{50}{110}
\begin{tikzpicture}[tdplot_main_coords]
\coordinate (0) at (0,0,0);
\coordinate (1) at (\Depth,0,\Height);
\coordinate (2) at (\Depth,\Width,0);
\coordinate (3) at (0,\Width,\Height);
\coordinate (4) at (\Depth,\Width,\Height);
\coordinate (5) at (0,\Width,0);
\coordinate (6) at (0,0,\Height);
\coordinate (7) at (\Depth,0,0);

\draw[gray!10,fill=gray!10] (0) -- (6) -- (1) -- (7) -- cycle;% Left Face
\draw[gray!10,fill=gray!30] (0) -- (7) -- (2) -- (5) -- cycle;% Bottom Face
\draw[gray!10,fill=gray!40] (0) -- (6) -- (3) -- (5) -- cycle;% Back Face
\draw[gray!10,fill=gray!20,opacity=0.6] (1) -- (4) -- (2) -- (7) -- cycle;% Front Face
\draw[gray!10,fill=gray!20,opacity=0.8] (1) -- (4) -- (3) -- (6) -- cycle;% Top Face
\draw[gray!10,fill=gray!20,opacity=0.8] (4) -- (3) -- (5) -- (2) -- cycle;% Right Face

\draw[gray!10,fill=gray!80,opacity=0.8] (0) -- (7) -- (4) -- (3) -- cycle;

\node at (1) {\small\(1\)};
\node at (2) {\small\(2\)};
\node at (3) {\small\(3\)};
\node at (4) {\small\(4\)};
\node at (5) {\small\(5\)};
\node at (6) {\small\(6\)};
\node at (7) {\small\(7\)};

%% Following is for debugging purposes so you can see where the points are
%% These are last so that they show up on top
%\foreach \xy in {O, A, B, C, D, E, F, G}{
%    \node at (\xy) {\xy};
%}
\end{tikzpicture}
}
\begin{tikzpicture}[tdplot_main_coords]
\coordinate (0) at (0,0,0);
\coordinate (1) at (\Depth,0,\Height);
\coordinate (2) at (\Depth,\Width,0);
\coordinate (3) at (0,\Width,\Height);
\coordinate (4) at (\Depth,\Width,\Height);
\coordinate (5) at (0,\Width,0);
\coordinate (6) at (0,0,\Height);
\coordinate (7) at (\Depth,0,0);

\draw[gray!10,fill=gray!10] (0) -- (6) -- (1) -- (7) -- cycle;% Left Face
\draw[gray!10,fill=gray!30] (0) -- (7) -- (2) -- (5) -- cycle;% Bottom Face
\draw[gray!10,fill=gray!40] (0) -- (6) -- (3) -- (5) -- cycle;% Back Face
\draw[gray!10,fill=gray!20,opacity=0.6] (1) -- (4) -- (2) -- (7) -- cycle;% Front Face
\draw[gray!10,fill=gray!20,opacity=0.8] (1) -- (4) -- (3) -- (6) -- cycle;% Top Face
\draw[gray!10,fill=gray!20,opacity=0.8] (4) -- (3) -- (5) -- (2) -- cycle;% Right Face

\draw[gray!10,fill=gray!80,opacity=0.8] (1) -- (3) -- (2) -- cycle;

\node at (1) {\small\(1\)};
\node at (2) {\small\(2\)};
\node at (3) {\small\(3\)};
%\node at (4) {\small\(4\)};
\node at (5) {\small\(5\)};
\node at (6) {\small\(6\)};
\node at (7) {\small\(7\)};
\end{tikzpicture}
\end{center}
Each line represents a copy of the quaternions living inside \(\Oct{}\), and if we label the lines with arrows, they recover our multiplication table.