Jean-Philippe Labbé's webpage - Development

Sage Days 128

2025-06-30T12:00:00+02:00

During the Sage Days 128 in Le Teich, I worked on the implementation of a function that I wanted to have in Sage for a long time!

Add deformation cones and checking for regularity for Point Configurations and normal fans of Polyhedra #39496

This functionalities are available in Sage version 10.6+.

You can check whether a (full-dimensional) fan is polytopal:

sage: def mother(epsilon=0):
....:     rays = [(4-epsilon,epsilon,0),(0,4-epsilon,epsilon),(epsilon,0,4-epsilon),(2,1,1),(1,2,1),(1,1,2),(-1,-1,-1)]
....:     L = [(0,1,4),(0,3,4),(1,2,5),(1,4,5),(0,2,3),(2,3,5),(3,4,5),(6,0,1),(6,1,2),(6,2,0)]
....:     S1 = [Cone([rays[i] for i in indices]) for indices in L]
....:     return Fan(S1)

When epsilon=0, it is not polytopal:

sage: epsilon = 0
sage: mother(epsilon).is_polytopal()
False

Doing a slight perturbation makes the same subdivision polytopal:

sage: epsilon = 1/2
sage: mother(epsilon).is_polytopal()
True

You can also obtain the deformation cone of a polytope using this functionality. The deformation cone provides the cone of so-called "right-hand-sides" of the H-representation of the starting polytope which provide a deformed polytope whose normal fan coarsens that of the starting polytope. The interior direction of the deformation cone provide the deformation of the polytope that preserve the normal fan (i.e. the space of all the normally equivalent polytopes). The output format follows the order of the normals of the starting polytope and it is not quotiented by the linear transformations (since the output would not be canonical, we run into representations issues here...).

Let's examine the deformation cone of the square with one truncated vertex:

sage: tc = Polyhedron([(1, -1), (1/3, 1), (1, 1/3), (-1, 1), (-1, -1)])
sage: dc = tc.deformation_cone()
sage: dc.an_element()
(2, 1, 1, 0, 0)
sage: [_.A() for _ in tc.Hrepresentation()]
[(1, 0), (0, 1), (0, -1), (-3, -3), (-1, 0)]
sage: P = Polyhedron(rays=[(1, 0, 2), (0, 1, 1), (0, -1, 1), (-3, -3, 0), (-1, 0, 0)])
sage: P.rays()
(A ray in the direction (-1, -1, 0),
 A ray in the direction (-1, 0, 0),
 A ray in the direction (0, -1, 1),
 A ray in the direction (0, 1, 1),
 A ray in the direction (1, 0, 2))

Now, let's compute the deformation cone of the pyramid over a square and verify that it is not full dimensional:

sage: py = Polyhedron([(0, -1, -1), (0, -1, 1), (0, 1, -1), (0, 1, 1), (1, 0, 0)])
sage: dc_py = py.deformation_cone(); dc_py
A 4-dimensional polyhedron in QQ^5 defined as the convex hull of 1 vertex, 1 ray, 3 lines
sage: [ineq.b() for ineq in py.Hrepresentation()]
[0, 1, 1, 1, 1]
sage: r = dc_py.rays()[0]
sage: l1,l2,l3 = dc_py.lines()
sage: r.vector()-l1.vector()/2-l2.vector()-l3.vector()/2
(0, 1, 1, 1, 1)

This function uses the Gale transform as described in Section 5.4 of the Triangulations' book of De Loera-Rambau-Santos.

Computer Algebra System citations in MSC2020 52-Convex and discrete geometry

2022-08-10T14:40:00+02:00

Some time ago, I had the idea to measure the quantity of citations of Sagemath and other computer algebra systems in my area of research work, that is the Mathematics Subject Classification (MSC) 2020: 52-Convex and discrete geometry.

Recent history

At the Sage Days 79, that I organized in November 2016, several Sage developers manifested their interests in pushing forward the capacities of Sagemath in polyhedral geometry, working to improve the interface with other open source projects, while enlarging the community and increase the communication between users and programmers.

This led to the Sage Days 84 in March 2017, an intensive 2-weeks long retreat where a dozen of Sage developers met and set the table for several years of work involved in making the polyhedral library of Sage both user-friendly and research-efficient. This workshop also involved the generous collaboration with developers of PARI/GP and polymake that helped to bring closer the communities.

Two coding sprints then occured during the Year of Coding Sprints at the IMA in 2017/18, where the collaboration with developers of Normaliz and pynormaliz was very fruitful.

Since then, the development of Optimization and Polyhedral Geometry in Sagemath is monitored on the Wiki Page, and the Release Tours describe the new developments in polyhedral geometry as well.

Research Works in Convex and Discrete Geometry refering to CAS softwares

As all developers of open source computer algebra system or specialized packages, it is difficult to measure the impact of the work involved in integrating, deploying, developing and maintaining a software. In the mathematical field, I am under the impression (and likely not alone having this impression) that Academic institutions (whether they are Governmental Funding institutions or Educational) and their members are slow in increasing the recognition of work in the implementation of theoretical methods to provide evidence whether certain assertions be valid or not. Perhaps this lies in the fact that softwares are often shortlived? It is hard to measure the impact in this case when a piece of software was written to obtain the results in a few papers...

Launched in 2005, Sage is now close to attain its age of majority (well, at least here in Québec). One may now answer confidently to those that doubted its viability involved in its mission statement: Creating a viable free open source alternative to Magma, Maple, Mathematica and Matlab. It surely is living its life and is now running on Windows, OSX and Linux.

But is it an alternative to Magma, Maple, Mathematica and Matlab?

I am particularly interested to the answer in the case of research in Convex and Discrete Geometry. So, I have harvested the data. from MathSciNet. I looked at the citations to known CAS softwares with 52 as 1) its primary classification 2) a secondary classification and 3) either of the two.

The softwares/packages that I used to compile the data are: sage, normaliz, ppl, cdd, topcom, latte, polymake, porta, qhull, cgal, mathematica, maple, magma, and matlab.

The first graphic shows the citations to CAS with a Primary classification in 52:

The second graphic shows the citations to CAS with a Secondary classification in 52:

The third graphic shows the total citations to CAS with a classification in 52:

In the 20 first years of the century, we may observe that the citations to CAS in 52 increased five-fold, from around a dozen to more than 60.

Below is the same data, now in relative terms:

The second graphic shows the citations to CAS with a Secondary classification in 52:

The third graphic shows the total citations to CAS with a classification in 52:

One may then observe that the bottom 7 softwares (which are either standard or optional packages in Sage) available account for 64% of the total number of citations, and citations to Sage form half of them, making Sage cited one out of three times when a CAS is cited in the 52-class.

Conclusion

At the beginning of the century, one could say that Magma, Maple, Mathematica, and matlab took about 50% of the share of citations. Currently, the four together are cited in around 32% of the cases and Sagemath winds up being cited also around 32%, and other Open Source discrete geometry projects collect the rest of the acknowledgements.

My intention in producing these graphics are to measure the growing impact of Open Source CAS in research in convex and discrete geometry. Further, to foster collaboration and also knowledge about softwares (perhaps I missed a software, let me know!). Sieving through articles, I discovered certain tools that could be used in certain cases and it is definitely useful to have them reported somewhere...

This post is also to acknowledge the huge amount of work that was put in the Open Source Projects listed above by their developers community. All the scientific work reported above could not have been possible without your contributions and we definitely should not forget to support each other through our projects.

Draw polytopes using TikZ

2020-11-06T11:00:00+01:00

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. Since version 9.2 of Sage it became even easier!

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

Install Sage 9.2 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)
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). You can paste the string here to save some typing.
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.tikz([674,108,-731],112) is

Then, I 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)

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

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

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!

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.

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

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

Introduction to Sage and Polyhedral Computations

2018-07-04T17:00:00+02:00

This page contains the relevant links and files for the introduction sessions of July 4-5-6 2018 at the University of Osnabrück.

Sage Jupyter Notebooks

First steps in Sage

First steps in Sage -- Solution

Introduction to imperative programming

Introduction to imperative programming -- Solution

Python object oriented

Python object oriented -- Solution

Polytope Playground

Introduction to Polytopes

Links to relevant pages

Sage Documentation

Draw polytopes in tikz

Recent tutorial on polyhedral geometry in Sage

Sage Support Google Group

Ask Sage

Sage Development Google Group

Sage Developer's Guide -- Writing code for Sage

Sage Days on Polytopes

2017-03-26T17:00:00+02:00

During the Sage Days 84 in Olot, I collected the tickets concerning the polyhedron class into a meta-ticket:

Meta-ticket: Polyhedron: new features and known bugs

Here are some notes that I took to help me.

To build the documentation without producing the plots:

make doc-html-no-plot

This command makes a doc-clean(!).

To build only a section of the doc:

sage -b && sage -docbuild thematic_tutorials html

You can also just select a folder.