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:
providing modularized installation with pip from binary wheels, - this major project was started in May 2020 in the Sage codebase and completed in passagemath 10.5.29 (May 2025),
establishing first-class membership in the scientific Python ecosystem,
providing independently usable Python interfaces to upstream libraries,
offering platform portability and integration testing services to upstream projects,
inviting collaborations with upstream projects,
empowering Sage users to participate in the scientific Python ecosystem by publishing packages,
developing a port to WebAssembly (Pyodide, emscripten-forge) for serverless deployment with Javascript,
developing a native Windows port - passagemath 10.6.1 (July 2025) published the first pip-installable wheel packages for native Windows on x86_64, - passagemath packages became available in the [MSYS2 software distribution](https://packages.msys2.org/search?t=pkg&q=passagemath) in November 2025.
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
Combinatorial and Discrete Geometry: Polyhedra, lattice polyhedra, lattice points in polyhedra, triangulations, fans, polyhedral complexes, hyperplane arrrangements
Parma Polyhedra Library (PPL) backends for rational polyhedra, lattice polygons, lattice polytopes; via pplpy
Linear, Mixed Integer Linear, and Semidefinite Optimization frontends
see https://github.com/passagemath/passagemath/blob/main/pkgs/sagemath-polyhedra/MANIFEST.in
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]"
-
$ 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]"
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
Metadata
Release files for passagemath-polyhedra 10.8.13
For a detailed explanation of source distributions (sdists) and built distributions (wheels), please see the package formats documentation.
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Built distributions (wheels)
Total release size: 316.6 MB
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Release files / passagemath_polyhedra-10.8.13-cp313-cp313-macosx_13_0_arm64.whl
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Release files / passagemath_polyhedra-10.8.13-cp312-cp312-musllinux_1_2_x86_64.whl
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| Size | 14.6 MB |
| Tags | CPython 3.12 Linux musl 1.2+ x86-64 |
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Release files / passagemath_polyhedra-10.8.13-cp312-cp312-musllinux_1_2_aarch64.whl
| Download URL | passagemath_polyhedra-10.8.13-cp312-cp312-musllinux_1_2_aarch64.whl |
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| Size | 14.1 MB |
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Release files / passagemath_polyhedra-10.8.13-cp312-cp312-manylinux_2_24_x86_64.manylinux_2_28_x86_64.whl
| Download URL | passagemath_polyhedra-10.8.13-cp312-cp312-manylinux_2_24_x86_64.manylinux_2_28_x86_64.whl |
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Release files / passagemath_polyhedra-10.8.13-cp312-cp312-manylinux_2_24_aarch64.manylinux_2_28_aarch64.whl
| Download URL | passagemath_polyhedra-10.8.13-cp312-cp312-manylinux_2_24_aarch64.manylinux_2_28_aarch64.whl |
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| Size | 13.5 MB |
| Tags | CPython 3.12 Linux glibc 2.24+ ARM64 Linux glibc 2.28+ ARM64 |
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Release files / passagemath_polyhedra-10.8.13-cp312-cp312-macosx_13_0_x86_64.whl
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| Size | 3.3 MB |
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Release files / passagemath_polyhedra-10.8.13-cp312-cp312-macosx_13_0_arm64.whl
| Download URL | passagemath_polyhedra-10.8.13-cp312-cp312-macosx_13_0_arm64.whl |
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| Size | 3.3 MB |
| Tags | CPython 3.12 macOS 13.0+ ARM64 |
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twine/7.0.0 CPython/3.13.14
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Release files / passagemath_polyhedra-10.8.13-cp311-cp311-musllinux_1_2_x86_64.whl
| Download URL | passagemath_polyhedra-10.8.13-cp311-cp311-musllinux_1_2_x86_64.whl |
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| Size | 14.9 MB |
| Tags | CPython 3.11 Linux musl 1.2+ x86-64 |
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twine/7.0.0 CPython/3.13.14
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Release files / passagemath_polyhedra-10.8.13-cp311-cp311-musllinux_1_2_aarch64.whl
| Download URL | passagemath_polyhedra-10.8.13-cp311-cp311-musllinux_1_2_aarch64.whl |
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| Size | 14.5 MB |
| Tags | CPython 3.11 Linux musl 1.2+ ARM64 |
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twine/7.0.0 CPython/3.13.14
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Release files / passagemath_polyhedra-10.8.13-cp311-cp311-manylinux_2_24_x86_64.manylinux_2_28_x86_64.whl
| Download URL | passagemath_polyhedra-10.8.13-cp311-cp311-manylinux_2_24_x86_64.manylinux_2_28_x86_64.whl |
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| Size | 13.9 MB |
| Tags | CPython 3.11 Linux glibc 2.24+ x86-64 Linux glibc 2.28+ x86-64 |
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twine/7.0.0 CPython/3.13.14
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Release files / passagemath_polyhedra-10.8.13-cp311-cp311-manylinux_2_24_aarch64.manylinux_2_28_aarch64.whl
| Download URL | passagemath_polyhedra-10.8.13-cp311-cp311-manylinux_2_24_aarch64.manylinux_2_28_aarch64.whl |
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| Size | 13.8 MB |
| Tags | CPython 3.11 Linux glibc 2.24+ ARM64 Linux glibc 2.28+ ARM64 |
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twine/7.0.0 CPython/3.13.14
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Release files / passagemath_polyhedra-10.8.13-cp311-cp311-macosx_13_0_x86_64.whl
| Download URL | passagemath_polyhedra-10.8.13-cp311-cp311-macosx_13_0_x86_64.whl |
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| Size | 3.3 MB |
| Tags | CPython 3.11 macOS 13.0+ x86-64 |
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No |
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twine/7.0.0 CPython/3.13.14
|
Release files / passagemath_polyhedra-10.8.13-cp311-cp311-macosx_13_0_arm64.whl
| Download URL | passagemath_polyhedra-10.8.13-cp311-cp311-macosx_13_0_arm64.whl |
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| Size | 3.3 MB |
| Tags | CPython 3.11 macOS 13.0+ ARM64 |
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twine/7.0.0 CPython/3.13.14
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