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passagemath: Convex polyhedra in arbitrary dimension, mixed integer linear optimization

passagemath is open source mathematical software in Python, released under the GNU General Public Licence GPLv2+.

It is a fork of SageMath, which has been developed 2005-2026 under the motto “Creating a Viable Open Source Alternative to Magma, Maple, Mathematica, and MATLAB”.

The passagemath fork uses the motto “Creating a Free Passage Between the Scientific Python Ecosystem and Mathematical Software Communities.” It was created in October 2024 with the following goals:

Moreover, the passagemath project:

  • provides a stable, frequently updated version of the Sage distribution,

  • integrates additional mathematical software, notably Macaulay2, a full set of GAP packages, and the Combinatorial Matrix Recognition library,

  • curates a library of Sage user packages.

Full documentation is available online.

passagemath attempts to support and provides binary wheels suitable for all major Linux distributions and recent versions of macOS.

Binary wheels for native Windows (x86_64, ARM) are are available for a subset of the passagemath distributions. Use of the full functionality of passagemath on Windows currently requires the use of Windows Subsystem for Linux (WSL) or virtualization.

The supported Python versions in the passagemath-10.8.x series are 3.11.x-3.14.x; the passagemath-10.6.x series (EOL 2026-10) still supports Python 3.10.x.

About this pip-installable distribution package

This pip-installable distribution passagemath-polyhedra is a distribution of a part of the Sage Library. It provides a small subset of the modules of the Sage library (“sagelib”, passagemath-standard), sufficient for computations with convex polyhedra in arbitrary dimension (in exact rational arithmetic), and linear and mixed integer linear optimization (in floating point arithmetic).

What is included

Examples

A quick way to try it out interactively:

$ pipx run --spec "passagemath-polyhedra[test]" ipython

In [1]: from passagemath_polyhedra import *

In [2]: P = Polyhedron(ieqs=[[0, 1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 1], [0, 0, 1, -1, -1, 1, 0], [0, 0, -1, 1, -1, 1, 0]], eqns=[[-31, 1, 1, 1, 1, 1, 1]]); P
Out[2]: A 5-dimensional polyhedron in QQ^6 defined as the convex hull of 7 vertices

In [3]: P.Vrepresentation()
Out[4]:
(A vertex at (31, 0, 0, 0, 0, 0),
 A vertex at (0, 0, 0, 0, 0, 31),
 A vertex at (0, 0, 0, 0, 31, 0),
 A vertex at (0, 0, 31/2, 0, 31/2, 0),
 A vertex at (0, 31/2, 31/2, 0, 0, 0),
 A vertex at (0, 31/2, 0, 0, 31/2, 0),
 A vertex at (0, 0, 0, 31/2, 31/2, 0))

Available as extras, from other distributions

Additional features

pip install "passagemath-polyhedra[graphs]"

Face lattices, combinatorial polyhedra, graph-theoretic constructions

$ pipx run --spec "passagemath-polyhedra[graphs,test]" ipython

In [1]: from passagemath_polyhedra import *

In [2]: c5_10 = Polyhedron(vertices = [[i, i**2, i**3, i**4, i**5] for i in range(1, 11)]); c5_10
Out[2]: A 5-dimensional polyhedron in ZZ^5 defined as the convex hull of 10 vertices

In [3]: c5_10_fl = c5_10.face_lattice(); [len(x) for x in c5_10_fl.level_sets()]
Out[3]: [1, 10, 45, 100, 105, 42, 1]
pip install "passagemath-polyhedra[graphs,groups]"

Constructing symmetric polyhedra, computing automorphisms, lattice point counting modulo group actions

$ pipx run --spec "passagemath-polyhedra[graphs,groups,test]" ipython

In [1]: from passagemath_polyhedra import *

In [2]: P24 = polytopes.twenty_four_cell(); P24
Out[2]: A 4-dimensional polyhedron in QQ^4 defined as the convex hull of 24 vertices

In [3]: AutP24 = P24.restricted_automorphism_group(); AutP24.order()
Out[3]: 1152
pip install "passagemath-polyhedra[toric]"

Toric varieties

$ pipx run --spec "passagemath-polyhedra[graphs,toric,test]" ipython

In [1]: from passagemath_polyhedra import *

In [2]: TV3 = ToricVariety(NormalFan(lattice_polytope.cross_polytope(3))); TV3
Out[2]: 3-d toric variety covered by 6 affine patches

In [3]: TV3.is_orbifold()
Out[3]: False
pip install "passagemath-polyhedra[latte]"

Installs LattE integrale for lattice point counting and volume computation using generating function techniques.

$ pipx run --spec "passagemath-polyhedra[latte,test]" ipython

In [1]: from passagemath_polyhedra import *

In [2]: P = polytopes.cube()

In [3]: P.integral_points_count()
Out[3]:
27

In [4]: (1000000000*P).integral_points_count(verbose=True)
This is LattE integrale...
...
Total time:...
Out[4]:
8000000012000000006000000001

Additional backends for polyhedral computations

pip install "passagemath-polyhedra[normaliz]"

Normaliz, via PyNormaliz, provides very fast computations in particular for polyhedra with data in algebraic number fields.

$ pipx run --spec "passagemath-polyhedra[normaliz,test]" ipython

In [1]: from passagemath_polyhedra import *

In [2]: gap_norm = polytopes.grand_antiprism(backend='normaliz'); gap_norm

In [3]: gap_norm.f_vector()
pip install "passagemath-polyhedra[cddlib]"

cddlib provides support for computations with polyhedra in floating-point arithmetic.

$ pipx run --spec "passagemath-polyhedra[cddlib,test]" ipython

In [1]: from passagemath_polyhedra import *

In [2]: P1 = polytopes.regular_polygon(5, exact=False); P1
Out[2]: A 2-dimensional polyhedron in RDF^2 defined as the convex hull of 5 vertices
pip install "passagemath-polyhedra[lrslib]"

lrslib can be used for polytope volume computations and for enumerating Nash equilibria.

$ pipx run --spec "passagemath-polyhedra[flint,lrslib,test]" ipython

In [1]: from passagemath_polyhedra import *

In [2]: A = matrix([[2, 1], [1, 5/2]]); B = matrix([[-1, 3], [2, 1]])

In [3]: g = NormalFormGame([A, B]); g.obtain_nash(algorithm='lrs')
Out[3]: [[(1/5, 4/5), (3/5, 2/5)]]
pip install "passagemath-polyhedra[polymake]"

Polymake, via passagemath-polymake

Optional backends for optimization

pip install "passagemath-polyhedra[cbc]"

COIN/OR CBC Mixed Integer Linear Optimization solver, via passagemath-coin-or-cbc

pip install "passagemath-polyhedra[cplex]"

CPLEX Mixed Integer Optimization solver (proprietary; requires licensed installation), via passagemath-cplex

pip install "passagemath-polyhedra[cvxpy]"

CVXPy as middle-end for various backends

pip install "passagemath-polyhedra[gurobi]"

Gurobi Mixed Integer Optimization solver (proprietary; requires licensed installation), via passagemath-gurobi

pip install "passagemath-polyhedra[highs]"

HiGHS

pip install "passagemath-polyhedra[scip]"

SCIP Mixed Integer Optimization solver, via PySCIPOpt

Development

$ git clone --origin passagemath https://github.com/passagemath/passagemath.git
$ cd passagemath
passagemath $ ./bootstrap
passagemath $ python3 -m venv polyhedra-venv
passagemath $ source polyhedra-venv/bin/activate
(polyhedra-venv) passagemath $ pip install -v -e pkgs/sagemath-polyhedra

Release files for passagemath-polyhedra 10.8.12rc2

For a detailed explanation of source distributions (sdists) and built distributions (wheels), please see the package formats documentation.

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Table of built distributions (wheels) for passagemath-polyhedra 10.8.12rc2
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passagemath_polyhedra-10.8.12rc2-cp314-cp314t-musllinux_1_2_aarch64.whl CPython 3.14 CPython 3.14 free-threading Linux musl 1.2+ ARM64 Details
passagemath_polyhedra-10.8.12rc2-cp314-cp314t-manylinux_2_24_x86_64.manylinux_2_28_x86_64.whl CPython 3.14 CPython 3.14 free-threading Linux glibc 2.24+ x86-64, Linux glibc 2.28+ x86-64 Details
passagemath_polyhedra-10.8.12rc2-cp314-cp314t-manylinux_2_24_aarch64.manylinux_2_28_aarch64.whl CPython 3.14 CPython 3.14 free-threading Linux glibc 2.24+ ARM64, Linux glibc 2.28+ ARM64 Details
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passagemath_polyhedra-10.8.12rc2-cp314-cp314t-macosx_13_0_arm64.whl CPython 3.14 CPython 3.14 free-threading macOS 13.0+ ARM64 Details
passagemath_polyhedra-10.8.12rc2-cp314-cp314-musllinux_1_2_x86_64.whl CPython 3.14 CPython 3.14 Linux musl 1.2+ x86-64 Details
passagemath_polyhedra-10.8.12rc2-cp314-cp314-musllinux_1_2_aarch64.whl CPython 3.14 CPython 3.14 Linux musl 1.2+ ARM64 Details
passagemath_polyhedra-10.8.12rc2-cp314-cp314-manylinux_2_24_x86_64.manylinux_2_28_x86_64.whl CPython 3.14 CPython 3.14 Linux glibc 2.24+ x86-64, Linux glibc 2.28+ x86-64 Details
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passagemath_polyhedra-10.8.12rc2-cp314-cp314-macosx_13_0_x86_64.whl CPython 3.14 CPython 3.14 macOS 13.0+ x86-64 Details
passagemath_polyhedra-10.8.12rc2-cp314-cp314-macosx_13_0_arm64.whl CPython 3.14 CPython 3.14 macOS 13.0+ ARM64 Details
passagemath_polyhedra-10.8.12rc2-cp313-cp313-musllinux_1_2_x86_64.whl CPython 3.13 CPython 3.13 Linux musl 1.2+ x86-64 Details
passagemath_polyhedra-10.8.12rc2-cp313-cp313-musllinux_1_2_aarch64.whl CPython 3.13 CPython 3.13 Linux musl 1.2+ ARM64 Details
passagemath_polyhedra-10.8.12rc2-cp313-cp313-manylinux_2_24_x86_64.manylinux_2_28_x86_64.whl CPython 3.13 CPython 3.13 Linux glibc 2.28+ x86-64, Linux glibc 2.24+ x86-64 Details
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passagemath_polyhedra-10.8.12rc2-cp313-cp313-macosx_13_0_x86_64.whl CPython 3.13 CPython 3.13 macOS 13.0+ x86-64 Details
passagemath_polyhedra-10.8.12rc2-cp313-cp313-macosx_13_0_arm64.whl CPython 3.13 CPython 3.13 macOS 13.0+ ARM64 Details
passagemath_polyhedra-10.8.12rc2-cp312-cp312-musllinux_1_2_x86_64.whl CPython 3.12 CPython 3.12 Linux musl 1.2+ x86-64 Details
passagemath_polyhedra-10.8.12rc2-cp312-cp312-musllinux_1_2_aarch64.whl CPython 3.12 CPython 3.12 Linux musl 1.2+ ARM64 Details
passagemath_polyhedra-10.8.12rc2-cp312-cp312-manylinux_2_24_x86_64.manylinux_2_28_x86_64.whl CPython 3.12 CPython 3.12 Linux glibc 2.24+ x86-64, Linux glibc 2.28+ x86-64 Details
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passagemath_polyhedra-10.8.12rc2-cp312-cp312-macosx_13_0_x86_64.whl CPython 3.12 CPython 3.12 macOS 13.0+ x86-64 Details
passagemath_polyhedra-10.8.12rc2-cp312-cp312-macosx_13_0_arm64.whl CPython 3.12 CPython 3.12 macOS 13.0+ ARM64 Details
passagemath_polyhedra-10.8.12rc2-cp311-cp311-musllinux_1_2_x86_64.whl CPython 3.11 CPython 3.11 Linux musl 1.2+ x86-64 Details
passagemath_polyhedra-10.8.12rc2-cp311-cp311-musllinux_1_2_aarch64.whl CPython 3.11 CPython 3.11 Linux musl 1.2+ ARM64 Details
passagemath_polyhedra-10.8.12rc2-cp311-cp311-manylinux_2_24_x86_64.manylinux_2_28_x86_64.whl CPython 3.11 CPython 3.11 Linux glibc 2.28+ x86-64, Linux glibc 2.24+ x86-64 Details
passagemath_polyhedra-10.8.12rc2-cp311-cp311-manylinux_2_24_aarch64.manylinux_2_28_aarch64.whl CPython 3.11 CPython 3.11 Linux glibc 2.24+ ARM64, Linux glibc 2.28+ ARM64 Details
passagemath_polyhedra-10.8.12rc2-cp311-cp311-macosx_13_0_x86_64.whl CPython 3.11 CPython 3.11 macOS 13.0+ x86-64 Details
passagemath_polyhedra-10.8.12rc2-cp311-cp311-macosx_13_0_arm64.whl CPython 3.11 CPython 3.11 macOS 13.0+ ARM64 Details

Total release size: 316.5 MB

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