Draw polytopes in LaTeX using TikZ

Author: Jean-Philippe Labbé <labbe@math.fu-berlin.de>

It is sometimes very helpful to draw 3-dimensional polytopes 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. Since version 9.8 of SageMath, the tikz output can be a TikzPicture object from the sage module sage.misc.latex_standalone. This short tutorial shows how it all works.

Instructions

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

  • Install SageTex (optional 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 for example, to Sage

  • Visualize the polytope P using the command P.show(aspect_ratio=1)

  • This will open an interactive view in your default browser, where you can rotate the polytope.

  • Once the desired view angle is found, click on the information icon in the lower right-hand corner and select Get Viewpoint. This will copy a string of the form ‘[x,y,z],angle’ to your local clipboard.

  • Go back to Sage and type Img = P.tikz([x,y,z],angle,output_type='LatexExpr'). You can paste the string here to save some typing.

  • Img now contains a Sage object of type LatexExpr containing the raw TikZ picture of your polytope.

Alternatively, you can save the tikz image to a file, by doing

Img = P.tikz([x,y,z], angle, output_type='TikzPicture')
Img.tex('Img_poly.tex')
Img.tex('Img_poly.tex', content_only=True)
Img.pdf('Img_poly.pdf')

Then in the pwd (present working directory of sage, the one of your article) you will have two files named Img_poly.tex and Img_poly.pdf containing the tikzpicture of the polytope P.

Customization

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.

  • output_type : string (default: None) None, 'LatexExpr' or 'TikzPicture', the type of the output. Since SageMath 9.8, the value None is deprecated as the default value will soon be changed from 'LatexExpr' to 'TikzPicture'.

Examples

Let’s say you 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, you 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, you visualize the polytope by typing P.show(aspect_ratio=1)

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

Img = P.tikz([674,108,-731], 112, output_type='TikzPicture')

Note: the output_type='TikzPicture' is necessary since SagMath 9.8 to avoid a deprecation warning message since the default output type will soon change from a LatexExpr (Python str) to a TikzPicture object (allowing more versatility, like being able to view it directly in the Jupyter notebook).

Or you may want to customize using the command

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

Further, you may want to edit deeper the style of the polytope, directly 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}]

It is also possible to 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}]

Finally, you may want to tweak your picture my adding labels, elements on vertices, edges, facets, etc.

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)).tikz([4,5,6],45,scale=0.75, facet_color='red',vertex_color='yellow',opacity=0.3, output_type='LatexExpr')}

  1. You may create the following tex commands

\newcommand{\polytopeimg}[4]{\sagestr{(#1).tikz(#2,#3,#4,output_type='LatexExpr')}}
\newcommand{\polytopeimgopt}[9]{\sagestr{(#1).tikz(#2,#3,#4,#5,#6,#7,#8,#9,output_type='LatexExpr')}}

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.

For more information on SageTeX, see the tutorial http://doc.sagemath.org/html/en/tutorial/sagetex.html.