It is sometimes very helpful to draw 3-dimensional polytope in a paper. TikZ is a very versatile tool to draw in scientific documents and Sage can deal easily with 3-dimensional polytopes. Finally sagetex makes everything work together nicely between Sage, tikz and latex. Since version 6.3 of Sage, there is a function for (projection of) polytopes to output a TikZ picture of the polytope. This short tutorial shows how it all works.

To put an image of a 3d-polytope in latex using TikZ and Sage, simply follow the instructions:

  • Install Sage 6.3 or higher
  • Install SageTex (optionnal but recommended!)
  • Put \usepackage{tikz} in the preamble of your article
  • Open Sage and change the directory to your article's by the command cd /path/to/article
  • Input your polytope, called P, to Sage
  • Visualize the polytope P using the command P.show(aspect_ratio=1)
  • Once the wished view angle is found, right click on the image and select Console
  • In the dialog box click the button State
  • Scroll up to the line starting with moveto
  • It reads something like moveto 0.0 {x y z angle} scale
  • Go back to Sage and type Img = P.projection().tikz([x,y,z],angle)
  • Img contains a Sage object of type LatexExpr containing the raw tikz picture of your polytope

Then, you can either copy-paste it to your article by typing Img in Sage or save it to a file, by doing

f=open('Img_poly.tex','w')
f.write(Img)
f.close()

Then in the pwd (present working directory of sage, the one of your article) you will have a file named Img_poly.tex containing the tikzpicture of your polytope

You can customize the polytope using the following options in the command P.tikz()

  • scale : positive number to scale the polytope
  • edge_color : string (default: blue!95!black) representing colors which tikz recognize
  • facet_color : string (default: blue!95!black) representing colors which tikz recognize
  • vertex_color : string (default: green) representing colors which tikz recognize
  • opacity : real number (default: 0.8) between 0 and 1 giving the opacity of the front facets.
  • axis : Boolean (default: False) draw the axes at the origin or not.

Example

Let's say I want to draw the polar dual of the following (nice!) polytope given by the following list of vertices:

[[1,0,1],[1,0,0],[1,1,0],[0,0,-1],[0,1,0],[-1,0,0],[0,1,1],[0,0,1],[0,-1,0]]

In Sage, I type:

P=Polyhedron(vertices=[[1,0,1],[1,0,0],[1,1,0],[0,0,-1],[0,1,0],[-1,0,0],[0,1,1],[0,0,1],[0,-1,0]]).polar()

Then, I visualize the polytope by typing P.show(aspect_ratio=1)

When I found a good angle, I follow the above procedure to obtain the values [674,108,-731] and angle=112

The image corresponding to the code Img=P.projection().tikz([674,108,-731],112) is

A 3-dim. associahedron

Then, I may want to customize using the command

Img=P.projection().tikz([674,108,-731],112,scale=2, edge_color='orange',facet_color='red',vertex_color='blue',opacity=0.4)

which gives the image (the scaling difference do not appear here)

A 3-dim. associahedron

Further, I may want to edit deeper the style of the polytope inside the tikzpicture. For example, line 6-9 in the tikzpicture reads:

back/.style={loosely dotted, thin},
edge/.style={color=orange, thick},
facet/.style={fill=red,fill opacity=0.400000},
vertex/.style={inner sep=1pt,circle,draw=blue!25!black,fill=blue!75!black,thick,anchor=base}]

I can replace it by the following 4 lines (and adding \usetikzlibrary{shapes} in the preamble)

back/.style={loosely dashed,line width=2pt},
edge/.style={color=yellow, line width=2pt},
facet/.style={fill=cyan,fill opacity=0.400000},
vertex/.style={inner sep=4pt,star,star points=7,draw=blue!75!white,fill=blue!85!white,thick,anchor=base}]

to give

A 3-dim. associahedron

Finally, you may want to tweak your picture my adding labels, elements on vertices, edges, facets, etc. Possibilities are unlimited with the possibilities of TikZ!

A 3-dim. associahedron

The source for the last image can be downloaded here.

Automatize using SageTex

For this you need to put

\usepackage{sagetex}

in the preamble of your article

There are different ways to use sagetex and you may create your own. Here are some possibilities.

  1. You can directly type in a sagestr in the article:
\sagestr{(polytopes.permutahedron(4)).projection().tikz([4,5,6],45,scale=0.75, facet_color='red',vertex_color='yellow',opacity=0.3)}

You should get

A 3-dim. permutahedron
  1. You may create the following tex commands
\newcommand{\polytopeimg}[4]{\sagestr{(#1).projection().tikz(#2,#3,#4)}}
\newcommand{\polytopeimgopt}[9]{\sagestr{(#1).projection().tikz(#2,#3,#4,#5,#6,#7,#8,#9)}}

in your preamble and use them with a sagesilent in your article:

\begin{sagesilent}
Polytope=polytopes.great_rhombicuboctahedron()
\end{sagesilent}
\polytopeimg{Polytope}{[276,-607,-746]}{102}{1}
\polytopeimgopt{Polytope}{view=[-907,379,183]}{angle=129}{scale=2}{edge_color='red'}{facet_color='yellow'}{vertex_color='blue'}{opacity=0.6}{axis=False}

Then, run pdflatex, execute Sage on the file article_name.sagetex.sage and run pdflatex again.

You should get

A great rhombicuboctahedron A great rhombicuboctahedron

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