Symbolic Geometric Algebra/Calculus package for SymPy.

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brombo/galgebra was originally written by Alan Bromborsky, but was no longer actively maintained, and as of 2019-11-25 no longer exists.

pygae/galgebra is a community fork, maintained by Pythonic Geometric Algebra Enthusiasts.

The fork supports Python 3, increases test coverage, sets up CI and linters, maintains releases to PyPI, improves docs and has many bug fixes, see Changelog.


Geometric Algebra

  • Arbitrary Vector Basis and Metric

  • Scalar, Vector, Bivector, Multivector, Pseudoscalar, Spinor, Blade

  • Basic Geometic Algebra Operations

    • Sum Difference

    • Geometric Product

    • Outer and Inner Products

    • Left and Right Contractions

    • Reverse, Dual, Exponential

    • Commutator

    • Projection, Reflection, Rotation

    • Reciprocal Frames

  • Inspecting Base/Blade Representation

  • Symbolic Manipulations

    • expand, factor, simplify, subs, trigsimp etc.

Overloaded Python operators for basic GA operations:

\[\begin{split}\begin{split}\begin{aligned} A+B &= \texttt{A+B} \\ A-B &= \texttt{A-B} \\ AB &= \texttt{A*B} \\ A \wedge B &= \mathtt{A \verb!^! B} \\ A \cdot B &= \texttt{A|B} \\ A \rfloor B &= \texttt{A<B} \\ A \lfloor B &= \texttt{A>B} \\ A/B &= \texttt{A/B} \\ \end{aligned}\end{split}\end{split}\]

Geometric Calculus

  • Geometric Derivative

  • Submanifolds

  • Linear Transformations

  • Differential Operators

The various derivatives of a multivector function is accomplished by multiplying the gradient operator vector with the function:

\[\begin{split}\begin{aligned} \nabla F &= \texttt{grad*F} \\ F \bar{\nabla} &= \texttt{F*rgrad} \\ \nabla {\wedge}F &= \mathtt{grad \verb!^! F} \\ F {\wedge}\bar{\nabla} &= \mathtt{F \verb!^! rgrad} \\ \nabla \cdot F &= \texttt{grad|F} \\ F \cdot \bar{\nabla} &= \texttt{F|rgrad} \\ \nabla \rfloor F &= \texttt{grad<F} \\ F \rfloor \bar{\nabla} &= \texttt{F<rgrad} \\ \nabla \lfloor F &= \texttt{grad>F} \\ F \lfloor \bar{\nabla} &= \texttt{F>rgrad} \end{aligned}\end{split}\]
\[\begin{split}\begin{aligned} F \nabla &= \texttt{F*grad} \\ \bar{\nabla} F &= \texttt{rgrad*F} \\ F {\wedge}\nabla &= \mathtt{F \verb!^! grad} \\ \bar{\nabla} {\wedge}F &= \mathtt{rgrad \verb!^! F} \\ F \cdot \nabla &= \texttt{F|grad} \\ \bar{\nabla}\cdot F &= \texttt{rgrad|F} \\ F \rfloor \nabla &= \texttt{F<grad} \\ \bar{\nabla} \rfloor F &= \texttt{rgrad<F} \\ F \lfloor \nabla &= \texttt{F>grad} \\ \bar{\nabla} \lfloor F &= \texttt{rgrad>F} \end{aligned}\end{split}\]

Tip: an example for getting grad and rgrad of a 3-d Euclidean geometric algebra in rectangular coordinates:

from sympy import symbols
from import Ga

o3d = Ga('e', g=[1,1,1], coords=symbols('x,y,z',real=True))
(grad,rgrad) = o3d.grads()


  • Enhanced Console Printing

  • Latex Printing

    • out-of-the-box support for Jupyter Notebook

    • PDF generation and croping support if you have pdflatex/pdfcrop installed

Getting Started

After installing GAlgebra (see section Installing GAlgebra below), in a Jupyter Notebook:

from sympy import symbols
from import Ga

from galgebra.printer import Format
Format(Fmode = False, Dmode = True)

st4coords = (t,x,y,z) = symbols('t x y z', real=True)
st4 = Ga('e',

M ='M','mv',f = True)

M.grade(3).Fmt(3,r'\langle \mathbf{M} \rangle _3')

You will see:

\[\begin{split}\begin{aligned} \langle \mathbf{M} \rangle _3 =& M^{txy} \boldsymbol{e}_{t}\wedge \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y} \\ & + M^{txz} \boldsymbol{e}_{t}\wedge \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{z} \\ & + M^{tyz} \boldsymbol{e}_{t}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \\ & + M^{xyz} \boldsymbol{e}_{x}\wedge \boldsymbol{e}_{y}\wedge \boldsymbol{e}_{z} \end{aligned}\end{split}\]

You may also check out more examples here.

For detailed documentation, please visit .

NOTE: If you are coming from sympy.galgebra or brombo/galgebra, please check out section Migration Guide below.

Installing GAlgebra


  • Works on Linux, Windows, Mac OSX

  • Python >= 3.8

    • 0.5.0 was the last supported release for Python 3.5-3.7

    • 0.4.x was the last supported release series for Python 2.7

  • SymPy >= 1.3

    • Only SymPy 1.12 is tested via CI, see .github/workflows/ci.yml for more details

    • 0.5.0 was the last supported release for SymPy 1.7

Migration Guide

Note: The APIs have changed since the era of sympy.galgebra and brombo/galgebra, some properties and methods are deprecated, the supported versions of Python and SymPy have also changed, please check Changelog and further update your scripts accordingly besides the following. If you encounter any problems, feel free to open an issue!

Migrating from sympy.galgebra

GAlgebra is no longer part of SymPy since 1.0.0, if you have an import like this in your source:

from import *

Simply remove the sympy. prefix before galgebra then you are good to go:

from import *

Migrating from brombo/galgebra

The way to install is now deprecated by pip install galgebra and all modules in GAlgebra should be imported from galgebra, for example:

from galgebra.printer import Format, Eprint, latex, GaPrinter
from import Ga
from import Mv, Nga

Bundled Resources

Note that in the doc/books directory there are:

  • BookGA.pdf which is a collection of notes on Geometric Algebra and Calculus based of “Geometric Algebra for Physicists” by Doran and Lasenby and on some papers by Lasenby and Hestenes.

  • galgebra.pdf which is the original main doc of GAlgebra in PDF format, while the math part is still valid, the part describing the installation and usage of GAlgebra is outdated, please read with caution or visit instead.

  • Macdonald which contains bundled supplementary materials for Linear and Geometric Algebra and Vector and Geometric Calculus by Alan Macdonald, see here and here for more information.

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